# Velocity versus "Velocity" (Dimensional Analysis branch from curvature threads)

[Martin Taylor 2016.07.20.14.28]

  Very early in my undergraduate engineering days, I was taught a


couple of basic approaches to problem solution: Be sure you are
calculating the right kind of thing, and if your carefully
computed result is very different from a back-of-the-envelope
calculation, your carefully computed result is probably the one
that’s wrong. One approach to “Calculating the right thing” is
dimensional analysis, and that is what has been missing in the
curvature discussion.

I have been wracking my brain to see why Rick calls V = Â |dXd2Y-d2XdY|Â 1/3Â *R 1/3
a “velocity” when it is clearly a distance to the 4/3 power.
I went back and looked at the Wikipedia article on curvature, and
found that there, they use an example to make it easier to
understand the curvature concept. They imagine a marker moving at
unit velocity along a track (ds/dt = 1 numerically) and then ask
about the rate of change of the tangential velocity vector as the
marker moves (infinitesimally) around the curve. Since the
tangential velocity is, by definition in the case of the
illustrative example, 1.0, the rate of change of direction of the
tangent – the angular velocity – is numerically the same. But
note the word “numerically”.

  Let's do a little dimensional analysis to see what's going on.


There are several basic “dimensions”, among which are Mass (M),
length (L) and time (T). Pure numbers have no dimension. A
distance, say “x” has dimension L, while an area has dimension L2 .
A velocity has dimension LT-1 (by convention we use
negative exponents rather than fractional signs, but either
notation is OK; you could say a velocity is L/T, but it’s easier
to keep track if you use negative exponents). So what dimension is
a length, and therefore of dimension L, so curvature has dimension
L-1 . What dimension is an angle? One way of looking at
an angle is to think of a bar of length x fixed as one end, and
see how far (y) the other end moves when you rotate the bar
through some angle t. Clearly it’s the same angle no matter how
long the bar, so angle has dimension LL-1 , which means
that it is dimensionless.

  How about derivatives? The differentials in the numerator and


denominator have the same dimensionalities as whatever they are
differentials of. So what is the dimension of the slope s = dy/dx
of a ramp? Y is of dimension L, and so is X, so a slope is
dimensionless. It’s just a number. But the rate of change of slope
as a function of x is different. It is often notated as d2x/dy2 ,
which might lead you to think it is also a pure number. But you
could also notate it as ds/dx, and since s is a pure number, its
dimension must be L-1.

  What if we are dealing with speed (velocity)? Speed of a car is


in km(or miles)/hour, and has dimension LT-1 . The same
is true when you are working with derivatives. Velocity always has
dimension LT-1 , so if you go through a complicated
calculation and come up with an expression that has a different
dimensionality, you know you have done something wrong.

  So what is wrong with Rick's "V"? We know something is wrong,


because if you work out the dimensionality of the expression |dXd2Y-d2XdY|Â 1/3Â *R1/3 ,
subtract things that have the same dimension, but not that have
different dimensions (you can’t add an area to a volume, or a
length to a mass, for example). The part within the “|” absolute
markers has dimension L3 , and it is taken to the 1/3
power, which yields something of dimension L. The other part of
Rick’s expression is R1/3 and R has dimension L, so the
entire expression has dimension L4/3 , which is not a
velocity, which would have dimension LT-1.

  How can we fix the problem? Going back to the Wikipedia article


marker is moving along the curve with unit along-track velocity,
or ds/dt =1. As so stated, it is numerically acceptable but
dimensionally wrong, because “1” is a pure number and ds/dt has
dimension LT-1 . It should be written ds/dt = 1 length
unit per time unit.Â The Wikipedia article continues:

-------quote------

Suppose that C is a twice continuously
differentiable
immersed plane curve, which here means that there exists a parametric
representation
of C by a pair of functions Î³(t) = (x(t), y(t))
such that the first and second derivatives of x and y
both exist and are continuous, and

throughout the domain. For such a plane curve, there exists a
reparametrization with respect to arc length s. This is a parametrization of C
such that

[5]
The velocity vector T(s ) is the unit tangent
vector. The unit normal vector N(s), the curvature
Îº(s), the oriented or signed curvature
k(s), and the radius of curvature R(s)
are given by

-------end quote------ So what are the dimensionalities of the parts of this analysis? The
first thing to note is that if the differentiation is with respect
to time apostrophe " ’ " represents a dimension T-1
x(s) (distance travelled per unit time) has a dimension L, so x’(s)
has a dimension LT-1 . The “1” of the first equation
therefor has dimension L2T-2 . This simply
expresses a velocity squared, which is correct because its square
root is the velocity in the Cartesian plane.

In what follows, don't mix up T(s), the tangent vector with the


dimension T (no s).

T(s) is a unit tangent vector, but what is the unit? It is a


velocity, and it’s a unit only because the speed along the curve has
been defined as one unit distance per unit time. In other words, the
speed defines the relation between L and T rather than being
defined by that relation as is usually the case, It’s as though your
speed in your car was always 1 by definition. So T(s) has dimension
LT-1, and T’(s) has dimension LT-2 . It’s an
acceleration (a lateral acceleration along the direction normal to
the curve tangent).

How then do we deal with the next line of the Wikipedia quote, which


seems to be a curvature (L-1) times a distance (L), which would make
T’(s) a pure number, which it is not. We have to go back to the
definition of k(s), as the magnitude of dT(s)/ds, which is an
acceleration, since s is described as a time parameter of the
definition of the curve at the unit speed. Curvature is hence
defined as a temporal acceleration, which is confusing since it is
also a measure of dimension L-1 . So how do we resolve
this confusion? By remembering that the idea of time and speed were
introduced only to make the ideas easier to visualise. When we are
talking about the geometry there is no time, but there is distance.

Let's say call the distance travelled per unit time at unit speed


“z”. Since in the parameterization s is taken to be time, dz/ds is
speed (linear velocity). Everything in the example could be
translated into units of z rather than s. Everything in the
derivation, including the definition of k(s), can be replaced by the
substitution of z for s, so where we have dimension T in the
foregoing, we can replace it with dimension L. The curvature k(s)
becomes k(z), with dimension LL-2, or L-1 ,
which is correct. The concept of “angular velocity”, the change in
angle of the tangent per unit time (dimension T-1 ),
becomes a change in tangent angle per unit along-curve distance
(dimension L-1 ), and the word “velocity” must be replaced
by some other word. I don’t know what the appropriate word is, so I
am coining a neologism “telocity” (meaning by derivation something
like “distanceness”). Curvature k(z) is then identified with
“angular telocity”, a purely geometric concept.

The Wikipedia example give an expression for curvature of a graph on


a plane, which bears some resemblance to what Rick calls “Velocity”.
Let’s analyze its dimensionality, remembering that the derivatives
are with respect to the arc length parameter that has dimension L.
The expression is

<img src="cid:part10.5297EBA7.437093D9@mmtaylor.net" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.838ex; width:18.779ex;
height:6.509ex;" alt="k =
\frac{x'y''-y'x''}{(x'^2+y'^2)^{3/2}}.">
In the numerator, a single prime has dimension LL<sup>-1</sup>      , in


other words it is a pure number like slope, while a double prime
has dimension LL-2, or L-1 , so the numerator
has dimension L-1 . The denominator is a power of a
dimensionless number, so it is dimensionless, The whole expression
has dimension L-1, as it should, since curvature k
is 1/R where R is radius of curvature.

    So we have a number of different expressions for curvature,


but when you do the dimensional analysis, they all have the correct
dimension L-1 , whether you define it as the inverse of
the radius of curvature, or as the angular telocity, they are all
the same thing.

But I still don't know how Rick came up with an expression for


velocity (or even for telocity) that has a dimension L4/3. Something
must have gone wrong with the algebra he called “kindergarted math”,
but I’m not willing to slog through it to find the mistake. That’s
his job. All we can be certain of is that there is a mistake
somewhere. Dimensional analysis can be very useful that way.

Martin

PS. I hope the superscripts come through OK, as Rick's did to me.


That’s what encouraged me to use them rather than the plain text L^2
to represent L2.

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Velocity versus Velocity (Dim (108 Bytes)

Velocity versus Velocity (Dim1 (109 Bytes)

···

On 2016/07/19 8:24 PM, Richard Marken
wrote:

[From Rick Marken (2016.07.19.1725)]

        On Mon, Jul 18, 2016 at 5:12 PM, Alex


Gomez-Marin agomezmarin@gmail.com
wrote:
Â

              AGM:


First, Rick, your demo hardly proves anything
because you inject ad hoc temporal dynamics in the
references whose lawful (or unlawful) properties will
simply be reflected by the control system, which does
hardly more than integrating them.Â

          RM: OK, let's start from the beginning. The goal of


this exercise is to develop a model that explains how
people (and other organisms) produce curved paths that
have the property that the velocity of movement while
producing the path (V) is proportional (by a power
function) to the curvature of the path at each instant. So
my first step was to develop a control model of someone
like an artist drawing a curved line. The model is
diagrammed below:

          RM: The phenomenon to be accounted for by the model is


the curved pattern (squiggle) produced by the artist. The
completed squiggle is shown at the bottom of the figure,
in the environment where the squiggle is actually
produced. The model has to produce this squiggle exactly
as the artist produced it, by varying the position of the
pen over time. According to PCT, the observed variations
in pen position (qi.x and qi .y)
are controlled results of variations in the reference
specifications (r.x and r.y) for the perception of those
positions (p.x and p.y). Therefore the model varies these
references in a way that would produce the observed
variations in qi.x and qi.y.Â

          RM: r.x and r.y are the references that you object to,


claiming that they contain ad hoc temporal
dynamics. I question whether they do, but I agree that it
looks trivial (and a lot like cheating) to put the desired
end result (the squiggly movements of the qi .x
and qi .y over time) into the model (in the
form of the identical squiggly movements of r.x and r.y).
But that is the way the PCT model works; controlled
(intended) results are results that match specifications
set autonomously by the organism itself.Â

          RM: But the reference signals in the control model are


not “cheating” any more than are the Â “command” signals in
motor control models of behavior, like that of Gribble/Ostry ,
which is shown below. The commands in theÂ model are
commands for output; forces that will produce movements of
the pen that will result in the observed squiggle. In the
PCT model, references, r.x and r.y, are commands for
input; perception that match the reference specifications.

          RM:Â So both models have to use internal commands in


order to produce the observed result (squiggle). The
difference is that the commands in the PCT model “look
like” the observed result; the commands in the Gribble/Ostry
don’t necessarily “look like” the result produced. But in
both cases the commands have to be carefully crafted to
produce the correct result. So the possibility of
introducing “ad hoc temporal dynamics” is
present in both models. But the PCT model can do something
that the Gribble/Ostry cannot do: it can
control. That is, it can produce the intended squiggle in
the face of disturbances.Â

          RM: Although the variations in the references (r.x,


Â r.y) in the PCT model correspond to the squiggle that is
produced (qi.x, qi .y) , I
didn’t expect this simple model to produce a squiggle with
a power function relationship between angular velocity (V)
and curvature (R). I assumed, like you, that the power law
relationship was either 1) a controlled result in itself
(the person controlling for speeding up through tighter
turns), which would require a whole extra control
organization in the model or 2) the result of complex
dynamic characteristics of muscle force production (the
functions of e.x and e.y in the control model diagram
above) and/or of the feedback function connecting force
output to pen movement input (k.f in that diagram).Â

          RM: So I was very surprised to find that the squiggle


produced by this simple control model showed a power
relationship between V and R . And when the squiggle was
an ellipse the coefficient of the power function was about
the same (~.31) as that found by Gribble/Ostry
for their ellipse production model Â – a model with a far
more dynamically complex method of generating the ellipse
than my control model. That’s when I realized that the
observed relationship between V and R might be a
mathematical property of all curved lines. And, indeed, it
turns out that it is. The relationship between V and R,
which can be found using kindergarten math, isÂ

            V


= Â |dXd2Y-d2XdY|Â 1/3Â *R1/3Â Â Â

RM: Â Note that the termÂ Â |dXd2Y-d2XdY |,
which I called D, implying that it was a constant, is a
variable. So the value of V for any curve is
proportional (exactly) to the 1/3 power ofÂ |dXd2Y-d2XdY |
and the 1/3 power of R.Â Â I
was as surprised by this as as anyone. So I wanted to
make sure it was true so I did the multiple regression
analysis using log (Â |dXd2Y-d2XdY |
) and log (R) as predictors of log (V) for many
different “squiggles” and always found that all the
variance in log (V) was accounted for by an equation of
the form:

            log


(V) = .33* log (|dXd2Y-d2 XdY|) +.33*
log(R)Â

              AGM:


Second, you are stubbornly confused about the
difference between a mathematical relation
(that allows to re-express curvature as a function of
speed, plus another non-constant term that you insist
in ignoring and treating like a constant),

          RM: I hope you see now that I do not treat the


variableÂ Â |dXd2Y-d2XdY |
Â as a constant. The mathematical relationship is as
flawless as my kindergarten math teacher;)

Â

              between


a physical realization (the fact that one can
in principle draw the same curved line at infinitely
different speeds),

          RM: Actually, I understand that the same curved line


(squiggle) can be produced at an infinity of different
speeds and by an infinity of different means (different
variations in o.x and o.y producing the same squiggle, qi .x,
qi .y).Â The same relationship between V and R
holds regardless of the speed with which the squiggle is
produced and and the means used to produce it. The
relationship between V and R is always:Â

            log


(V) = .33* log (|dXd2Y-d2 XdY|) +.33*
log(R)Â

              RM: This same


relationship between V and R even holds for all the
different squiggly patterns made by o.x and o.y to
Â produce the same squiggly pattern --qi.x, qi.y.Â

              AGM:


and between a biological fact (that out of all
possible combinations of speed and curvature, living
beings are, for yet some unknown reason —but therre are
tens if not hundreds of papers making proposals—
constrained following the power law,

          RM: But now we know the reason. It's a result of the


fact that the relationship between V and R for any curved
line is

            V


= Â |dXd2Y-d2XdY|Â 1/3Â *R1/3Â Â Â

            RM:


There is no way to draw a curved line so that this
equation does not hold. So there is no biological
constraint that creates the observed power law; it’s a
mathematical constraint.Â

              RM: The reason people


have found different power coefficients for the
relationship between V and R is because they have left
the variableÂ Â |dXd2Y-d2XdY | Â out
of the analysis. When you leaveÂ Â |dXd2Y-d2XdY | Â out
of the analysis, variations in that variable can lead
to different estimates of the power coefficient of R.
The amount by which the value of the power coefficient
of R is affected by leavingÂ Â |dXd2Y-d2XdY | Â out
of the analysis depends on the shape of the squiggle
that Â is
drawn. LeavingÂ Â |dXd2Y-d2 XdY| out of the
analysis of the V/R relationship for an ellipse
affects the actual power coefficient of R (.33) very
little, so the value obtained is around .31 (see
Gribble and Ostry, Table 1). Other squiggles can bring
the power coefficient of R down as low as .2.Â

            RM:


On that note, Martin Taylor noted that the power
coefficient for R, which is around .33 for a curved
figure drawn in the air, is closer to .25 when the same
figure is drawn Â in a viscous medium (like water). It
turns out that this can be explained in terms of a
difference in the feedback function (k.f in the diagram)
in the two cases.Â

            RM:


In the PCT model, the feedback function is a simple,
linear coefficient. When the model traces out an ellipse
with a feedback function of k.f = 1.0, the power
coefficient of R is .32; when the feedback function is
changed to k.f = .5 – equivalent to trying to move a
pen through a more resistive medium – theÂ power
coefficient of R is .26. This happens simply because the
ellipse drawn in the resistive medium is a little
sloppier than the one drawn in the air. The change in
feedback function changes the loop gain of the control
system.Â

              AGM:


But you will now reply for the n-th time saying that
everybody that has ever worked on the power-law miss es the point of control
systems and that your toy demo proves they don’t get
it.

