# Tangential Velocity and the Radius of Curvature

[From Bruce Abbott (2017.11.1640 EST)]

In the following I hope to clarify why in the computation of the power law for curved movements, V and R are mathematically independent, so that the observed power-law relationship must emerge from other constraints.

The radius of curvature R of a line at a given point P is the radius of a circle whose curvature best fits the curvature of the line at that point, as shown in the diagram below:

If the line is being drawn by a pen, the velocity V of the pen at point P is the instantaneous velocity tangent to the circle at point P, otherwise known as the tangential velocity.

Gribble and Ostry (1996) used the following formulas to compute V and R from data consisting of a series of X,Y points taken at regular time intervals during the drawing of a figure:

V = (Xdot2 + Ydot2)1/2

R = [(Xdot2 + Ydot2)3/2]/ |XdotYdotdot – XdotdotYdot|

“Xdot” and “Ydot” are the velocities of the point along the X axis and Y axis, respectively. These are the differences between adjacent points along their respective axes, e.g., Xdot = X1 – X0

“Xdotdot” and “Ydotdot” are the corresponding accelerations, taken as the difference between adjacent velocities.

The formula for V recognizes that V is the hypotenuse of a right triangle for which Xdot and Ydot are the opposite sides. According to the Pythagorean Theorem, “The square of the hypotenuse of a right triangle is equal to the sum of the squares of the opposite sides.”

This form appears again in the numerator of the formula for R: [(Xdot2 + Ydot2)3/2], which therefore can be rewritten as V3. The denominator of the formula for R, stated in English, says to take the velocity of the point along the X-axis times its acceleration along the Y axis and from this product, subtract the acceleration of the point along the X-axis times its velocity along the Y-axis, and then take the absolute value of the result. I will refer to this result as the “cross product.”

Now it may seem rather strange that the radius of curvature R should be calculated based on a numerator that is the cube of the point’s tangential velocity. Based on this, you might think that R is some function of tangential velocity. It is not. The reason it is not is found in the cross product.

To explain what the cross product does in the computation of R, I am going to lay out two scenarios. In the first, the line is being traced at a variable velocity along a portion of the curve for which the curvature is constant. In the second, the line is being traced at a constant velocity along a portion of the curve in which the curvature is getting tighter, or in other words, where R is getting smaller.

Case 1: Varying velocity, constant curvature

Let us assume that V is increasing. Consequently, the V-cubed in the numerator of the formula for R is increasing. But so are the velocities and accelerations of the point along the respective X and Y axes increasing. These changes increase the cross product proportionately. The consequence is that although V increases, R remains unchanged. This agrees with our condition that the point is moving around a curve of constant curvature.

Case 2: Constant velocity, variable curvature

Let us assume that the curve is tightening as the point moves long it at a constant velocity. In the formula for R, V-cubed is now constant. However, as the curve tightens, the velocities and accelerations along the X and Y axes are now changing. (How much each changes depends on the current direction of the tangent along which the point is traveling. Given that the curve is tightening, the cross product will be increasing, and as a result, R, the radius of curvature, will grow smaller.

Taken together, these two cases demonstrate the complete mathematical independence of V and R. Given any R whatsoever, V can take any value whatsoever. This agrees with the common-sense notion that the shape of a curve is unrelated to how fast the curve is being traced.

The Power Law for curved movements states that such movements are governed by a power-law relationship between V and R. The law may be expressed as follows: V = kRβ, where k is a constant of proportionality and β is the power exponent. Taking the logarithms on both sides yields this:

Log V = Log k + β Log R

This is a handy form in which to express the law because it makes Log V a linear function of R, which plots as a straight line with Log k as the intercept and β as the slope. Any systematic deviations from linearity are easily spotted on the plot. In many cases, such as freehand scribbles or drawing ellipses, the slope coefficient β has been observed to be close to 1/3.

