[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 |dX*d ^{2}Y-d^{2}X*dY|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 L^{2} .

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 d^{2}x/dy^{2} ,

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 |dX*d ^{2}Y-d^{2}X*dY|Â

^{1/3Â }*R

^{1/3},

you start with |L3 - L3|, which is fine, because you 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 L

^{3}, and it is taken to the 1/3

power, which yields something of dimension L. The other part of

Rick’s expression is R

^{1/3}and R has dimension L, so the

entire expression has dimension L

^{4/3}, which is not a

velocity, which would have dimension LT

^{-1}.

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

on “Curvature”, we can start with 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

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} .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 L^{2}T^{-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 L^{2}.

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## ···

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 |dX

ddY|Â^{2}Y-d^{2}X^{1/3Â }*R^{1/3Â Â }Â

RM: Â Note that the termÂ Â |dX

ddY |,^{2}Y-d^{2}X

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 |dXddY |^{2}Y-d^{2}X

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 |dXddY |^{2}Y-d^{2}X

) and log (R) as predictors of log (V) for many

different “squiggles” and always found thatallthe

variance in log (V) was accounted for by an equation of

the form:

`log`

(V) = .33* log (|dX

ddY|) +.33*^{2}Y-d^{2}X

log(R)Â

`AGM:`

Second, you are stubbornly confused about the

difference betweena 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 |dX

ddY |^{2}Y-d^{2}X

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 (|dX

ddY|) +.33*^{2}Y-d^{2}X

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 |dX

ddY|Â^{2}Y-d^{2}X^{1/3Â }*R^{1/3Â 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 |dXddY | Â out^{2}Y-d^{2}X

of the analysis. When you leaveÂ Â |dXddY | Â out^{2}Y-d^{2}X

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 |dXddY | Â out^{2}Y-d^{2}X

of the analysis depends on the shape of the squiggle

that Â is

drawn. LeavingÂ Â |dXddY| out of the^{2}Y-d^{2}X

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) tosay 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