          RM: Yes. But they have been fooled by a rather


convincing illusion. It’s hard not to see the observed
relationship between V and R as a situation where the
agent purposefully changes speed through curves (the power
law actually suggests that agents increase their speed as
the curve increases, but this increase in speed decreases
as curvature increases; it does not suggest that control
systems tend to slow down around sharp curves).

            RM: So I agree that it


is very surprising (and, perhaps, disappointing) that
the relationship between V and R tells us nothing about
how people draw curves. But that doesn’t mean that
research on how people (and other organisms) produce
curved paths should come to an end. To the contrary, it
opens up new and fruitful questions about exactly how
this is done. For example, the integral output function
that I use in the existing model is obviously an over
simplification. Something like theÂ Â Gribble/Ostry
model pictured above is probably a closed approximation.
A clever experimenter should be able to design
behavioral (and/or physiological) studies to determine
what the best model of the output function is.Â

            RM:


Going “up a level” (so to speak) research could also be
aimed at determining how what are presumably higher
level control systems set the references for the x,y
coordinates of the figure being drawn (assuming that the
figure is a controlled result and not a side effect of
controlling other variables, as in the CROWD demo).Â

            RM:


So there is really a lot of very important and
challenging research to be done in order to understand
how people draw figures. But this research must be based
on an understanding of the fact that the figure drawn is
a controlled variable – an intended result. And so any
research aimed at understanding how figures are drawn
must be based on an understanding of how control works.Â

              AGM:


But, again, ** you gloss over serious flaws
interpreting** the difference between mathematical
equations, physical conditions and biological
constraints as facts, ** and you magnify the
relevance of a toy demos** that, I wish could
shed new light, but so far don’t shed much new to the
problem.

          RM: I hope this post helps. I don't believe I have


glossed over flaws. But if there are substantive flaws
please point them out. As I said above, if my analysis is
correct that doesn’t mean all is lost. Indeed, I think it
actually opens up many new and very productive
possibilities for research.Â

Best regards

Rick

              So


I encourage you (and everyone still reading these
email exchanges) to say something new and relevant ,
because I still believe that asking what is being
perceived and what is being controlled is worth-while
in figuring out why speed and curvature are
constrained they way they are.

Â

Alex

Richard S. MarkenÂ

                                    "The childhood of the human


race is far from over. We
have a long way to go before
most people will understand that
what they do for
others is just as important to
their well-being as what they do
for
themselves." – William T. Powers

[From Rick Marken (2016.07.20.1440)]

Re Velocity versus Velocity (106 Bytes)

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···

Martin Taylor (2016.07.20.14.28]) –

MT: I have been wracking my brain to see why Rick calls V = Â |dXd2Y-d2XdY|Â 1/3Â *R 1/3
a “velocity” when it is clearly a distance to the 4/3 power.

RM: Boy, what a lot of work for nothing. The formula

V = Â |dXd2Y-d2XdY|Â 1/3Â *R1/3

is the relationship between V, as computed fromÂ

and R, as computed from

RM: The equation for V and/or R may be wrong but those are the equations used to compute these variables in studies of the “power law”.Â

RM: Now can you pull yourself together and start working with me in stead of against me. I feel like Obama trying to work with the damn Republican congress.Â

BestÂ

Rick

Â

  I went back and looked at the Wikipedia article on curvature, and


found that there, they use an example to make it easier to
understand the curvature concept. They imagine a marker moving at
unit velocity along a track (ds/dt = 1 numerically) and then ask
about the rate of change of the tangential velocity vector as the
marker moves (infinitesimally) around the curve. Since the
tangential velocity is, by definition in the case of the
illustrative example, 1.0, the rate of change of direction of the
tangent – the angular velocity – is numerically the same. But
note the word “numerically”.

  Let's do a little dimensional analysis to see what's going on.


There are several basic “dimensions”, among which are Mass (M),
length (L) and time (T). Pure numbers have no dimension. A
distance, say “x” has dimension L, while an area has dimension L2 .
A velocity has dimension LT-1 (by convention we use
negative exponents rather than fractional signs, but either
notation is OK; you could say a velocity is L/T, but it’s easier
to keep track if you use negative exponents). So what dimension is
a length, and therefore of dimension L, so curvature has dimension
L-1 . What dimension is an angle? One way of looking at
an angle is to think of a bar of length x fixed as one end, and
see how far (y) the other end moves when you rotate the bar
through some angle t. Clearly it’s the same angle no matter how
long the bar, so angle has dimension LL-1 , which means
that it is dimensionless.

  How about derivatives? The differentials in the numerator and


denominator have the same dimensionalities as whatever they are
differentials of. So what is the dimension of the slope s = dy/dx
of a ramp? Y is of dimension L, and so is X, so a slope is
dimensionless. It’s just a number. But the rate of change of slope
as a function of x is different. It is often notated as d2x/dy2 ,
which might lead you to think it is also a pure number. But you
could also notate it as ds/dx, and since s is a pure number, its
dimension must be L-1.

  What if we are dealing with speed (velocity)? Speed of a car is


in km(or miles)/hour, and has dimension LT-1 . The same
is true when you are working with derivatives. Velocity always has
dimension LT-1 , so if you go through a complicated
calculation and come up with an expression that has a different
dimensionality, you know you have done something wrong.

  So what is wrong with Rick's "V"? We know something is wrong,


because if you work out the dimensionality of the expression |dXd2Y-d2XdY|Â 1/3Â *R1/3 ,
subtract things that have the same dimension, but not that have
different dimensions (you can’t add an area to a volume, or a
length to a mass, for example). The part within the “|” absolute
markers has dimension L3 , and it is taken to the 1/3
power, which yields something of dimension L. The other part of
Rick’s expression is R1/3 and R has dimension L, so the
entire expression has dimension L4/3 , which is not a
velocity, which would have dimension LT-1.

  How can we fix the problem? Going back to the Wikipedia article


marker is moving along the curve with unit along-track velocity,
or ds/dt =1. As so stated, it is numerically acceptable but
dimensionally wrong, because “1” is a pure number and ds/dt has
dimension LT-1 . It should be written ds/dt = 1 length
unit per time unit.Â The Wikipedia article continues:

-------quote------

Suppose that C is a twice continuously
differentiable
immersed plane curve, which here means that there exists a parametric
representation
of C by a pair of functions Î³(t) = (x(t), y(t))
such that the first and second derivatives of x and y
both exist and are continuous, and

                âˆ¥

Î³
’

âˆ¥

2
=

x
’

(
t

)

2
+

y
’

(
t

)

2
â‰
0
{\displaystyle


|\gamma ‘|^{2}=x’(t)^{2}+y’(t)^{2}\not =0}

  throughout the domain. For such a plane curve, there exists a


reparametrization with respect to arc length s. This is a parametrization of C
such that

                âˆ¥

Î³
’

âˆ¥

2
=

x
’

(
s

)

2
+

y
’

(
s

)

2
=
1.
{\displaystyle


|\gamma ‘|^{2}=x’(s)^{2}+y’(s)^{2}=1.}

        <sup>[[5]](https://urldefense.proofpoint.com/v2/url?u=https-3A__en.wikipedia.org_wiki_Curvature-23cite-5Fnote-2D5&d=CwMFaQ&c=8hUWFZcy2Z-Za5rBPlktOQ&r=-dJBNItYEMOLt6aj_KjGi2LMO_Q8QB-ZzxIZIF8DGyQ&m=WAhBTXr6Rti0qHXdDP23u-EMeMMXweY-Z6pEue3R25Q&s=2mjh1qJQr74wfQvyhrgOGkYajXxcndy7lFKIZHgVsy8&e=)</sup>


The velocity vector T(s ) is the unit tangent
vector. The unit normal vector N(s), the curvature
Îº(s), the oriented or signed curvature
k(s), and the radius of curvature R(s)
are given by

                          T
(
s
)

=

Î³
’

(
s
)
,

T
’

(
s
)

=
k
(
s
)

N
(
s
)
,

Îº
(
s
)

=
âˆ¥

T
’

(
s
)
âˆ¥
=
âˆ¥

Î³
”

(
s
)
âˆ¥
=

|
k
(
s
)
|
,

R
(
s
)

=

1

Îº
(
s
)

.

{\displaystyle


{\begin{aligned}\mathbf {T} (s)&=\gamma
'(s),\\mathbf {T} '(s)&=k(s)\mathbf {N}
(s),\\kappa (s)&=|\mathbf {T} '(s)|=|\gamma
‘’(s)|=\left|k(s)\right|,\R(s)&={\frac
{1}{\kappa (s)}}.\end{aligned}}}

-------end quote------ So what are the dimensionalities of the parts of this analysis? The
first thing to note is that if the differentiation is with respect
to time apostrophe " ’ " represents a dimension T-1
x(s) (distance travelled per unit time) has a dimension L, so x’(s)
has a dimension LT-1 . The “1” of the first equation
therefor has dimension L2T-2 . This simply
expresses a velocity squared, which is correct because its square
root is the velocity in the Cartesian plane.

In what follows, don't mix up T(s), the tangent vector with the


dimension T (no s).

T(s) is a unit tangent vector, but what is the unit? It is a


velocity, and it’s a unit only because the speed along the curve has
been defined as one unit distance per unit time. In other words, the
speed defines the relation between L and T rather than being
defined by that relation as is usually the case, It’s as though your
speed in your car was always 1 by definition. So T(s) has dimension
LT-1, and T’(s) has dimension LT-2 . It’s an
acceleration (a lateral acceleration along the direction normal to
the curve tangent).

How then do we deal with the next line of the Wikipedia quote, which


seems to be a curvature (L-1) times a distance (L), which would make
T’(s) a pure number, which it is not. We have to go back to the
definition of k(s), as the magnitude of dT(s)/ds, which is an
acceleration, since s is described as a time parameter of the
definition of the curve at the unit speed. Curvature is hence
defined as a temporal acceleration, which is confusing since it is
also a measure of dimension L-1 . So how do we resolve
this confusion? By remembering that the idea of time and speed were
introduced only to make the ideas easier to visualise. When we are
talking about the geometry there is no time, but there is distance.

Let's say call the distance travelled per unit time at unit speed


“z”. Since in the parameterization s is taken to be time, dz/ds is
speed (linear velocity). Everything in the example could be
translated into units of z rather than s. Everything in the
derivation, including the definition of k(s), can be replaced by the
substitution of z for s, so where we have dimension T in the
foregoing, we can replace it with dimension L. The curvature k(s)
becomes k(z), with dimension LL-2, or L-1 ,
which is correct. The concept of “angular velocity”, the change in
angle of the tangent per unit time (dimension T-1 ),
becomes a change in tangent angle per unit along-curve distance
(dimension L-1 ), and the word “velocity” must be replaced
by some other word. I don’t know what the appropriate word is, so I
am coining a neologism “telocity” (meaning by derivation something
like “distanceness”). Curvature k(z) is then identified with
“angular telocity”, a purely geometric concept.

The Wikipedia example give an expression for curvature of a graph on


a plane, which bears some resemblance to what Rick calls “Velocity”.
Let’s analyze its dimensionality, remembering that the derivatives
are with respect to the arc length parameter that has dimension L.
The expression is

  In the numerator, a single prime has dimension LL<sup>-1</sup>      , in


other words it is a pure number like slope, while a double prime
has dimension LL-2, or L-1 , so the numerator
has dimension L-1 . The denominator is a power of a
dimensionless number, so it is dimensionless, The whole expression
has dimension L-1, as it should, since curvature k
is 1/R where R is radius of curvature.

    So we have a number of different expressions for curvature,


but when you do the dimensional analysis, they all have the correct
dimension L-1 , whether you define it as the inverse of
the radius of curvature, or as the angular telocity, they are all
the same thing.

But I still don't know how Rick came up with an expression for


velocity (or even for telocity) that has a dimension L4/3. Something
must have gone wrong with the algebra he called “kindergarted math”,
but I’m not willing to slog through it to find the mistake. That’s
his job. All we can be certain of is that there is a mistake
somewhere. Dimensional analysis can be very useful that way.