As noted above, V and R are mathematically independent variables. That V turns out to be a function of R in these cases therefore must be due to some constraints that emerge when tracing such curves. For example, racecar drivers try to go as fast as possible on the straightaways but slow on the curves owing the centrifugal forces acting on the tires during a turn, which threaten to overwhelm the adhesion of the tires to the road surface. This naturally imposes a relationship between the car’s velocity and the racecourse’s curvature.

A contrary opinion has been offered by Marken and Shaffer (2017), who have argued that the power law is a mathematical artifact of the way V and R are computed. Noting that the term in the denominator of the formula for R is V-cubed, they rewrite the formula for R as follows

R = V3/D, where D symbolizes the cross product, |XdotYdotdot – XdotdotYdot|. They then solve the formula in terms of V:

V = D1/3R1/3

Taking the logarithms on both sides yields:

Log V = 1/3 Log D + 1/3 Log R

From this formula they conclude that “power laws may be a statistical artifact inasmuch as the results of the regression analysis used to determine whether the law holds simply reflect a mathematical relationship between the measures of curvature and velocity used in the analysis. In particular, these equations suggest that a regression analysis that includes log(D) as a predictor variable would always find the coefficient of the curvature variable, which corresponds to the estimate of the exponent of the power law, to be exactly equal to its true value, 1/3.”

This analysis is seriously mistaken, as I hope to demonstrate clearly in what follows.

Despite the superficial similarity, the “power law” arrived at based on the formula for R has no bearing on the empirical power law. In the Marken-Shaffer power law, R is a function of V as a consequence of the way in which R is computed. But this formula treats the cross product that appears in the denominator of the formula for R as if it were an independent predictor of V. Yet R itself depends on that same cross product. This logic is equivalent to logic of the following example:

Researchers have been investigating how a certain length variable L varies with area A of a rectangle. As the a point moves along one side of a rectangle, the area of the rectangle increases. Empirically it has been found that, in a particular environment,

L = kA2

Now the area involved is the area of a rectangle, so A = length L times width W. Thus, solving for L,

L = A/W

Following the Marken-Shaffer logic, I conclude that the relationship shown empirically is a mathematical consequence of the fact that area is a function of length.

The flaw in this logic is the assumption that the width W is constant. If the width of the rectangle is changing as its length increases, the area of the rectangle could be related to its length in any number of ways; for example, as the length increases linearly, the width could be increasing as the square root of the length. It is even possible that that the area could remain constant or even decrease as length increases. The Marken-Shaffer analysis would lead to the inclusion of W as the “D” factor in a regression analysis to discover the “hidden variable” due to which the observed relationship departs from the expected one (based on the formula for area) of strict proportionality. This while still asserting that the dependence of length on area is a mathematical consequence of the way in which the area of a rectangle is computed.

The mathematical independence of V and R can be appreciated in another way. Recall that D is the cross product that appears in the denominator of the formula for computing the radius of curvature, R. Its function in the formula is to remove the dependence of R on V so that R becomes a pure measure of shape, independent of the velocity with which the curve is drawn. An easy way to see this is to perform a dimensional analysis to make sure that the units of measure come out right. To make this example concrete, assume that distance is being measured in millimeters (m) and time in seconds (s). Plugging these units into the formula for R yields the following:

V3 = (mm/s)3 = mm3/s3

Xdot = mm/s

Xdotdot = mm/s2

Ydot = mm/s

Ydotdot = mm/s2

R = (mm3/s3)/ |(mm/s)(mm/s2) - (mm/s)(mm/s2)| = (mm3/s3)/(mm2/s3) = mm, a length, as it should be for a radius.

Dividing V3 by the cross product removes time from the measurement of R (the seconds all cancel out), yielding a pure measure of length. Therefore R is independent of V and not of necessity linked to R by a 1/3 (or other exponent) power law (or indeed, any other law). That it is apparently so linked under a variety of circumstances in which curves are being drawn or followed, is the question that researchers in this area have been trying to answer.

Bruce

[From Rick Marken (2017.11.12.1740)]

···

Â

Bruce Abbott (2017.11.1640 EST)–.Â

BA: As noted above, V and R are mathematically independent variables.Â

RM: I don’t see how that is consistent with the easily shown fact thatÂ

log(V) = 1/3 log® + 1/3 log (D)

BA: That V turns out to be a function of R in these cases therefore must be due to some constraints that emerge when tracing such curves.Â

RM: The relationship between V and R that is found using regression analysis depends on the nature of the trajectory itself (specifically, the extent to which variance in D – affine velocity – covaries with R–curvature) not on “some constraints that emerge when tracing such curves”, whatever those are. If the traced curve is produced intentionally (controlled) then one can say that the observed relationship between V and R is a side effect of carrying out that intention; a side effect that is perfectly predicted by an OVB analysis of that traced curve.Â

Â

BA: For example, racecar drivers try to go as fast as possible on the straightaways but slow on the curves owing the centrifugal forces acting on the tires during a turn, which threaten to overwhelm the adhesion of the tires to the road surface.Â This naturally imposes a relationship between the carâ€™s velocity and the racecourseâ€™s curvature.

RM: This and your reference to “tracing” made me realize that the problem with power law research, from a control theory perspective, stems from the fact that the measures of velocity and curvature that are used as the variables in the regression used to determine the power law are measures of properties of the same trajectory; that is, measures of V and RÂ are based on the same x,y values from the same movement trajectory. Zago et al mention a couple studies (Lacquaniti et al. 1983; Catavitello et al. 2016) where “the movement was guided by asking participants to follow with the pen tip the inner edge of a Plexiglas template cut by a numerical control milling machine.” This is the kind of study where it is possible to get measures of curvature and velocity that are truly independent of each other. This would be the case if the measures of curvature at each point in the movement trajectory were obtained from the template and the measures of velocity at each point in the movement trajectoryÂ were obtained from the pen tip. Call the coordinates of the template from the starting to the ending position of the pen x,y and the coordinates of the pen tip from its starting to ending position x’,y’. So C would be a function of x,y and V would be a function of x’,y’. Now you could determine whether people are controlling a variable, such as the one you mention, centrifugal force, that is a function of two independent variables C and V. C is now a legitimate measure of a possible disturbance variable and V is a measure of an aspect of the output variable. The CV (such as angular momentum or centrifugal force) would be a joint function of these two variables. Indeed, the best way to do the research would be to see if you could find a variable that is a function of C and V that remains nearly constant throughout the movement.Â

RM: I don’t knowÂ whether Lacquaniti et al. or Catavitello et al.did this. But if they did, any relationship they found between their measures of curvature and velocity would provide hints as to what variable is controlled when people move through such curved paths.Â

Â

BA:Â A contrary opinion has been offered by Marken and Shaffer (2017), who have argued that the power law is a mathematical artifact of the way V and R are computed.Â Noting that the term in the denominator of the formula for R is V-cubed, they rewrite the formula for R as follows

Â

R = V3/D, where D symbolizes the cross product, |XdotYdotdot – XdotdotYdot|.Â They then solve the formula in teerms of V:

Â

V = D1/3R1/3

Â

Taking the logarithms on both sides yields:

Â

Log V = 1/3 Log D + 1/3 Log R

RM: As I noted, we were apparently not the first to notice this simple fact; it was also noted by some power law researchers (Maoz, Portugaly, Flash & Weiss, 2006)Â to whom Zago et al refer in their rebuttal to our paper.Â

Â

BA:Â This analysis is seriously mistaken, as I hope to demonstrate clearly in what follows…

Â

BA: This logic is equivalent to logic of the following example:

Â

BA: Researchers have been investigating how a certain length variable L varies with area A of a rectangle.Â As the a point moves along one side of a rectangle, the area of the rectangle increases.Â Empirically it has been found that, in a particular environment,

Â

L = kA2

Â

BA: Now the area involved is the area of a rectangle, so A = length L times width W.Â Thus, solving for L,

Â

L = A/W

Â

BA: Following the Marken-Shaffer logic, I conclude that the relationship shown empirically is a mathematical consequence of the fact that area is a function of length.