Martin

PS. I hope the superscripts come through OK, as Rick's did to me.


That’s what encouraged me to use them rather than the plain text L^2
to represent L2.

  On 2016/07/19 8:24 PM, Richard Marken


wrote:

[From Rick Marken (2016.07.19.1725)]

Richard S. MarkenÂ

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

        On Mon, Jul 18, 2016 at 5:12 PM, Alex


Gomez-Marin agomezmarin@gmail.com
wrote:
Â

              AGM:


First, Rick, your demo hardly proves anything
because you inject ad hoc temporal dynamics in the
references whose lawful (or unlawful) properties will
simply be reflected by the control system, which does
hardly more than integrating them.Â

          RM: OK, let's start from the beginning. The goal of


this exercise is to develop a model that explains how
people (and other organisms) produce curved paths that
have the property that the velocity of movement while
producing the path (V) is proportional (by a power
function) to the curvature of the path at each instant. So
my first step was to develop a control model of someone
like an artist drawing a curved line. The model is
diagrammed below:

          RM: The phenomenon to be accounted for by the model is


the curved pattern (squiggle) produced by the artist. The
completed squiggle is shown at the bottom of the figure,
in the environment where the squiggle is actually
produced. The model has to produce this squiggle exactly
as the artist produced it, by varying the position of the
pen over time. According to PCT, the observed variations
in pen position (qi.x and qi .y)
are controlled results of variations in the reference
specifications (r.x and r.y) for the perception of those
positions (p.x and p.y). Therefore the model varies these
references in a way that would produce the observed
variations in qi.x and qi.y.Â

          RM: r.x and r.y are the references that you object to,


claiming that they contain ad hoc temporal
dynamics. I question whether they do, but I agree that it
looks trivial (and a lot like cheating) to put the desired
end result (the squiggly movements of the qi .x
and qi .y over time) into the model (in the
form of the identical squiggly movements of r.x and r.y).
But that is the way the PCT model works; controlled
(intended) results are results that match specifications
set autonomously by the organism itself.Â

          RM: But the reference signals in the control model are


not “cheating” any more than are the Â “command” signals in
motor control models of behavior, like that of Gribble/Ostry ,
which is shown below. The commands in theÂ model are
commands for output; forces that will produce movements of
the pen that will result in the observed squiggle. In the
PCT model, references, r.x and r.y, are commands for
input; perception that match the reference specifications.

          RM:Â So both models have to use internal commands in


order to produce the observed result (squiggle). The
difference is that the commands in the PCT model “look
like” the observed result; the commands in the Gribble/Ostry
don’t necessarily “look like” the result produced. But in
both cases the commands have to be carefully crafted to
produce the correct result. So the possibility of
introducing “ad hoc temporal dynamics” is
present in both models. But the PCT model can do something
that the Gribble/Ostry cannot do: it can
control. That is, it can produce the intended squiggle in
the face of disturbances.Â

          RM: Although the variations in the references (r.x,


Â r.y) in the PCT model correspond to the squiggle that is
produced (qi.x, qi .y) , I
didn’t expect this simple model to produce a squiggle with
a power function relationship between angular velocity (V)
and curvature (R). I assumed, like you, that the power law
relationship was either 1) a controlled result in itself
(the person controlling for speeding up through tighter
turns), which would require a whole extra control
organization in the model or 2) the result of complex
dynamic characteristics of muscle force production (the
functions of e.x and e.y in the control model diagram
above) and/or of the feedback function connecting force
output to pen movement input (k.f in that diagram).Â

          RM: So I was very surprised to find that the squiggle


produced by this simple control model showed a power
relationship between V and R . And when the squiggle was
an ellipse the coefficient of the power function was about
the same (~.31) as that found by Gribble/Ostry
for their ellipse production model Â – a model with a far
more dynamically complex method of generating the ellipse
than my control model. That’s when I realized that the
observed relationship between V and R might be a
mathematical property of all curved lines. And, indeed, it
turns out that it is. The relationship between V and R,
which can be found using kindergarten math, isÂ

            V


= Â |dXd2Y-d2XdY|Â 1/3Â *R1/3Â Â Â

RM: Â Note that the termÂ Â |dXd2Y-d2XdY |,
which I called D, implying that it was a constant, is a
variable. So the value of V for any curve is
proportional (exactly) to the 1/3 power ofÂ |dXd2Y-d2XdY |
and the 1/3 power of R.Â Â I
was as surprised by this as as anyone. So I wanted to
make sure it was true so I did the multiple regression
analysis using log (Â |dXd2Y-d2XdY |
) and log (R) as predictors of log (V) for many
different “squiggles” and always found that all the
variance in log (V) was accounted for by an equation of
the form:

            log


(V) = .33* log (|dXd2Y-d2 XdY|) +.33*
log(R)Â

              AGM:


Second, you are stubbornly confused about the
difference between a mathematical relation
(that allows to re-express curvature as a function of
speed, plus another non-constant term that you insist
in ignoring and treating like a constant),

          RM: I hope you see now that I do not treat the


variableÂ Â |dXd2Y-d2XdY |
Â as a constant. The mathematical relationship is as
flawless as my kindergarten math teacher;)

Â

              between


a physical realization (the fact that one can
in principle draw the same curved line at infinitely
different speeds),

          RM: Actually, I understand that the same curved line


(squiggle) can be produced at an infinity of different
speeds and by an infinity of different means (different
variations in o.x and o.y producing the same squiggle, qi .x,
qi .y).Â The same relationship between V and R
holds regardless of the speed with which the squiggle is
produced and and the means used to produce it. The
relationship between V and R is always:Â

            log


(V) = .33* log (|dXd2Y-d2 XdY|) +.33*
log(R)Â

              RM: This same


relationship between V and R even holds for all the
different squiggly patterns made by o.x and o.y to
Â produce the same squiggly pattern --qi.x, qi.y.Â

              AGM:


and between a biological fact (that out of all
possible combinations of speed and curvature, living
beings are, for yet some unknown reason —but therre are
tens if not hundreds of papers making proposals—
constrained following the power law,

          RM: But now we know the reason. It's a result of the


fact that the relationship between V and R for any curved
line is

            V


= Â |dXd2Y-d2XdY|Â 1/3Â *R1/3Â Â Â

            RM:


There is no way to draw a curved line so that this
equation does not hold. So there is no biological
constraint that creates the observed power law; it’s a
mathematical constraint.Â

              RM: The reason people


have found different power coefficients for the
relationship between V and R is because they have left
the variableÂ Â |dXd2Y-d2XdY | Â out
of the analysis. When you leaveÂ Â |dXd2Y-d2XdY | Â out
of the analysis, variations in that variable can lead
to different estimates of the power coefficient of R.
The amount by which the value of the power coefficient
of R is affected by leavingÂ Â |dXd2Y-d2XdY | Â out
of the analysis depends on the shape of the squiggle
that Â is
drawn. LeavingÂ Â |dXd2Y-d2 XdY| out of the
analysis of the V/R relationship for an ellipse
affects the actual power coefficient of R (.33) very
little, so the value obtained is around .31 (see
Gribble and Ostry, Table 1). Other squiggles can bring
the power coefficient of R down as low as .2.Â

            RM:


On that note, Martin Taylor noted that the power
coefficient for R, which is around .33 for a curved
figure drawn in the air, is closer to .25 when the same
figure is drawn Â in a viscous medium (like water). It
turns out that this can be explained in terms of a
difference in the feedback function (k.f in the diagram)
in the two cases.Â

            RM:


In the PCT model, the feedback function is a simple,
linear coefficient. When the model traces out an ellipse
with a feedback function of k.f = 1.0, the power
coefficient of R is .32; when the feedback function is
changed to k.f = .5 – equivalent to trying to move a
pen through a more resistive medium – theÂ power
coefficient of R is .26. This happens simply because the
ellipse drawn in the resistive medium is a little
sloppier than the one drawn in the air. The change in
feedback function changes the loop gain of the control
system.Â

              AGM:


But you will now reply for the n-th time saying that
everybody that has ever worked on the power-law miss es the point of control
systems and that your toy demo proves they don’t get
it.

          RM: Yes. But they have been fooled by a rather


convincing illusion. It’s hard not to see the observed
relationship between V and R as a situation where the
agent purposefully changes speed through curves (the power
law actually suggests that agents increase their speed as
the curve increases, but this increase in speed decreases
as curvature increases; it does not suggest that control
systems tend to slow down around sharp curves).

            RM: So I agree that it


is very surprising (and, perhaps, disappointing) that
the relationship between V and R tells us nothing about
how people draw curves. But that doesn’t mean that
research on how people (and other organisms) produce
curved paths should come to an end. To the contrary, it
opens up new and fruitful questions about exactly how
this is done. For example, the integral output function
that I use in the existing model is obviously an over
simplification. Something like theÂ Â Gribble/Ostry
model pictured above is probably a closed approximation.
A clever experimenter should be able to design
behavioral (and/or physiological) studies to determine
what the best model of the output function is.Â

            RM:


Going “up a level” (so to speak) research could also be
aimed at determining how what are presumably higher
level control systems set the references for the x,y
coordinates of the figure being drawn (assuming that the
figure is a controlled result and not a side effect of
controlling other variables, as in the CROWD demo).Â

            RM:


So there is really a lot of very important and
challenging research to be done in order to understand
how people draw figures. But this research must be based
on an understanding of the fact that the figure drawn is
a controlled variable – an intended result. And so any
research aimed at understanding how figures are drawn
must be based on an understanding of how control works.Â

              AGM:


But, again, ** you gloss over serious flaws
interpreting** the difference between mathematical
equations, physical conditions and biological
constraints as facts, ** and you magnify the
relevance of a toy demos** that, I wish could
shed new light, but so far don’t shed much new to the
problem.

          RM: I hope this post helps. I don't believe I have


glossed over flaws. But if there are substantive flaws
please point them out. As I said above, if my analysis is
correct that doesn’t mean all is lost. Indeed, I think it
actually opens up many new and very productive
possibilities for research.Â

Best regards

Rick

              So


I encourage you (and everyone still reading these
email exchanges) to say something new and relevant ,
because I still believe that asking what is being
perceived and what is being controlled is worth-while
in figuring out why speed and curvature are
constrained they way they are.

Â

Alex

Richard S. MarkenÂ

                                    "The childhood of the human


race is far from over. We
have a long way to go before
most people will understand that
what they do for
others is just as important to
their well-being as what they do
for
themselves." – William T. Powers

      --


I wish this was the only mistake and misunderstanding we have to deal with… It is easy to solve:

Please read the definitions of curvature in any of the motor control papers I have sent you. They all got it right.Â

The problem is that when Rick discretises differential equations and also expressions to be Â able to simulate them numerically, he drops (without saying, and most likely without noticing) the “delta t” that come with speed and acceleration (see an early email where I told him that):

velocity component is vx=dx/dt but he writes dx.Â

acceleration component is ax=d2x/dt2 but he writes ax=d2x.

If you don’t drop time units (1/T for speed and 1/T^2 for acceleration), then the units of curvature (1/L) match with the rest of the terms in its definition because things cancel out.

Period.

69c2e85ced1c675e5042dee2cf53a1d3e7c81a182 (120 Bytes)

Re Velocity versus Velocity2 (107 Bytes)

63e39608b8716a40e6806d00618dc57a65a8031e2 (120 Bytes)

c537cb3d17fcfedb16bc4e0ca77ffe9755a7c9af2 (120 Bytes)

e1259d44a4df0270a15624bb4b275c754084ac5d2 (120 Bytes)

Re Velocity versus Velocity3 (107 Bytes)

···

On Wed, Jul 20, 2016 at 11:21 PM, Martin Taylor mmt-csg@mmtaylor.net wrote:

[Martin Taylor 2016.07.20.14.28]

  Very early in my undergraduate engineering days, I was taught a


couple of basic approaches to problem solution: Be sure you are
calculating the right kind of thing, and if your carefully
computed result is very different from a back-of-the-envelope
calculation, your carefully computed result is probably the one
that’s wrong. One approach to “Calculating the right thing” is
dimensional analysis, and that is what has been missing in the
curvature discussion.

I have been wracking my brain to see why Rick calls V = Â |dXd2Y-d2XdY|Â 1/3Â *R 1/3
a “velocity” when it is clearly a distance to the 4/3 power.
I went back and looked at the Wikipedia article on curvature, and
found that there, they use an example to make it easier to
understand the curvature concept. They imagine a marker moving at
unit velocity along a track (ds/dt = 1 numerically) and then ask
about the rate of change of the tangential velocity vector as the
marker moves (infinitesimally) around the curve. Since the
tangential velocity is, by definition in the case of the
illustrative example, 1.0, the rate of change of direction of the
tangent – the angular velocity – is numerically the same. But
note the word “numerically”.

  Let's do a little dimensional analysis to see what's going on.


There are several basic “dimensions”, among which are Mass (M),
length (L) and time (T). Pure numbers have no dimension. A
distance, say “x” has dimension L, while an area has dimension L2 .
A velocity has dimension LT-1 (by convention we use
negative exponents rather than fractional signs, but either
notation is OK; you could say a velocity is L/T, but it’s easier
to keep track if you use negative exponents). So what dimension is
a length, and therefore of dimension L, so curvature has dimension
L-1 . What dimension is an angle? One way of looking at
an angle is to think of a bar of length x fixed as one end, and
see how far (y) the other end moves when you rotate the bar
through some angle t. Clearly it’s the same angle no matter how
long the bar, so angle has dimension LL-1 , which means
that it is dimensionless.

  How about derivatives? The differentials in the numerator and


denominator have the same dimensionalities as whatever they are
differentials of. So what is the dimension of the slope s = dy/dx
of a ramp? Y is of dimension L, and so is X, so a slope is
dimensionless. It’s just a number. But the rate of change of slope
as a function of x is different. It is often notated as d2x/dy2 ,
which might lead you to think it is also a pure number. But you
could also notate it as ds/dx, and since s is a pure number, its
dimension must be L-1.