RM: I don’t think that is quite the correct analogy. A more accurate one would be this:Â

Researchers are studying how people make the movements that create variations in the size of a square. The changes in size are produced by a mouse that simultaneously affects the length (x) and width (y) of the square so that x = m and y = m. It is found that variations in x and y over time follow a linear relationship: y = k*x where k is found to be equal to 1.0. This is called the “linear law” and it is assumed to say something about how people produce the movements that cause variations in the size of the square.Â

Â

BA: The mathematical independence of V and R can be appreciated in another way.Â

RM: V and R are related by the following formula

Log V = 1/3 Log D + 1/3 Log R

RM: when they are measures of the instantaneous velocity and curvature at the same points on the same trajectory. However, V and R are mathematically and physically independent when they are measures of the velocity and curvature of two different trajectories. I think this could be the basis for some interesting PCT-based research on the variables controlledÂ when people move along pre-determined curved paths, like roads and milled templates. So let’s get cracking doing some PCT research on how people move through curves!

Best regards

Rick

Â

Recall that D is the cross product that appears in the denominator of the formula for computing the radius of curvature, R.Â Its function in the formula is to remove the dependence of R on V so that R becomes a pure measure of shape, independent of the velocity with which the curve is drawn.Â An easy way to see this is to perform a dimensional analysis to make sure that the units of measure come out right.Â To make this example concrete, assume that distance is being measured in millimeters (m) Â and time in seconds (s).Â Plugging these units into the formula for R yields the following:

Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â V3 = (mm/s)3 = mm3/s3

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Xdot = mm/s

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Xdotdot = mm/s2

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Ydot = mm/s

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Ydotdot = mm/s2

Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â R = (mm3/s3)/ |(mm/s)(mm/s2) - (mm/s)(mm/s2)| = (mm3/s3)/(mm2/s3) = mm,Â a length, as it should be for a radius.

Â

Dividing V3 by the cross product removes time from the measurement of R (the seconds all cancel out), yielding a pure measure of length.Â Therefore R is independent of V and not of necessity linked to R by a 1/3 (or other exponent) power law (or indeed, any other law).Â That it is apparently so linked under a variety of circumstances in which curves are being drawn or followed, is the question that researchers in this area have been trying to answer.

Â

Bruce

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â€?
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â --Antoine de Saint-Exupery

[Eetu Pikkarainen 2017-11-13 9:28]

[From Bruce Abbott (2017.11.1640 EST)]

Bruce, thanks, now I realized better how the calculations are done. I wonder whether this could make it still more simple and plain:

In this diagram the trajectory produced during an experiment / observation is marked by the light blue line. The points 0 to 5 are the
series of X,Y points taken at regular time intervals. The momentary velocity at each point is the distance from the previous point to the current point divided by the time interval. These distances are depicted in the diagram as red
vectors from point to point. Now both the velocity and the curvature must be computed from these same vectors (i.e. series of X, Y pairs). However, this does not make velocity and curvature mathematically dependent on each other. The vector has two independent
properties: length and direction. The velocity is computed from the lengths of the vectors and curvature from the directions of them. As little as you can predict the length of a vector from its angle can you predict the velocity from the curvature.

(There is no mathematical dependence but there seems to a practical dependence visible in my random diagram: To give an accurate vector drawing of the changing curve I can use longer
vectors (i.e. higher speed) in flat parts and I must use shorter (i.e. slower speed of shorter time intervals) in sharper parts.)

Eetu

Please, regard all my statements as questions,

no matter how they are formulated.

[From Bruce Abbott (2017.11.1640 EST)]

In the following I hope to clarify why in the computation of the power law for curved movements, V and R are mathematically independent, so that the observed power-law relationship must emerge from other constraints.