  What if we are dealing with speed (velocity)? Speed of a car is


in km(or miles)/hour, and has dimension LT-1 . The same
is true when you are working with derivatives. Velocity always has
dimension LT-1 , so if you go through a complicated
calculation and come up with an expression that has a different
dimensionality, you know you have done something wrong.

  So what is wrong with Rick's "V"? We know something is wrong,


because if you work out the dimensionality of the expression |dXd2Y-d2XdY|Â 1/3Â *R1/3 ,
subtract things that have the same dimension, but not that have
different dimensions (you can’t add an area to a volume, or a
length to a mass, for example). The part within the “|” absolute
markers has dimension L3 , and it is taken to the 1/3
power, which yields something of dimension L. The other part of
Rick’s expression is R1/3 and R has dimension L, so the
entire expression has dimension L4/3 , which is not a
velocity, which would have dimension LT-1.

  How can we fix the problem? Going back to the Wikipedia article


marker is moving along the curve with unit along-track velocity,
or ds/dt =1. As so stated, it is numerically acceptable but
dimensionally wrong, because “1” is a pure number and ds/dt has
dimension LT-1 . It should be written ds/dt = 1 length
unit per time unit.Â The Wikipedia article continues:

-------quote------

Suppose that C is a twice continuously
differentiable
immersed plane curve, which here means that there exists a parametric
representation
of C by a pair of functions Î³(t) = (x(t), y(t))
such that the first and second derivatives of x and y
both exist and are continuous, and

                âˆ¥

Î³
’

âˆ¥

2
=

x
’

(
t

)

2
+

y
’

(
t

)

2
â‰
0
{\displaystyle


|\gamma ‘|^{2}=x’(t)^{2}+y’(t)^{2}\not =0}

  throughout the domain. For such a plane curve, there exists a


reparametrization with respect to arc length s. This is a parametrization of C
such that

                âˆ¥

Î³
’

âˆ¥

2
=

x
’

(
s

)

2
+

y
’

(
s

)

2
=
1.
{\displaystyle


|\gamma ‘|^{2}=x’(s)^{2}+y’(s)^{2}=1.}

        <sup>[[5]](https://urldefense.proofpoint.com/v2/url?u=https-3A__en.wikipedia.org_wiki_Curvature-23cite-5Fnote-2D5&d=CwMFaQ&c=8hUWFZcy2Z-Za5rBPlktOQ&r=-dJBNItYEMOLt6aj_KjGi2LMO_Q8QB-ZzxIZIF8DGyQ&m=WAhBTXr6Rti0qHXdDP23u-EMeMMXweY-Z6pEue3R25Q&s=2mjh1qJQr74wfQvyhrgOGkYajXxcndy7lFKIZHgVsy8&e=)</sup>


The velocity vector T(s ) is the unit tangent
vector. The unit normal vector N(s), the curvature
Îº(s), the oriented or signed curvature
k(s), and the radius of curvature R(s)
are given by

                          T
(
s
)

=

Î³
’

(
s
)
,

T
’

(
s
)

=
k
(
s
)

N
(
s
)
,

Îº
(
s
)

=
âˆ¥

T
’

(
s
)
âˆ¥
=
âˆ¥

Î³
”

(
s
)
âˆ¥
=

|
k
(
s
)
|
,

R
(
s
)

=

1

Îº
(
s
)

.

{\displaystyle


{\begin{aligned}\mathbf {T} (s)&=\gamma
'(s),\\mathbf {T} '(s)&=k(s)\mathbf {N}
(s),\\kappa (s)&=|\mathbf {T} '(s)|=|\gamma
‘’(s)|=\left|k(s)\right|,\R(s)&={\frac
{1}{\kappa (s)}}.\end{aligned}}}

-------end quote------ So what are the dimensionalities of the parts of this analysis? The
first thing to note is that if the differentiation is with respect
to time apostrophe " ’ " represents a dimension T-1
x(s) (distance travelled per unit time) has a dimension L, so x’(s)
has a dimension LT-1 . The “1” of the first equation
therefor has dimension L2T-2 . This simply
expresses a velocity squared, which is correct because its square
root is the velocity in the Cartesian plane.

In what follows, don't mix up T(s), the tangent vector with the


dimension T (no s).

T(s) is a unit tangent vector, but what is the unit? It is a


velocity, and it’s a unit only because the speed along the curve has
been defined as one unit distance per unit time. In other words, the
speed defines the relation between L and T rather than being
defined by that relation as is usually the case, It’s as though your
speed in your car was always 1 by definition. So T(s) has dimension
LT-1, and T’(s) has dimension LT-2 . It’s an
acceleration (a lateral acceleration along the direction normal to
the curve tangent).

How then do we deal with the next line of the Wikipedia quote, which


seems to be a curvature (L-1) times a distance (L), which would make
T’(s) a pure number, which it is not. We have to go back to the
definition of k(s), as the magnitude of dT(s)/ds, which is an
acceleration, since s is described as a time parameter of the
definition of the curve at the unit speed. Curvature is hence
defined as a temporal acceleration, which is confusing since it is
also a measure of dimension L-1 . So how do we resolve
this confusion? By remembering that the idea of time and speed were
introduced only to make the ideas easier to visualise. When we are
talking about the geometry there is no time, but there is distance.

Let's say call the distance travelled per unit time at unit speed


“z”. Since in the parameterization s is taken to be time, dz/ds is
speed (linear velocity). Everything in the example could be
translated into units of z rather than s. Everything in the
derivation, including the definition of k(s), can be replaced by the
substitution of z for s, so where we have dimension T in the
foregoing, we can replace it with dimension L. The curvature k(s)
becomes k(z), with dimension LL-2, or L-1 ,
which is correct. The concept of “angular velocity”, the change in
angle of the tangent per unit time (dimension T-1 ),
becomes a change in tangent angle per unit along-curve distance
(dimension L-1 ), and the word “velocity” must be replaced
by some other word. I don’t know what the appropriate word is, so I
am coining a neologism “telocity” (meaning by derivation something
like “distanceness”). Curvature k(z) is then identified with
“angular telocity”, a purely geometric concept.

The Wikipedia example give an expression for curvature of a graph on


a plane, which bears some resemblance to what Rick calls “Velocity”.
Let’s analyze its dimensionality, remembering that the derivatives
are with respect to the arc length parameter that has dimension L.
The expression is

  In the numerator, a single prime has dimension LL<sup>-1</sup>      , in


other words it is a pure number like slope, while a double prime
has dimension LL-2, or L-1 , so the numerator
has dimension L-1 . The denominator is a power of a
dimensionless number, so it is dimensionless, The whole expression
has dimension L-1, as it should, since curvature k
is 1/R where R is radius of curvature.

    So we have a number of different expressions for curvature,


but when you do the dimensional analysis, they all have the correct
dimension L-1 , whether you define it as the inverse of
the radius of curvature, or as the angular telocity, they are all
the same thing.

But I still don't know how Rick came up with an expression for


velocity (or even for telocity) that has a dimension L4/3. Something
must have gone wrong with the algebra he called “kindergarted math”,
but I’m not willing to slog through it to find the mistake. That’s
his job. All we can be certain of is that there is a mistake
somewhere. Dimensional analysis can be very useful that way.

Martin

PS. I hope the superscripts come through OK, as Rick's did to me.


That’s what encouraged me to use them rather than the plain text L^2
to represent L2.

  On 2016/07/19 8:24 PM, Richard Marken


wrote:

[From Rick Marken (2016.07.19.1725)]

        On Mon, Jul 18, 2016 at 5:12 PM, Alex


Gomez-Marin agomezmarin@gmail.com
wrote:
Â

              AGM:


First, Rick, your demo hardly proves anything
because you inject ad hoc temporal dynamics in the
references whose lawful (or unlawful) properties will
simply be reflected by the control system, which does
hardly more than integrating them.Â

          RM: OK, let's start from the beginning. The goal of


this exercise is to develop a model that explains how
people (and other organisms) produce curved paths that
have the property that the velocity of movement while
producing the path (V) is proportional (by a power
function) to the curvature of the path at each instant. So
my first step was to develop a control model of someone
like an artist drawing a curved line. The model is
diagrammed below:

          RM: The phenomenon to be accounted for by the model is


the curved pattern (squiggle) produced by the artist. The
completed squiggle is shown at the bottom of the figure,
in the environment where the squiggle is actually
produced. The model has to produce this squiggle exactly
as the artist produced it, by varying the position of the
pen over time. According to PCT, the observed variations
in pen position (qi.x and qi .y)
are controlled results of variations in the reference
specifications (r.x and r.y) for the perception of those
positions (p.x and p.y). Therefore the model varies these
references in a way that would produce the observed
variations in qi.x and qi.y.Â

          RM: r.x and r.y are the references that you object to,


claiming that they contain ad hoc temporal
dynamics. I question whether they do, but I agree that it
looks trivial (and a lot like cheating) to put the desired
end result (the squiggly movements of the qi .x
and qi .y over time) into the model (in the
form of the identical squiggly movements of r.x and r.y).
But that is the way the PCT model works; controlled
(intended) results are results that match specifications
set autonomously by the organism itself.Â

          RM: But the reference signals in the control model are


not “cheating” any more than are the Â “command” signals in
motor control models of behavior, like that of Gribble/Ostry ,
which is shown below. The commands in theÂ model are
commands for output; forces that will produce movements of
the pen that will result in the observed squiggle. In the
PCT model, references, r.x and r.y, are commands for
input; perception that match the reference specifications.

          RM:Â So both models have to use internal commands in


order to produce the observed result (squiggle). The
difference is that the commands in the PCT model “look
like” the observed result; the commands in the Gribble/Ostry
don’t necessarily “look like” the result produced. But in
both cases the commands have to be carefully crafted to
produce the correct result. So the possibility of
introducing “ad hoc temporal dynamics” is
present in both models. But the PCT model can do something
that the Gribble/Ostry cannot do: it can
control. That is, it can produce the intended squiggle in
the face of disturbances.Â

          RM: Although the variations in the references (r.x,


Â r.y) in the PCT model correspond to the squiggle that is
produced (qi.x, qi .y) , I
didn’t expect this simple model to produce a squiggle with
a power function relationship between angular velocity (V)
and curvature (R). I assumed, like you, that the power law
relationship was either 1) a controlled result in itself
(the person controlling for speeding up through tighter
turns), which would require a whole extra control
organization in the model or 2) the result of complex
dynamic characteristics of muscle force production (the
functions of e.x and e.y in the control model diagram
above) and/or of the feedback function connecting force
output to pen movement input (k.f in that diagram).Â

          RM: So I was very surprised to find that the squiggle


produced by this simple control model showed a power
relationship between V and R . And when the squiggle was
an ellipse the coefficient of the power function was about
the same (~.31) as that found by Gribble/Ostry
for their ellipse production model Â – a model with a far
more dynamically complex method of generating the ellipse
than my control model. That’s when I realized that the
observed relationship between V and R might be a
mathematical property of all curved lines. And, indeed, it
turns out that it is. The relationship between V and R,
which can be found using kindergarten math, isÂ

            V


= Â |dXd2Y-d2XdY|Â 1/3Â *R1/3Â Â Â

RM: Â Note that the termÂ Â |dXd2Y-d2XdY |,
which I called D, implying that it was a constant, is a
variable. So the value of V for any curve is
proportional (exactly) to the 1/3 power ofÂ |dXd2Y-d2XdY |
and the 1/3 power of R.Â Â I
was as surprised by this as as anyone. So I wanted to
make sure it was true so I did the multiple regression
analysis using log (Â |dXd2Y-d2XdY |
) and log (R) as predictors of log (V) for many
different “squiggles” and always found that all the
variance in log (V) was accounted for by an equation of
the form:

            log


(V) = .33* log (|dXd2Y-d2 XdY|) +.33*
log(R)Â

              AGM:


Second, you are stubbornly confused about the
difference between a mathematical relation
(that allows to re-express curvature as a function of
speed, plus another non-constant term that you insist
in ignoring and treating like a constant),

          RM: I hope you see now that I do not treat the


variableÂ Â |dXd2Y-d2XdY |
Â as a constant. The mathematical relationship is as
flawless as my kindergarten math teacher;)

Â

              between


a physical realization (the fact that one can
in principle draw the same curved line at infinitely
different speeds),

          RM: Actually, I understand that the same curved line


(squiggle) can be produced at an infinity of different
speeds and by an infinity of different means (different
variations in o.x and o.y producing the same squiggle, qi .x,
qi .y).Â The same relationship between V and R
holds regardless of the speed with which the squiggle is
produced and and the means used to produce it. The
relationship between V and R is always:Â

            log


(V) = .33* log (|dXd2Y-d2 XdY|) +.33*
log(R)Â

              RM: This same


relationship between V and R even holds for all the
different squiggly patterns made by o.x and o.y to
Â produce the same squiggly pattern --qi.x, qi.y.Â

              AGM:


and between a biological fact (that out of all
possible combinations of speed and curvature, living
beings are, for yet some unknown reason —but therre are
tens if not hundreds of papers making proposals—
constrained following the power law,

          RM: But now we know the reason. It's a result of the


fact that the relationship between V and R for any curved
line is

            V


= Â |dXd2Y-d2XdY|Â 1/3Â *R1/3Â Â Â

            RM:


There is no way to draw a curved line so that this
equation does not hold. So there is no biological
constraint that creates the observed power law; it’s a
mathematical constraint.Â

              RM: The reason people


have found different power coefficients for the
relationship between V and R is because they have left
the variableÂ Â |dXd2Y-d2XdY | Â out
of the analysis. When you leaveÂ Â |dXd2Y-d2XdY | Â out
of the analysis, variations in that variable can lead
to different estimates of the power coefficient of R.
The amount by which the value of the power coefficient
of R is affected by leavingÂ Â |dXd2Y-d2XdY | Â out
of the analysis depends on the shape of the squiggle
that Â is
drawn. LeavingÂ Â |dXd2Y-d2 XdY| out of the
analysis of the V/R relationship for an ellipse
affects the actual power coefficient of R (.33) very
little, so the value obtained is around .31 (see
Gribble and Ostry, Table 1). Other squiggles can bring
the power coefficient of R down as low as .2.Â

            RM:


On that note, Martin Taylor noted that the power
coefficient for R, which is around .33 for a curved
figure drawn in the air, is closer to .25 when the same
figure is drawn Â in a viscous medium (like water). It
turns out that this can be explained in terms of a
difference in the feedback function (k.f in the diagram)
in the two cases.Â

            RM:


In the PCT model, the feedback function is a simple,
linear coefficient. When the model traces out an ellipse
with a feedback function of k.f = 1.0, the power
coefficient of R is .32; when the feedback function is
changed to k.f = .5 – equivalent to trying to move a
pen through a more resistive medium – theÂ power
coefficient of R is .26. This happens simply because the
ellipse drawn in the resistive medium is a little
sloppier than the one drawn in the air. The change in
feedback function changes the loop gain of the control
system.Â

              AGM:


But you will now reply for the n-th time saying that
everybody that has ever worked on the power-law miss es the point of control
systems and that your toy demo proves they don’t get
it.

          RM: Yes. But they have been fooled by a rather


convincing illusion. It’s hard not to see the observed
relationship between V and R as a situation where the
agent purposefully changes speed through curves (the power
law actually suggests that agents increase their speed as
the curve increases, but this increase in speed decreases
as curvature increases; it does not suggest that control
systems tend to slow down around sharp curves).

            RM: So I agree that it


is very surprising (and, perhaps, disappointing) that
the relationship between V and R tells us nothing about
how people draw curves. But that doesn’t mean that
research on how people (and other organisms) produce
curved paths should come to an end. To the contrary, it
opens up new and fruitful questions about exactly how
this is done. For example, the integral output function
that I use in the existing model is obviously an over
simplification. Something like theÂ Â Gribble/Ostry
model pictured above is probably a closed approximation.
A clever experimenter should be able to design
behavioral (and/or physiological) studies to determine
what the best model of the output function is.Â

            RM:


Going “up a level” (so to speak) research could also be
aimed at determining how what are presumably higher
level control systems set the references for the x,y
coordinates of the figure being drawn (assuming that the
figure is a controlled result and not a side effect of
controlling other variables, as in the CROWD demo).Â

            RM:


So there is really a lot of very important and
challenging research to be done in order to understand
how people draw figures. But this research must be based
on an understanding of the fact that the figure drawn is
a controlled variable – an intended result. And so any
research aimed at understanding how figures are drawn
must be based on an understanding of how control works.Â

              AGM:


But, again, ** you gloss over serious flaws
interpreting** the difference between mathematical
equations, physical conditions and biological
constraints as facts, ** and you magnify the
relevance of a toy demos** that, I wish could
shed new light, but so far don’t shed much new to the
problem.

          RM: I hope this post helps. I don't believe I have


glossed over flaws. But if there are substantive flaws
please point them out. As I said above, if my analysis is
correct that doesn’t mean all is lost. Indeed, I think it
actually opens up many new and very productive
possibilities for research.Â

Best regards

Rick

              So


I encourage you (and everyone still reading these
email exchanges) to say something new and relevant ,
because I still believe that asking what is being
perceived and what is being controlled is worth-while
in figuring out why speed and curvature are
constrained they way they are.

Â

Alex

Richard S. MarkenÂ

                                    "The childhood of the human


race is far from over. We
have a long way to go before
most people will understand that
what they do for
others is just as important to
their well-being as what they do
for
themselves." – William T. Powers

      --


[From Rick Marken (2016.07.20.1540)]

c537cb3d17fcfedb16bc4e0ca77ffe9755a7c9af3 (120 Bytes)

69c2e85ced1c675e5042dee2cf53a1d3e7c81a183 (120 Bytes)

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Re Velocity versus Velocity4 (107 Bytes)

e1259d44a4df0270a15624bb4b275c754084ac5d3 (120 Bytes)

Re Velocity versus Velocity5 (107 Bytes)

···

On Wed, Jul 20, 2016 at 2:45 PM, Alex Gomez-Marin agomezmarin@gmail.com wrote:

AGM: I wish this was the only mistake and misunderstanding we have to deal with… It is easy to solve:

AGM: The problem is that when Rick discretises differential equations and also expressions to be Â able to simulate them numerically, he drops (without saying, and most likely without noticing) the “delta t” that come with speed and acceleration (see an early email where I told him that):

RM: Actually, I fixed that problem when you told me of the error. So now my derivatives are computed as dx/dt (actually delta.x/delta.t) But since dt is a constant it made little difference in the results.Â

RM: Again, the best way to show me that I am wrong is to send me paths taken by fish or mice (mice would probably be better inasmuch as they are probably 2 D paths) that don’t follow a power law between V and R (or A and C) and, if they really don’t, then I will admit that something is wrong with my conclusions. If you could send the X,Y coordinates of the path in a spreadsheet Â that would be great!

Thanks so much.Â

BestÂ

Rick

Â

velocity component is vx=dx/dt but he writes dx.Â

acceleration component is ax=d2x/dt2 but he writes ax=d2x.

If you don’t drop time units (1/T for speed and 1/T^2 for acceleration), then the units of curvature (1/L) match with the rest of the terms in its definition because things cancel out.

Period.

On Wed, Jul 20, 2016 at 11:21 PM, Martin Taylor mmt-csg@mmtaylor.net wrote:

[Martin Taylor 2016.07.20.14.28]

  Very early in my undergraduate engineering days, I was taught a


couple of basic approaches to problem solution: Be sure you are
calculating the right kind of thing, and if your carefully
computed result is very different from a back-of-the-envelope
calculation, your carefully computed result is probably the one
that’s wrong. One approach to “Calculating the right thing” is
dimensional analysis, and that is what has been missing in the
curvature discussion.

I have been wracking my brain to see why Rick calls V = Â |dXd2Y-d2XdY|Â 1/3Â *R 1/3
a “velocity” when it is clearly a distance to the 4/3 power.
I went back and looked at the Wikipedia article on curvature, and
found that there, they use an example to make it easier to
understand the curvature concept. They imagine a marker moving at
unit velocity along a track (ds/dt = 1 numerically) and then ask
about the rate of change of the tangential velocity vector as the
marker moves (infinitesimally) around the curve. Since the
tangential velocity is, by definition in the case of the
illustrative example, 1.0, the rate of change of direction of the
tangent – the angular velocity – is numerically the same. But
note the word “numerically”.

  Let's do a little dimensional analysis to see what's going on.


There are several basic “dimensions”, among which are Mass (M),
length (L) and time (T). Pure numbers have no dimension. A
distance, say “x” has dimension L, while an area has dimension L2 .
A velocity has dimension LT-1 (by convention we use
negative exponents rather than fractional signs, but either
notation is OK; you could say a velocity is L/T, but it’s easier
to keep track if you use negative exponents). So what dimension is
a length, and therefore of dimension L, so curvature has dimension
L-1 . What dimension is an angle? One way of looking at
an angle is to think of a bar of length x fixed as one end, and
see how far (y) the other end moves when you rotate the bar
through some angle t. Clearly it’s the same angle no matter how
long the bar, so angle has dimension LL-1 , which means
that it is dimensionless.

  How about derivatives? The differentials in the numerator and


denominator have the same dimensionalities as whatever they are
differentials of. So what is the dimension of the slope s = dy/dx
of a ramp? Y is of dimension L, and so is X, so a slope is
dimensionless. It’s just a number. But the rate of change of slope
as a function of x is different. It is often notated as d2x/dy2 ,
which might lead you to think it is also a pure number. But you
could also notate it as ds/dx, and since s is a pure number, its
dimension must be L-1.

  What if we are dealing with speed (velocity)? Speed of a car is


in km(or miles)/hour, and has dimension LT-1 . The same
is true when you are working with derivatives. Velocity always has
dimension LT-1 , so if you go through a complicated
calculation and come up with an expression that has a different
dimensionality, you know you have done something wrong.

  So what is wrong with Rick's "V"? We know something is wrong,


because if you work out the dimensionality of the expression |dXd2Y-d2XdY|Â 1/3Â *R1/3 ,
subtract things that have the same dimension, but not that have
different dimensions (you can’t add an area to a volume, or a
length to a mass, for example). The part within the “|” absolute
markers has dimension L3 , and it is taken to the 1/3
power, which yields something of dimension L. The other part of
Rick’s expression is R1/3 and R has dimension L, so the
entire expression has dimension L4/3 , which is not a
velocity, which would have dimension LT-1.

  How can we fix the problem? Going back to the Wikipedia article


marker is moving along the curve with unit along-track velocity,
or ds/dt =1. As so stated, it is numerically acceptable but
dimensionally wrong, because “1” is a pure number and ds/dt has
dimension LT-1 . It should be written ds/dt = 1 length
unit per time unit.Â The Wikipedia article continues:

-------quote------

Suppose that C is a twice continuously
differentiable
immersed plane curve, which here means that there exists a parametric
representation
of C by a pair of functions Î³(t) = (x(t), y(t))
such that the first and second derivatives of x and y
both exist and are continuous, and

                âˆ¥

Î³
’

âˆ¥

2
=

x
’

(
t

)

2
+

y
’

(
t

)

2
â‰
0
{\displaystyle


|\gamma ‘|^{2}=x’(t)^{2}+y’(t)^{2}\not =0}

  throughout the domain. For such a plane curve, there exists a


reparametrization with respect to arc length s. This is a parametrization of C
such that

                âˆ¥

Î³
’

âˆ¥

2
=

x
’

(
s

)

2
+

y
’

(
s

)

2
=
1.
{\displaystyle


|\gamma ‘|^{2}=x’(s)^{2}+y’(s)^{2}=1.}

        <sup>[[5]](https://urldefense.proofpoint.com/v2/url?u=https-3A__en.wikipedia.org_wiki_Curvature-23cite-5Fnote-2D5&d=CwMFaQ&c=8hUWFZcy2Z-Za5rBPlktOQ&r=-dJBNItYEMOLt6aj_KjGi2LMO_Q8QB-ZzxIZIF8DGyQ&m=WAhBTXr6Rti0qHXdDP23u-EMeMMXweY-Z6pEue3R25Q&s=2mjh1qJQr74wfQvyhrgOGkYajXxcndy7lFKIZHgVsy8&e=)</sup>


The velocity vector T(s ) is the unit tangent
vector. The unit normal vector N(s), the curvature
Îº(s), the oriented or signed curvature
k(s), and the radius of curvature R(s)
are given by

                          T
(
s
)

=

Î³
’

(
s
)
,

T
’

(
s
)

=
k
(
s
)

N
(
s
)
,

Îº
(
s
)

=
âˆ¥

T
’

(
s
)
âˆ¥
=
âˆ¥

Î³
”

(
s
)
âˆ¥
=

|
k
(
s
)
|
,

R
(
s
)

=

1

Îº
(
s
)

.

{\displaystyle


{\begin{aligned}\mathbf {T} (s)&=\gamma
'(s),\\mathbf {T} '(s)&=k(s)\mathbf {N}
(s),\\kappa (s)&=|\mathbf {T} '(s)|=|\gamma
‘’(s)|=\left|k(s)\right|,\R(s)&={\frac
{1}{\kappa (s)}}.\end{aligned}}}

-------end quote------ So what are the dimensionalities of the parts of this analysis? The
first thing to note is that if the differentiation is with respect
to time apostrophe " ’ " represents a dimension T-1
x(s) (distance travelled per unit time) has a dimension L, so x’(s)
has a dimension LT-1 . The “1” of the first equation
therefor has dimension L2T-2 . This simply
expresses a velocity squared, which is correct because its square
root is the velocity in the Cartesian plane.

In what follows, don't mix up T(s), the tangent vector with the


dimension T (no s).

T(s) is a unit tangent vector, but what is the unit? It is a


velocity, and it’s a unit only because the speed along the curve has
been defined as one unit distance per unit time. In other words, the
speed defines the relation between L and T rather than being
defined by that relation as is usually the case, It’s as though your
speed in your car was always 1 by definition. So T(s) has dimension
LT-1, and T’(s) has dimension LT-2 . It’s an
acceleration (a lateral acceleration along the direction normal to
the curve tangent).

How then do we deal with the next line of the Wikipedia quote, which


seems to be a curvature (L-1) times a distance (L), which would make
T’(s) a pure number, which it is not. We have to go back to the
definition of k(s), as the magnitude of dT(s)/ds, which is an
acceleration, since s is described as a time parameter of the
definition of the curve at the unit speed. Curvature is hence
defined as a temporal acceleration, which is confusing since it is
also a measure of dimension L-1 . So how do we resolve
this confusion? By remembering that the idea of time and speed were
introduced only to make the ideas easier to visualise. When we are
talking about the geometry there is no time, but there is distance.

Let's say call the distance travelled per unit time at unit speed


“z”. Since in the parameterization s is taken to be time, dz/ds is
speed (linear velocity). Everything in the example could be
translated into units of z rather than s. Everything in the
derivation, including the definition of k(s), can be replaced by the
substitution of z for s, so where we have dimension T in the
foregoing, we can replace it with dimension L. The curvature k(s)
becomes k(z), with dimension LL-2, or L-1 ,
which is correct. The concept of “angular velocity”, the change in
angle of the tangent per unit time (dimension T-1 ),
becomes a change in tangent angle per unit along-curve distance
(dimension L-1 ), and the word “velocity” must be replaced
by some other word. I don’t know what the appropriate word is, so I
am coining a neologism “telocity” (meaning by derivation something
like “distanceness”). Curvature k(z) is then identified with
“angular telocity”, a purely geometric concept.