The radius of curvature R of a line at a given point P is the radius of a circle whose curvature best fits the curvature of the line at that point, as shown in the diagram below:

If the line is being drawn by a pen, the velocity V of the pen at point P is the instantaneous velocity tangent to the circle at point P, otherwise known as the tangential velocity.

Gribble and Ostry (1996) used the following formulas to compute V and R from data consisting of a series of X,Y points taken at regular time intervals during the drawing of a figure:

V = (Xdot2 + Ydot2)1/2

R = [(Xdot2 + Ydot2)3/2]/ |XdotYdotdot – XdotdotYdot|

“Xdot” and “Ydot” are the velocities of the point along the X axis and Y axis, respectively. These are the differences between adjacent points along their respective axes, e.g., Xdot = X1 – X0

“Xdotdot” and “Ydotdot” are the corresponding accelerations, taken as the difference between adjacent velocities.

The formula for V recognizes that V is the hypotenuse of a right triangle for which Xdot and Ydot are the opposite sides. According to the Pythagorean Theorem, “The square of the hypotenuse of a right triangle is equal
to the sum of the squares of the opposite sides.”

This form appears again in the numerator of the formula for R: [(Xdot2 + Ydot2)3/2], which therefore can be rewritten as V3 . The denominator of the formula for R, stated in
English, says to take the velocity of the point along the X-axis times its acceleration along the Y axis and from this product, subtract the acceleration of the point along the X-axis times its velocity along the Y-axis, and then take the absolute value of
the result. I will refer to this result as the “cross product.”

Now it may seem rather strange that the radius of curvature R should be calculated based on a numerator that is the cube of the point’s tangential velocity. Based on this, you might think that R is some function of
tangential velocity. It is not. The reason it is not is found in the cross product.

To explain what the cross product does in the computation of R, I am going to lay out two scenarios. In the first, the line is being traced at a variable velocity along a portion of the curve for which the curvature
is constant. In the second, the line is being traced at a constant velocity along a portion of the curve in which the curvature is getting tighter, or in other words, where R is getting smaller.

Case 1: Varying velocity, constant curvature

Let us assume that V is increasing. Consequently, the V-cubed in the numerator of the formula for R is increasing. But so are the velocities and accelerations of the point along the respective X and Y axes increasing.
These changes increase the cross product proportionately. The consequence is that although V increases, R remains unchanged. This agrees with our condition that the point is moving around a curve of constant curvature.

Case 2: Constant velocity, variable curvature

Let us assume that the curve is tightening as the point moves long it at a constant velocity. In the formula for R, V-cubed is now constant. However, as the curve tightens, the velocities and accelerations along the
X and Y axes are now changing. (How much each changes depends on the current direction of the tangent along which the point is traveling. Given that the curve is tightening, the cross product will be increasing, and as a result, R, the radius of curvature,
will grow smaller.

Taken together, these two cases demonstrate the complete mathematical independence of V and R. Given any R whatsoever, V can take any value whatsoever. This agrees with the common-sense notion that the shape of a curve
is unrelated to how fast the curve is being traced.

The Power Law for curved movements states that such movements are governed by a power-law relationship between V and R. The law may be expressed as follows: V = kRβ , where k is a constant of proportionality
and β is the power exponent. Taking the logarithms on both sides yields this:

Log V = Log k + β Log R

This is a handy form in which to express the law because it makes Log V a linear function of R, which plots as a straight line with Log k as the intercept and β as the slope. Any systematic deviations from linearity
are easily spotted on the plot. In many cases, such as freehand scribbles or drawing ellipses, the slope coefficient β has been observed to be close to 1/3.

As noted above, V and R are mathematically independent variables. That V turns out to be a function of R in these cases therefore must be due to some constraints that emerge when tracing such curves. For example, racecar
drivers try to go as fast as possible on the straightaways but slow on the curves owing the centrifugal forces acting on the tires during a turn, which threaten to overwhelm the adhesion of the tires to the road surface. This naturally imposes a relationship
between the car’s velocity and the racecourse’s curvature.