The Wikipedia example give an expression for curvature of a graph on


a plane, which bears some resemblance to what Rick calls “Velocity”.
Let’s analyze its dimensionality, remembering that the derivatives
are with respect to the arc length parameter that has dimension L.
The expression is

  In the numerator, a single prime has dimension LL<sup>-1</sup>      , in


other words it is a pure number like slope, while a double prime
has dimension LL-2, or L-1 , so the numerator
has dimension L-1 . The denominator is a power of a
dimensionless number, so it is dimensionless, The whole expression
has dimension L-1, as it should, since curvature k
is 1/R where R is radius of curvature.

    So we have a number of different expressions for curvature,


but when you do the dimensional analysis, they all have the correct
dimension L-1 , whether you define it as the inverse of
the radius of curvature, or as the angular telocity, they are all
the same thing.

But I still don't know how Rick came up with an expression for


velocity (or even for telocity) that has a dimension L4/3. Something
must have gone wrong with the algebra he called “kindergarted math”,
but I’m not willing to slog through it to find the mistake. That’s
his job. All we can be certain of is that there is a mistake
somewhere. Dimensional analysis can be very useful that way.

Martin

PS. I hope the superscripts come through OK, as Rick's did to me.


That’s what encouraged me to use them rather than the plain text L^2
to represent L2.

  On 2016/07/19 8:24 PM, Richard Marken


wrote:

[From Rick Marken (2016.07.19.1725)]

        On Mon, Jul 18, 2016 at 5:12 PM, Alex


Gomez-Marin agomezmarin@gmail.com
wrote:
Â

              AGM:


First, Rick, your demo hardly proves anything
because you inject ad hoc temporal dynamics in the
references whose lawful (or unlawful) properties will
simply be reflected by the control system, which does
hardly more than integrating them.Â

          RM: OK, let's start from the beginning. The goal of


this exercise is to develop a model that explains how
people (and other organisms) produce curved paths that
have the property that the velocity of movement while
producing the path (V) is proportional (by a power
function) to the curvature of the path at each instant. So
my first step was to develop a control model of someone
like an artist drawing a curved line. The model is
diagrammed below:

          RM: The phenomenon to be accounted for by the model is


the curved pattern (squiggle) produced by the artist. The
completed squiggle is shown at the bottom of the figure,
in the environment where the squiggle is actually
produced. The model has to produce this squiggle exactly
as the artist produced it, by varying the position of the
pen over time. According to PCT, the observed variations
in pen position (qi.x and qi .y)
are controlled results of variations in the reference
specifications (r.x and r.y) for the perception of those
positions (p.x and p.y). Therefore the model varies these
references in a way that would produce the observed
variations in qi.x and qi.y.Â

          RM: r.x and r.y are the references that you object to,


claiming that they contain ad hoc temporal
dynamics. I question whether they do, but I agree that it
looks trivial (and a lot like cheating) to put the desired
end result (the squiggly movements of the qi .x
and qi .y over time) into the model (in the
form of the identical squiggly movements of r.x and r.y).
But that is the way the PCT model works; controlled
(intended) results are results that match specifications
set autonomously by the organism itself.Â

          RM: But the reference signals in the control model are


not “cheating” any more than are the Â “command” signals in
motor control models of behavior, like that of Gribble/Ostry ,
which is shown below. The commands in theÂ model are
commands for output; forces that will produce movements of
the pen that will result in the observed squiggle. In the
PCT model, references, r.x and r.y, are commands for
input; perception that match the reference specifications.

          RM:Â So both models have to use internal commands in


order to produce the observed result (squiggle). The
difference is that the commands in the PCT model “look
like” the observed result; the commands in the Gribble/Ostry
don’t necessarily “look like” the result produced. But in
both cases the commands have to be carefully crafted to
produce the correct result. So the possibility of
introducing “ad hoc temporal dynamics” is
present in both models. But the PCT model can do something
that the Gribble/Ostry cannot do: it can
control. That is, it can produce the intended squiggle in
the face of disturbances.Â

          RM: Although the variations in the references (r.x,


Â r.y) in the PCT model correspond to the squiggle that is
produced (qi.x, qi .y) , I
didn’t expect this simple model to produce a squiggle with
a power function relationship between angular velocity (V)
and curvature (R). I assumed, like you, that the power law
relationship was either 1) a controlled result in itself
(the person controlling for speeding up through tighter
turns), which would require a whole extra control
organization in the model or 2) the result of complex
dynamic characteristics of muscle force production (the
functions of e.x and e.y in the control model diagram
above) and/or of the feedback function connecting force
output to pen movement input (k.f in that diagram).Â

          RM: So I was very surprised to find that the squiggle


produced by this simple control model showed a power
relationship between V and R . And when the squiggle was
an ellipse the coefficient of the power function was about
the same (~.31) as that found by Gribble/Ostry
for their ellipse production model Â – a model with a far
more dynamically complex method of generating the ellipse
than my control model. That’s when I realized that the
observed relationship between V and R might be a
mathematical property of all curved lines. And, indeed, it
turns out that it is. The relationship between V and R,
which can be found using kindergarten math, isÂ

            V


= Â |dXd2Y-d2XdY|Â 1/3Â *R1/3Â Â Â

RM: Â Note that the termÂ Â |dXd2Y-d2XdY |,
which I called D, implying that it was a constant, is a
variable. So the value of V for any curve is
proportional (exactly) to the 1/3 power ofÂ |dXd2Y-d2XdY |
and the 1/3 power of R.Â Â I
was as surprised by this as as anyone. So I wanted to
make sure it was true so I did the multiple regression
analysis using log (Â |dXd2Y-d2XdY |
) and log (R) as predictors of log (V) for many
different “squiggles” and always found that all the
variance in log (V) was accounted for by an equation of
the form:

            log


(V) = .33* log (|dXd2Y-d2 XdY|) +.33*
log(R)Â

              AGM:


Second, you are stubbornly confused about the
difference between a mathematical relation
(that allows to re-express curvature as a function of
speed, plus another non-constant term that you insist
in ignoring and treating like a constant),

          RM: I hope you see now that I do not treat the


variableÂ Â |dXd2Y-d2XdY |
Â as a constant. The mathematical relationship is as
flawless as my kindergarten math teacher;)

Â

              between


a physical realization (the fact that one can
in principle draw the same curved line at infinitely
different speeds),

          RM: Actually, I understand that the same curved line


(squiggle) can be produced at an infinity of different
speeds and by an infinity of different means (different
variations in o.x and o.y producing the same squiggle, qi .x,
qi .y).Â The same relationship between V and R
holds regardless of the speed with which the squiggle is
produced and and the means used to produce it. The
relationship between V and R is always:Â

            log


(V) = .33* log (|dXd2Y-d2 XdY|) +.33*
log(R)Â

              RM: This same


relationship between V and R even holds for all the
different squiggly patterns made by o.x and o.y to
Â produce the same squiggly pattern --qi.x, qi.y.Â

              AGM:


and between a biological fact (that out of all
possible combinations of speed and curvature, living
beings are, for yet some unknown reason —but therre are
tens if not hundreds of papers making proposals—
constrained following the power law,

          RM: But now we know the reason. It's a result of the


fact that the relationship between V and R for any curved
line is

            V


= Â |dXd2Y-d2XdY|Â 1/3Â *R1/3Â Â Â

            RM:


There is no way to draw a curved line so that this
equation does not hold. So there is no biological
constraint that creates the observed power law; it’s a
mathematical constraint.Â

              RM: The reason people


have found different power coefficients for the
relationship between V and R is because they have left
the variableÂ Â |dXd2Y-d2XdY | Â out
of the analysis. When you leaveÂ Â |dXd2Y-d2XdY | Â out
of the analysis, variations in that variable can lead
to different estimates of the power coefficient of R.
The amount by which the value of the power coefficient
of R is affected by leavingÂ Â |dXd2Y-d2XdY | Â out
of the analysis depends on the shape of the squiggle
that Â is
drawn. LeavingÂ Â |dXd2Y-d2 XdY| out of the
analysis of the V/R relationship for an ellipse
affects the actual power coefficient of R (.33) very
little, so the value obtained is around .31 (see
Gribble and Ostry, Table 1). Other squiggles can bring
the power coefficient of R down as low as .2.Â

            RM:


On that note, Martin Taylor noted that the power
coefficient for R, which is around .33 for a curved
figure drawn in the air, is closer to .25 when the same
figure is drawn Â in a viscous medium (like water). It
turns out that this can be explained in terms of a
difference in the feedback function (k.f in the diagram)
in the two cases.Â

            RM:


In the PCT model, the feedback function is a simple,
linear coefficient. When the model traces out an ellipse
with a feedback function of k.f = 1.0, the power
coefficient of R is .32; when the feedback function is
changed to k.f = .5 – equivalent to trying to move a
pen through a more resistive medium – theÂ power
coefficient of R is .26. This happens simply because the
ellipse drawn in the resistive medium is a little
sloppier than the one drawn in the air. The change in
feedback function changes the loop gain of the control
system.Â

              AGM:


But you will now reply for the n-th time saying that
everybody that has ever worked on the power-law miss es the point of control
systems and that your toy demo proves they don’t get
it.

          RM: Yes. But they have been fooled by a rather


convincing illusion. It’s hard not to see the observed
relationship between V and R as a situation where the
agent purposefully changes speed through curves (the power
law actually suggests that agents increase their speed as
the curve increases, but this increase in speed decreases
as curvature increases; it does not suggest that control
systems tend to slow down around sharp curves).

            RM: So I agree that it


is very surprising (and, perhaps, disappointing) that
the relationship between V and R tells us nothing about
how people draw curves. But that doesn’t mean that
research on how people (and other organisms) produce
curved paths should come to an end. To the contrary, it
opens up new and fruitful questions about exactly how
this is done. For example, the integral output function
that I use in the existing model is obviously an over
simplification. Something like theÂ Â Gribble/Ostry
model pictured above is probably a closed approximation.
A clever experimenter should be able to design
behavioral (and/or physiological) studies to determine
what the best model of the output function is.Â

            RM:


Going “up a level” (so to speak) research could also be
aimed at determining how what are presumably higher
level control systems set the references for the x,y
coordinates of the figure being drawn (assuming that the
figure is a controlled result and not a side effect of
controlling other variables, as in the CROWD demo).Â

            RM:


So there is really a lot of very important and
challenging research to be done in order to understand
how people draw figures. But this research must be based
on an understanding of the fact that the figure drawn is
a controlled variable – an intended result. And so any
research aimed at understanding how figures are drawn
must be based on an understanding of how control works.Â

              AGM:


But, again, ** you gloss over serious flaws
interpreting** the difference between mathematical
equations, physical conditions and biological
constraints as facts, ** and you magnify the
relevance of a toy demos** that, I wish could
shed new light, but so far don’t shed much new to the
problem.

          RM: I hope this post helps. I don't believe I have


glossed over flaws. But if there are substantive flaws
please point them out. As I said above, if my analysis is
correct that doesn’t mean all is lost. Indeed, I think it
actually opens up many new and very productive
possibilities for research.Â

Best regards

Rick

              So


I encourage you (and everyone still reading these
email exchanges) to say something new and relevant ,
because I still believe that asking what is being
perceived and what is being controlled is worth-while
in figuring out why speed and curvature are
constrained they way they are.

Â

Alex

Richard S. MarkenÂ

                                    "The childhood of the human


race is far from over. We
have a long way to go before
most people will understand that
what they do for
others is just as important to
their well-being as what they do
for
themselves." – William T. Powers

      --


Richard S. MarkenÂ

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[Martin Taylor 16.07.20.21.19]

[From Rick Marken (2016.07.20.1440)]

Clearly you are following your usual practice of NOT reading what


you reply to. Here’s the nub of the problem. If “V” is a true
velocity, then the dots signify derivatives with respect to time. If
“R” is the radius of curvature, it’s a length and the dots signify
derivatives with respect to arc length along the curve. They aren’t
the same thing, but you use your “kindergarten math” as though they
are. No wonder you get weird results that have no internal
consistency.

The difference is that you keep wasting space jundering along


telling us over and over again that the description of a curve is
sufficient to tell us how fast someone or something will trace that
curve. It’s nonsense. I may not be able to trace the curve
accurately at unlimited speed, but I sure can go as slow and as
irregularly as I want, regardless of its shape. Obama had clearly
defined policies that would have begun to “make America Great Again”
after the Bush disaster. You just have a policy of repeating “shape
is speed”, as the Republicans say “More guns mean safer streets”.
Actually, they make more sense, because there probably is a relation
between their two concepts, even if the actual relation is the
opposite of what they say. Your mantra doesn’t have even that cover
of respectability.

I'm leaving the rest of my message in this one so you won't have to


look back a few hours in the archive if you ever decide you want to
read the explanation of why I am quite certain your V = Â |dXd2Y-d2XdY|Â 1/3Â *R1/3
formula not just wrong, but just so nonsensical as to be neither
right nor wrong.

  Martin

<sup></sup>


Re Velocity versus Velocity6 (107 Bytes)

Re Velocity versus Velocity7 (107 Bytes)

Re Velocity versus Velocity8 (107 Bytes)

Re Velocity versus Velocity9 (107 Bytes)

Re Velocity versus Velocity10 (108 Bytes)

Re Velocity versus Velocity11 (108 Bytes)

Re Velocity versus Velocity12 (108 Bytes)

Re Velocity versus Velocity13 (108 Bytes)

···

Martin Taylor (2016.07.20.14.28]) –

              MT: I have been wracking my brain to see why Rick


calls V = Â |dXd2Y-d2XdY|Â 1/3Â *R 1/3
a “velocity” when it is clearly a distance to
the 4/3 power.