A contrary opinion has been offered by Marken and Shaffer (2017), who have argued that the power law is a mathematical artifact of the way V and R are computed. Noting that the term in the denominator of the formula
for R is V-cubed, they rewrite the formula for R as follows

R = V3/D, where D symbolizes the cross product, |XdotYdotdot – XdotdotYdot|. They then solve the formula in terms of V:

V = D1/3R1/3

Taking the logarithms on both sides yields:

Log V = 1/3 Log D + 1/3 Log R

From this formula they conclude that “power laws may be a statistical artifact inasmuch as the results of the regression analysis used to determine whether the law holds simply reflect a mathematical
relationship between the measures of curvature and velocity used in the analysis. In particular, these equations suggest that a regression analysis that includes log(D) as a predictor variable would always find the coefficient of the curvature variable, which
corresponds to the estimate of the exponent of the power law, to be exactly equal to its true value, 1/3.”

This analysis is seriously mistaken, as I hope to demonstrate clearly in what follows.

Despite the superficial similarity, the “power law” arrived at based on the formula for R has no bearing on the empirical power law. In the Marken-Shaffer power law, R is a function of V as a consequence of the way in
which R is computed. But this formula treats the cross product that appears in the denominator of the formula for R as if it were an independent predictor of V. Yet R itself depends on that same cross product. This logic is equivalent to logic of the following
example:

Researchers have been investigating how a certain length variable L varies with area A of a rectangle. As the a point moves along one side of a rectangle, the area of the rectangle increases. Empirically it has been
found that, in a particular environment,

L = kA2

Now the area involved is the area of a rectangle, so A = length L times width W. Thus, solving for L,

L = A/W

Following the Marken-Shaffer logic, I conclude that the relationship shown empirically is a mathematical consequence of the fact that area is a function of length.

The flaw in this logic is the assumption that the width W is constant. If the width of the rectangle is changing as its length increases, the area of the rectangle could be related to its length in any number of ways;
for example, as the length increases linearly, the width could be increasing as the square root of the length. It is even possible that that the area could remain constant or even decrease as length increases. The Marken-Shaffer analysis would lead to the
inclusion of W as the “D” factor in a regression analysis to discover the “hidden variable” due to which the observed relationship departs from the expected one (based on the formula for area) of strict proportionality. This while still asserting that the
dependence of length on area is a mathematical consequence of the way in which the area of a rectangle is computed.

The mathematical independence of V and R can be appreciated in another way. Recall that D is the cross product that appears in the denominator of the formula for computing the radius of curvature, R. Its function in
the formula is to remove the dependence of R on V so that R becomes a pure measure of shape, independent of the velocity with which the curve is drawn. An easy way to see this is to perform a dimensional analysis to make sure that the units of measure come
out right. To make this example concrete, assume that distance is being measured in millimeters (m) and time in seconds (s). Plugging these units into the formula for R yields the following:

V3 = (mm/s)3 = mm3/s3

Xdot = mm/s

Xdotdot = mm/s2

Ydot = mm/s

Ydotdot = mm/s2

R = (mm3/s3)/ |(mm/s)(mm/s2) - (mm/s)(mm/s2)| = (mm3/s3)/(mm2/s3) = mm, a length, as it should be for a radius.

Dividing V3 by the cross product removes time from the measurement of R (the seconds all cancel out), yielding a pure measure of length. Therefore R is independent of V and
not of necessity linked to R by a 1/3 (or other exponent) power law (or indeed, any other law). That
it is apparently so linked under a variety of circumstances in which curves are being drawn or followed, is the question that researchers in this area have been trying to answer.

Bruce

[Bruce Abbott (2017.11.13.0900 EST)]

[From Rick Marken (2017.11.12.1740)]

Bruce Abbott (2017.11.1640 EST)–.

BA: As noted above, V and R are mathematically independent variables.

RM: I don’t see how that is consistent with the easily shown fact that

log(V) = 1/3 log® + 1/3 log (D)

I explained all that, but I donâ€™t see any evidence that you read that explanation – and you excised it from your reply.Â The second proof I gave, in the form of a dimensional analysis of the units that appear in the formula for R, similarly went unmentioned.Â But letâ€™s have a look at your analysis.

Your equation solves for V in the equation for computing the radius of curvature:

R = V3/D, where D is the cross-product term |XdotYdotdot – XdotdotYdot||

Solving for V gives

V3 = RD.Â Taking the cube root on both sides yields

V = R1/3D1/3Â .Â Taking the logarithms on both sides then gives

log V = 1/3 log R + 1/3 log D

This is in the form of a power law with an exponent of 1/3.Â It is in part this formal similarity to the empirical power law, with its commonly found exponent of 1/3, that has led you into the mathematical blunder that I (and several others) have been trying to disabuse you of.

If this analysis were true, then the data relating V and R should always yield a power law relation with an exponent of 1/3.Â Empirically, however, it doesnâ€™t; in fact the exponent can vary widely across experimental circumstances.Â To account for this you introduce the idea that any deviations from an exponent of 1/3 are due to a â€œhidden variable bias,â€? which you then explore by using D as a predictor variable in a linear regression analysis.

The problem with all this is that in the formula for R, D functions as a suppressor variable.Â Depending on the degree of curvature and the tangential velocity at a given point reached at time t, D removes the effect of velocity from the computation of radius.Â You can solve for V (as you have) in the formula for R, but all this tells you is what the velocity of the point was at the moment that R, a pure measure of shape, was computed.Â A moment later both V and R may have changed, but the cross-product in the denominator of the formula for R will also have changed in such a way as to remove Vâ€™s correlation with R.

When you plug the observed values of V and R into your regression equation, using D as a predictor variable, what you are asking is â€œwhat value of D will give me the observed relation between V and R?â€?Â However, the correct value of D, obtainable from the data, is the one that removes the correlation between V and R in the computation of R.Â Your â€œhidden variable,â€? if it differs from D1/3, simply introduces a spurious correlation between V and R by making sure that D does not vary in a way that compensates for changes in V in the computation of R.

Worse, this new, empirical value of D (found through linear regression) implies a value for curvature (if this value is plugged into the computation of R) that in general will not agree with the actual curvature of the shape!

Now, Iâ€™ve dealt directly with your analysis and shown why it is wrong.Â You have not attempted to do the same with my proof that V and R are mathematically independent measures.

Bruce

[Martin Taylor 2017.11.13.09.39]

[From Rick Marken (2017.11.12.1740)]

``````Rick, you say you are going to write a rebuttal if (and by
``````

implication only if) my comment gets published. Given this comment
to Bruce, I assume you are able to refute my proof that D (defined
as xdotydotdot - ydotxdotdot) is V3 times a function of
purely spatial variables we can call S. Could you put that disproof
on CSGnet (or if you prefer, send it to me privately), please?

``````If my proof holds, then your equation can be rewritten

log(V) = 1/3 log(R) + log (V) + log (S)

Hence your equation is entirely consistent with V and R being
``````

mathematically independent variables, so your disproof of my
analysis of D is essential. To make it easy, so you don’t have to
(re?)read my comment, here is my analysis of D:

``````![Dequation.jpg|303x163](upload://bj651NxdMdYvrHJrDS7PYocedWV.jpeg)

Martin
``````
···
``````                Bruce Abbott (2017.11.1640
``````

EST)–.

``````                BA: As noted above, V and R are
``````

mathematically independent variables.

``````          RM: I don't see how that is consistent with the easily
``````

shown fact that

log(V) = 1/3 log® + 1/3 log (D)

[From Rick Marken (2017.11.13.2200)]

···

Martin Taylor (2017.11.13.09.39)-

``````MT: Rick, you say you are going to write a rebuttal if (and by
``````

implication only if) my comment gets published.