RM: Boy, what a lot of work for nothing. The formula

V = Â |dXd2Y-d2XdY|Â 1/3Â *R1/3

            </sup>


is the relationship between V, as computed fromÂ

and R, as computed from

          RM: The equation for V and/or R may be wrong but those


are the equations used to compute these variables in
studies of the “power law”.Â

          RM: Now can you pull yourself together and start


working with me in stead of against me. I feel like Obama
trying to work with the damn Republican congress.Â

Â

              I went back and looked at the Wikipedia article on


curvature, and found that there, they use an example
to make it easier to understand the curvature concept.
They imagine a marker moving at unit velocity along a
rate of change of the tangential velocity vector as
the marker moves (infinitesimally) around the curve.
Since the tangential velocity is, by definition in the
case of the illustrative example, 1.0, the rate of
change of direction of the tangent – the angular
velocity – is numerically the same. But note the word
“numerically”.

              Let's do a little dimensional analysis to see what's


going on. There are several basic “dimensions”, among
which are Mass (M), length (L) and time (T). Pure
numbers have no dimension. A distance, say “x” has
dimension L, while an area has dimension L2 .
A velocity has dimension LT-1 (by
convention we use negative exponents rather than
fractional signs, but either notation is OK; you could
say a velocity is L/T, but it’s easier to keep track
if you use negative exponents). So what dimension is a
“curvature”. Curvature is 1/radius-of-curvature, and a
radius is a length, and therefore of dimension L, so
curvature has dimension L-1 . What dimension
is an angle? One way of looking at an angle is to
think of a bar of length x fixed as one end, and see
how far (y) the other end moves when you rotate the
bar through some angle t. Clearly it’s the same angle
no matter how long the bar, so angle has dimension LL-1 ,
which means that it is dimensionless.

              How about derivatives? The differentials in the


numerator and denominator have the same
dimensionalities as whatever they are differentials
of. So what is the dimension of the slope s = dy/dx of
a ramp? Y is of dimension L, and so is X, so a slope
is dimensionless. It’s just a number. But the rate of
change of slope as a function of x is different. It is
often notated as d2x/dy2 , which
might lead you to think it is also a pure number. But
you could also notate it as ds/dx, and since s is a
pure number, its dimension must be L-1.

              What if we are dealing with speed (velocity)? Speed


of a car is in km(or miles)/hour, and has dimension LT-1 .
The same is true when you are working with
derivatives. Velocity always has dimension LT-1 ,
so if you go through a complicated calculation and
come up with an expression that has a different
dimensionality, you know you have done something
wrong.

              So what is wrong with Rick's "V"? We know something


is wrong, because if you work out the dimensionality
of the expression |dXd2Y-d2XdY|Â 1/3Â *R1/3 ,
can add and subtract things that have the same
dimension, but not that have different dimensions (you
can’t add an area to a volume, or a length to a mass,
for example). The part within the “|” absolute markers
has dimension L3 , and it is taken to the
1/3 power, which yields something of dimension L. The
other part of Rick’s expression is R1/3 and
R has dimension L, so the entire expression has
dimension L4/3 , which is not a velocity,
which would have dimension LT-1.

              How can we fix the problem? Going back to the


the assertion that the virtual marker is moving along
the curve with unit along-track velocity, or ds/dt =1.
As so stated, it is numerically acceptable but
dimensionally wrong, because “1” is a pure number and
ds/dt has dimension LT-1 . It should be
written ds/dt = 1 length unit per time unit.Â The
Wikipedia article continues:

-------quote------

Suppose that C is a twice continuously
differentiable
immersed
plane curve ,
which here means that there exists a parametric
representation
of C by a pair of
functions Î³(t) = (x(t), y(t))
such that the first and second derivatives of x
and y both exist and are continuous, and

âˆ¥ Î³ ’ âˆ¥ 2 = x ’ ( t ) 2 + y ’ ( t ) 2
2
â‰ 0 {\displaystyle |\gamma
‘|^{2}=x’(t)^{2}+y’(t)^{2}\not =0}
throughout the domain. For such a plane curve, there
exists a reparametrization with respect to arc length s .
This is a parametrization of C such that

âˆ¥ Î³ ’ âˆ¥ 2 = x ’ ( s ) 2 + y ’ ( s ) 2
2
= 1. {\displaystyle |\gamma
‘|^{2}=x’(s)^{2}+y’(s)^{2}=1.} [5]
The velocity vector T(s ) is the unit
tangent vector. The unit normal vector N(s ),
the curvature Îº(s), the oriented
or signed curvature k(s ), and
the radius of curvature R(s) are given
by

T ( s ) = Î³ ’ ( s ) , T ’ ( s ) = k (
s ) N ( s ) , Îº ( s ) = âˆ¥ T â€² ( s ) âˆ¥ = âˆ¥ Î³ ” (
s ) âˆ¥ = | k ( s ) | , R ( s ) = 1 Îº ( s ) .
{\displaystyle {\begin{aligned}\mathbf {T}
(s)&=\gamma '(s),\\mathbf {T}
'(s)&=k(s)\mathbf {N} (s),\\kappa
(s)&=|\mathbf {T} ‘(s)|=|\gamma
‘’(s)|=\left|k(s)\right|,\R(s)&={\frac
{1}{\kappa (s)}}.\end{aligned}}}
-------end quote------ So what are the dimensionalities of the parts of this
analysis? The first thing to note is that if the
differentiation is with respect to time apostrophe " ’ "
represents a dimension T-1 .Â x(s) (distance
travelled per unit time) has a dimension L, so x’(s) has
a dimension LT-1 . The “1” of the first
equation therefor has dimension L2T-2 .
This simply expresses a velocity squared, which is
correct because its square root is the velocity in the
Cartesian plane.

            In what follows, don't mix up T(s), the tangent vector


with the dimension T (no s).

            T(s) is a unit tangent vector, but what is the unit? It


is a velocity, and it’s a unit only because the speed
along the curve has been defined as one unit distance
per unit time. In other words, the speed defines the
relation between L and T rather than being defined by
that relation as is usually the case, It’s as though
T(s) has dimension LT-1 , and T’(s) has
dimension LT-2 . It’s an acceleration (a
lateral acceleration along the direction normal to the
curve tangent).

            How then do we deal with the next line of the Wikipedia


quote, which seems to be a curvature (L-1) times a
distance (L), which would make T’(s) a pure number,
which it is not. We have to go back to the definition of
k(s), as the magnitude of dT(s)/ds, which is an
acceleration, since s is described as a time parameter
of the definition of the curve at the unit speed.
Curvature is hence defined as a temporal acceleration,
which is confusing since it is also a measure of
dimension L-1 . So how do we resolve this
confusion? By remembering that the idea of time and
speed were introduced only to make the ideas easier to
visualise. When we are talking about the geometry there
is no time, but there is distance.

            Let's say call the distance travelled per unit time at


unit speed “z”. Since in the parameterization s is taken
to be time, dz/ds is speed (linear velocity). Everything
in the example could be translated into units of z
rather than s. Everything in the derivation, including
the definition of k(s), can be replaced by the
substitution of z for s, so where we have dimension T in
the foregoing, we can replace it with dimension L. The
curvature k(s) becomes k(z), with dimension LL-2 ,
or L-1 , which is correct. The concept of
“angular velocity”, the change in angle of the tangent
per unit time (dimension T-1 ), becomes a
change in tangent angle per unit along-curve distance
(dimension L-1 ), and the word “velocity” must
be replaced by some other word. I don’t know what the
appropriate word is, so I am coining a neologism
“telocity” (meaning by derivation something like
“distanceness”). Curvature k(z) is then identified with
“angular telocity”, a purely geometric concept.

            The Wikipedia example give an expression for curvature


of a graph on a plane, which bears some resemblance to
what Rick calls “Velocity”. Let’s analyze its
dimensionality, remembering that the derivatives are
with respect to the arc length parameter that has
dimension L. The expression is

              In the numerator, a single prime has dimension LL<sup>-1</sup>                  ,


in other words it is a pure number like slope, while a
double prime has dimension LL-2, or L-1 ,
so the numerator has dimension L-1 . The
denominator is a power of a dimensionless number, so
it is dimensionless, The whole expression has
dimension L-1 , as it should, since
curvature k is 1/R where R is radius of
curvature.

                            So we have a number of different expressions for


curvature, but when you do the dimensional analysis,
they all have the correct dimension L-1 ,
whether you define it as the inverse of the radius of
curvature, or as the angular telocity, they are all the
same thing.

            But I still don't know how Rick came up with an


expression for velocity (or even for telocity) that has
a dimension L4/3. Something must have gone wrong with
the algebra he called “kindergarted math”, but I’m not
willing to slog through it to find the mistake. That’s
his job. All we can be certain of is that there is a
mistake somewhere. Dimensional analysis can be very
useful that way.

            Martin

PS. I hope the superscripts come through OK, as Rick's


did to me. That’s what encouraged me to use them rather
than the plain text L^2 to represent L2.

On 2016/07/19 8:24 PM, Richard Marken wrote:

[From Rick Marken
(2016.07.19.1725)]

Richard S. MarkenÂ

                                    "The childhood of the human


race is far from over. We
have a long way to go before
most people will understand that
what they do for
others is just as important to
their well-being as what they do
for
themselves." – William T.
Powers

                    On Mon, Jul 18, 2016 at


5:12 PM, Alex Gomez-Marin agomezmarin@gmail.com
wrote:
Â

                          AGM:


First, Rick, ** your demo hardly proves
anything** because you inject ad hoc
temporal dynamics in the references whose
lawful (or unlawful) properties will
simply be reflected by the control system,
which does hardly more than integrating
them.Â

                      RM: OK, let's start from the beginning. The


goal of this exercise is to develop a model
that explains how people (and other organisms)
produce curved paths that have the property
that the velocity of movement while producing
the path (V) is proportional (by a power
function) to the curvature of the path at each
instant. So my first step was to develop a
control model of someone like an artist
drawing a curved line. The model is diagrammed
below:

                      RM: The phenomenon to be accounted for by


the model is the curved pattern (squiggle)
produced by the artist. The completed squiggle
is shown at the bottom of the figure, in the
environment where the squiggle is actually
produced. The model has to produce this
squiggle exactly as the artist produced it, by
varying the position of the pen over time.
According to PCT, the observed variations in
pen position (qi.x and qi .y)
are controlled results of variations in the
reference specifications (r.x and r.y) for the
perception of those positions (p.x and p.y).
Therefore the model varies these references in
a way that would produce the observed
variations in qi.x and qi.y.Â

                      RM: r.x and r.y are the references that you


object to, claiming that they contain ad hoc
temporal dynamics. I question whether they do,
but I agree that it looks trivial (and a lot
like cheating) to put the desired end result
(the squiggly movements of the qi .x
and qi .y over time) into the
model (in the form of the identical squiggly
movements of r.x and r.y). But that is the way
the PCT model works; controlled (intended)
results are results that match specifications
set autonomously by the organism itself.Â

                      RM: But the reference signals in the


control model are not “cheating” any more than
are the Â “command” signals in motor control
models of behavior, like that of Gribble/Ostry ,
which is shown below. The commands in
theÂ model are commands for output; forces that
will produce movements of the pen that will
result in the observed squiggle. In the PCT
model, references, r.x and r.y, are commands
for input; perception that match the reference
specifications.

                      RM:Â So both models have to use internal


commands in order to produce the observed
result (squiggle). The difference is that the
commands in the PCT model “look like” the
observed result; the commands in the Gribble/Ostry
don’t necessarily “look like” the result
produced. But in both cases the commands have
to be carefully crafted to produce the correct
result. So the possibility of introducing “ad
hoc temporal dynamics” is present
in both models. But the PCT model can do
something that the Gribble/Ostry
cannot do: it can control. That is, it can
produce the intended squiggle in the face of
disturbances.Â

                      RM: Although the variations in the


references (r.x, Â r.y) in the PCT model
correspond to the squiggle that is produced (qi .x,
qi .y) , I didn’t expect this
simple model to produce a squiggle with a
power function relationship between angular
velocity (V) and curvature (R). I assumed,
like you, that the power law relationship was
either 1) a controlled result in itself (the
person controlling for speeding up through
tighter turns), which would require a whole
extra control organization in the model or 2)
the result of complex dynamic characteristics
of muscle force production (the functions of
e.x and e.y in the control model diagram
above) and/or of the feedback function
connecting force output to pen movement input
(k.f in that diagram).Â

                      RM: So I was very surprised to find that


the squiggle produced by this simple control
model showed a power relationship between V
and R . And when the squiggle was an ellipse
the coefficient of the power function was
about the same (~.31) as that found by
Gribble/Ostry for their ellipse
production model Â – a model with a far more
dynamically complex method of generating the
ellipse than my control model. That’s when I
realized that the observed relationship
between V and R might be a mathematical
property of all curved lines. And, indeed, it
turns out that it is. The relationship between
V and R, which can be found using kindergarten
math, isÂ

                        V


= Â |dXd2Y-d2XdY|Â 1/3Â *R1/3Â Â Â

RM: Â Note that the termÂ Â |dXd2Y-d2XdY |,
which I called D, implying that it was a
constant, is a variable. So the value of V
for any curve is proportional (exactly) to
the 1/3 power ofÂ |dXd2Y-d2XdY |
and the 1/3 power of R.Â Â I
was as surprised by this as as anyone. So I
wanted to make sure it was true so I did the
multiple regression analysis using log (Â |dXd2Y-d2XdY |
) and log (R) as predictors of log (V) for
many different “squiggles” and always found
that all the variance in log (V)
was accounted for by an equation of the
form:

                        log


(V) = .33* log (|dXd2Y-d2 XdY|) +.33*
log(R)Â

                          AGM:


Second, you are stubbornly confused about
the difference between ** a mathematical
relation** (that allows to re-express
curvature as a function of speed, plus
another non-constant term that you insist
in ignoring and treating like a constant),

                      RM: I hope you see now that I do not treat


the variableÂ Â |dXd2Y-d2XdY |
Â as a constant. The mathematical
relationship is as flawless as my
kindergarten math teacher;)

Â

                          between


a physical realization (the fact
that one can in principle draw the same
curved line at infinitely different
speeds),

                      RM: Actually, I understand that the same


curved line (squiggle) can be produced at an
infinity of different speeds and by an
infinity of different means (different
variations in o.x and o.y producing the same
squiggle, qi.x, qi .y).Â
The same relationship between V and R holds
regardless of the speed with which the
squiggle is produced and and the means used to
produce it. The relationship between V and R
is always:Â

                        log


(V) = .33* log (|dXd2Y-d2 XdY|) +.33*
log(R)Â

                          RM: This


same relationship between V and R even
holds for all the different squiggly
patterns made by o.x and o.y to Â produce
the same squiggly pattern --qi.x, qi.y.Â

                          AGM:


and between a biological fact
(that out of all possible combinations of
speed and curvature, living beings are,
for yet some unknown reason —but therre are
tens if not hundreds of papers making
proposals— constrained following the power
law,

                      RM: But now we know the reason. It's a


result of the fact that the relationship
between V and R for any curved line is

                        V


= Â |dXd2Y-d2XdY|Â 1/3Â *R1/3Â Â Â

                        RM:


There is no way to draw a curved line so
that this equation does not hold. So there
is no biological constraint that creates the
observed power law; it’s a mathematical
constraint.Â

                          RM: The


reason people have found different power
coefficients for the relationship between
V and R is because they have left the
variableÂ Â |dXd2Y-d2XdY |
Â out of the analysis. When you leaveÂ Â |dXd2Y-d2XdY |
Â out of the analysis, variations in that
variable can lead to different estimates
of the power coefficient of R. The amount
by which the value of the power
coefficient of R is affected by leavingÂ Â |dXd2Y-d2XdY |
Â out of the analysis depends on the shape
of the squiggle thatÂ is drawn. LeavingÂ Â |dXd2Y-d2 XdY| out of
the analysis of the V/R relationship for
an ellipse affects the actual power
coefficient of R (.33) very little, so the
value obtained is around .31 (see Gribble
and Ostry, Table 1). Other squiggles can
bring the power coefficient of R down as
low as .2.Â

                        RM:


On that note, Martin Taylor noted that the
power coefficient for R, which is around .33
for a curved figure drawn in the air, is
closer to .25 when the same figure is drawn
Â in a viscous medium (like water). It turns
out that this can be explained in terms of a
difference in the feedback function (k.f in
the diagram) in the two cases.Â

                        RM:


In the PCT model, the feedback function is a
simple, linear coefficient. When the model
traces out an ellipse with a feedback
function of k.f = 1.0, the power coefficient
of R is .32; when the feedback function is
changed to k.f = .5 – equivalent to trying
to move a pen through a more resistive
medium – theÂ power
coefficient of R is .26. This happens simply
because the ellipse drawn in the resistive
medium is a little sloppier than the one
drawn in the air. The change in feedback
function changes the loop gain of the
control system.Â

                          AGM:


But you will now reply for the n-th time
saying that everybody that has ever worked
on the power-law miss es the point of
control systems and that your toy demo
proves they don’t get it.

                      RM: Yes. But they have been fooled by a


rather convincing illusion. It’s hard not to
see the observed relationship between V and R
as a situation where the agent purposefully
changes speed through curves (the power law
actually suggests that agents increase their
speed as the curve increases, but this
increase in speed decreases as curvature
increases; it does not suggest that control
systems tend to slow down around sharp
curves).

                        RM: So I


agree that it is very surprising (and,
perhaps, disappointing) that the
relationship between V and R tells us
nothing about how people draw curves. But
that doesn’t mean that research on how
people (and other organisms) produce curved
paths should come to an end. To the
contrary, it opens up new and fruitful
questions about exactly how this is done.
For example, the integral output function
that I use in the existing model is
obviously an over simplification. Something
like theÂ Â Gribble/Ostry
model pictured above is probably a closed
approximation. A clever experimenter should
be able to design behavioral (and/or
physiological) studies to determine what the
best model of the output function is.Â

                        RM:


Going “up a level” (so to speak) research
could also be aimed at determining how what
are presumably higher level control systems
set the references for the x,y coordinates
of the figure being drawn (assuming that the
figure is a controlled result and not a side
effect of controlling other variables, as in
the CROWD demo).Â

                        RM:


So there is really a lot of very important
and challenging research to be done in order
to understand how people draw figures. But
this research must be based on an
understanding of the fact that the figure
drawn is a controlled variable – an
intended result. And so any research aimed
at understanding how figures are drawn must
be based on an understanding of how control
works.Â

                          AGM:


But, again, ** you gloss over serious
flaws interpreting** the difference
between mathematical equations, physical
conditions and biological constraints as
facts, ** and you magnify the relevance
of a toy demos** that, I wish could
shed new light, but so far don’t shed much
new to the problem.

                      RM: I hope this post helps. I don't believe


I have glossed over flaws. But if there are
substantive flaws please point them out. As I
said above, if my analysis is correct that
doesn’t mean all is lost. Indeed, I think it
actually opens up many new and very productive
possibilities for research.Â

Best regards

Rick

                          So


I encourage you (and everyone still
reading these email exchanges) to ** say
something new and relevant** , because
I still believe that asking what is being
perceived and what is being controlled is
worth-while in figuring out why speed and
curvature are constrained they way they
are.

Â

Alex

                                                Richard


S. MarkenÂ

                                                    "The childhood


of the human
race is far from
over. We have a
long way to go
before most
people will
understand that
what they do for
others is just
as important to
their well-being
as what they do
for themselves."
– William T. Powers

[From Rick Marken (2016.07.20.1920)]

···

Martin Taylor (16.07.20.21.19)

MT: Clearly you are following your usual practice of NOT reading what


you reply to. Here’s the nub of the problem. If “V” is a true
velocity, then the dots signify derivatives with respect to time. If
“R” is the radius of curvature, it’s a length and the dots signify
derivatives with respect to arc length along the curve. They aren’t
the same thing, but you use your “kindergarten math” as though they
are. No wonder you get weird results that have no internal
consistency.

RM: Actually, I get amazingly clear (internally consistent) results that are not weird at all but completely expected. But all this will become clear to you once I explain the analysis that I did. I’'ll have to write up such a description for the paper I plan to write anyway so it’s certainly worth the trouble.

Best regards

Rick

          RM: The equation for V and/or R may be wrong but those


are the equations used to compute these variables in
studies of the “power law”.

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

Frankfurt starts his short interview like this:

âI was disturbed by the lack of respect and concern for the truth that I seem to observe in much of the speech and writing that was being producedâ?

Short wonderful interview here:

http://press.princeton.edu/titles/7929.html

Some more excerpts that exactly transcribe what is in my mind after such an exhausting exchange of emails:

Bullshit is a lack of concern for the difference between truth and falsity.

**The motivitation of the bullshiter is not to say things that are true or even to say things that are false; but he is serving some other purpose; and the question of whether what he says is true and false is irrevelant to his pursuit of that ambition. The bullshiter does not necessary need to be a liar. He may not think that what he says is false.Â **

**Bullshit is a more insidious threath to the value of truth, than lying. The liar wants you to say away from the truth, he wants to substitute for the truth, so the difference between truth and flasehood is very important for the liar. The bullshitter does not care at all.Â ****The liar has a constraint, he is limited to inserting a false for the true one.Â **The bullshitter, can say whatever he wants.

Why are we so tolerant of bullshit, whereas we are not tolerant to the liar?

**The increase in the amount of bullshit in contemporary life is because of the intensity of the marketing motive: selling products, candidates, people, ideas, books, programs, etc.Â **Once you settle that your object is to sell something, then your object is not to tell the truth about it but to make people believe what you want them to believe about it.

The tendency to bullshit is encouraged and promoted by the view that the responsible citizen (or scientist) must have an opinion about everything.

Highly educated people have the gifts that enable them to create bullshit. A lot of highly educated people acquire a kind of arrogance that encourages the production of bullshit.

Despite not being actively promoted, bullshit is protected.

()

If anyone wants to have a serious, careful and not showing-off discussion about the speed-curvature power law, from now on I am available at agomezmarin@gmail.com

Have a hopefully non-bullshit day,

Alex

Â

···

On Thu, Jul 21, 2016 at 4:22 AM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.20.1920)]

Martin Taylor (16.07.20.21.19)

MT: Clearly you are following your usual practice of NOT reading what


you reply to. Here’s the nub of the problem. If “V” is a true
velocity, then the dots signify derivatives with respect to time. If
“R” is the radius of curvature, it’s a length and the dots signify
derivatives with respect to arc length along the curve. They aren’t
the same thing, but you use your “kindergarten math” as though they
are. No wonder you get weird results that have no internal
consistency.

RM: Actually, I get amazingly clear (internally consistent) results that are not weird at all but completely expected. But all this will become Â clear to you once I explain the analysis that I did. I’'ll have to write up such a description for the paper I plan to write anyway so it’s certainly worth the trouble.Â

Best regards

Rick

Richard S. MarkenÂ

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

          RM: The equation for V and/or R may be wrong but those


are the equations used to compute these variables in
studies of the “power law”.Â

[From Rick Marken (2016.07.21.2100)]

···

On Thu, Jul 21, 2016 at 1:32 AM, Alex Gomez-Marin agomezmarin@gmail.com wrote:

AGM: Frankfurt starts his short interview like this:

âI was disturbed by the lack of respect and concern for the truth that I seem to observe in much of the speech and writing that was being producedâ?

RM: Â This statement looks like an excellent example of what I would call bullshit: it sounds good at first until you realize that behind it is the implication that the author knows the truth and anyone who doesn’t show respect for that truth is doing something bad – bullshitting. This statement is what you hear from religious fanatics, not scientists. A good scientist constantly disrespects currently held truths (theories) by testing them against data. Religious fanatics are respectful of the “truth” and they expect others to be respectful of them too, or else. Scientists are skeptical of what is currently considered to be truth and they don’t get upset when they or others challenge those truths. What scientists do get upset about is people claiming as truth alternative truths that have been rejected by test – truths like flat earth, geocentric universe, phlogiston, immuniization causes autism, cutting top marginal tax rates spurs growth and human activity has no effect on climate change. Now that’s bullshit.

RM: The power law is not an inviolable truth about how movement is produced. The fact that it is apparently supported by a great deal of research certainly increases one’s confidence that it is close to the truth. But it still may not be true and I have presented evidence (and I will present more) that suggests that it is not.Â

RM: My proposal is not bullshit (in the sense I describe above) because I am not trying to get you to respect a truth that I “know” is true. I have presented a rejectable proposal. It can be rejected by showing that there are movement patterns that do not follow the 1/3 power relationship between V and R (or the 2/3 power relationship between A and C). This evidence Â would be particularly convincing if the power relationship were found only for movements made by organisms since the power relationship is supposed to be a biological constraint.Â

RM: So I’m really sorry if my proposal seems like bullshit to you (and others). It would have been nice to have you along at least as the loyal opposition. You might have even been able to convince me, with evidence and mathematics, that I am wrong. I do change my mind when I have been shown evidence that convinces me that I an wrong. This happened in at least one case that I remember quite well. I was arguing that people were uncontrollable because they can autonomously vary their references for the state of a controlled variable. Bruce Abbott said that was not true; that they were still controllable. So I set up a little simulation to prove Bruce wrong and, lo and behold, he was right. So I do respond well to contrary evidence and I will happily admit that I am wrong if that is what the evidence unambiguously shows. Â

RM: So I hope you will stick around and continue fighting for your position; it’s good that you are skeptical of mine. But it would be best if you could prove me wrong using evidence rather than just talk.

Best regards

Rick

Â

Short wonderful interview here:

http://press.princeton.edu/titles/7929.html

Some more excerpts that exactly transcribe what is in my mind after such an exhausting exchange of emails:

Bullshit is a lack of concern for the difference between truth and falsity.

**The motivitation of the bullshiter is not to say things that are true or even to say things that are false; but he is serving some other purpose; and the question of whether what he says is true and false is irrevelant to his pursuit of that ambition. The bullshiter does not necessary need to be a liar. He may not think that what he says is false.Â **

**Bullshit is a more insidious threath to the value of truth, than lying. The liar wants you to say away from the truth, he wants to substitute for the truth, so the difference between truth and flasehood is very important for the liar. The bullshitter does not care at all.Â ****The liar has a constraint, he is limited to inserting a false for the true one.Â **The bullshitter, can say whatever he wants.

Why are we so tolerant of bullshit, whereas we are not tolerant to the liar?

**The increase in the amount of bullshit in contemporary life is because of the intensity of the marketing motive: selling products, candidates, people, ideas, books, programs, etc.Â **Once you settle that your object is to sell something, then your object is not to tell the truth about it but to make people believe what you want them to believe about it.

The tendency to bullshit is encouraged and promoted by the view that the responsible citizen (or scientist) must have an opinion about everything.

Highly educated people have the gifts that enable them to create bullshit. A lot of highly educated people acquire a kind of arrogance that encourages the production of bullshit.

Despite not being actively promoted, bullshit is protected.

()

If anyone wants to have a serious, careful and not showing-off discussion about the speed-curvature power law, from now on I am available at agomezmarin@gmail.com

Have a hopefully non-bullshit day,

Alex

Â

Richard S. MarkenÂ

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

On Thu, Jul 21, 2016 at 4:22 AM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.20.1920)]

Martin Taylor (16.07.20.21.19)

MT: Clearly you are following your usual practice of NOT reading what


you reply to. Here’s the nub of the problem. If “V” is a true
velocity, then the dots signify derivatives with respect to time. If
“R” is the radius of curvature, it’s a length and the dots signify
derivatives with respect to arc length along the curve. They aren’t
the same thing, but you use your “kindergarten math” as though they
are. No wonder you get weird results that have no internal
consistency.

RM: Actually, I get amazingly clear (internally consistent) results that are not weird at all but completely expected. But all this will become Â clear to you once I explain the analysis that I did. I’'ll have to write up such a description for the paper I plan to write anyway so it’s certainly worth the trouble.Â

Best regards

Rick

Richard S. MarkenÂ

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

          RM: The equation for V and/or R may be wrong but those


are the equations used to compute these variables in
studies of the “power law”.Â