RM: No I’ll write a rebuttal no matter what. If your comment gets published I’ll be astonished and disappointed in the judgement of the editors. But I’ll write a rebuttal to your comment as well if it actually gets published. But you asked for advanced comments on your comment so here is mine: I find your comment sickening, unprofessional and wrong. If your “proof” that log(V) = 1/3 log® + log (V) + log (S) were correct (which it’s not) then it would be power law researchers who have been doing things the wrong way for the last 20+ years, not me.Â

RM: The analyses in the Marken/Shaffer paper was done using the formulas for calculating velocity and curvature that have used by power law researchers for decades; the formulas published in their articles. The clearest formulas are those in Gribble and Ostry:

These are the ones you don’t like because you say that the derivatives for R are with respect to spatial rather than time differentials. But, in fact, the differentials in (8) and (9) are all with respect to time, just as the equations say. Contrary to your contention these equations are correct (Gribble and Ostry are much better mathematicians than you think you are). They are derived from the following formulas from Viviani (1992):

RM: It can be proved (if you could do the math) that (8) and (9), using time differentials, are equivalent to (A5) and (A6), using spatial and time differentials. It’s easier to calculate V and R using (8) and (9) rather than (A5) and (A6).Â

RM:Â I will now go wash my mind out with Bach.

BestÂ

RickÂ

Â

``````Given this comment
``````

to Bruce, I assume you are able to refute my proof that D (defined
as xdotydotdot - ydotxdotdot) is V3 times a function of
purely spatial variables we can call S. Could you put that disproof
on CSGnet (or if you prefer, send it to me privately), please?

``````If my proof holds, then your equation can be rewritten

log(V) = 1/3 log(R) + log (V) + log (S)

Hence your equation is entirely consistent with V and R being
``````

mathematically independent variables, so your disproof of my
analysis of D is essential. To make it easy, so you don’t have to
(re?)read my comment, here is my analysis of D:

``````Martin
``````

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you
have nothing left to take away.â€?
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â --Antoine de Saint-Exupery

[Martin Taylor 2017.11.14.11.16]

``````I have no opinion on that. I never wanted it to be published in the
``````

first place, but each of the three section editors asked me to
submit a version for publication, which the only reason I wrote it.
Are you going to show in what way it is wrong? I await your
counter-proof. Since everything else in my comment hinges on that, I
think it is rather important to the discussion.
Since you seem to defer to Richard Kennaway’s mathematical ability,
while asserting that Gribble and Ostry, who you do not know, are
better mathematicians than me, I ask Richard whether he thinks that
my derivation
is wrong, and whether, based on this, equating 1/3 log (D) to log(V)

• log(S) where S is an expression in purely spatial variables (x, y
and s, s in this case being interpreted as distance along the curve,
though it could be any variable at all) is therefore also wrong.
But really, it would be much better if you simply did the disproof
yourself. It’s easy, since the derivation uses only middle
high-school maths (at least it was in my Scottish high school nearly
70 years ago).
Incidentally, are you aware of the chain rule of differentiation
according to which dx/dt = dx/ds*ds/dt? In what way would they have been doing anything wrong (apart from
not finding the reason for their power-law findings)? Simply
reporting observations cannot be too wrong.
I suggest you read what I wrote rather than what you wish I had
written. As I said, those formulae are correct. It is only your
interpretation of the relationship between them that is wrong.
They certainly can be, as I wrote. Since the experimenters who do
this work already have the measures, using the time-derivatives
makes radius calculations easy. Just try what Bruce Abbott
suggested: compute R in that expression using different values of V.
Doubling V doesn’t change the calculation of R at all.
Since I said that both equations are correct, in what way is it
contrary to my contention when you say that they are correct?
Correct. Do you realize that Viviani’s “phi” parameter is the
generic parameter that resolves to arc distance along the curve from
an arbitrary zero – the “s” in the derivation you don’t like or the
“t” in Gribble and Ostry?
And then would you come back with a cleaner mind and answer my
request that you show how my analysis of D is wrong, without using
your idiosyncratic interpretations of appeals to other authorities?
Martin

···

On 2017/11/14 1:06 AM, Richard Marken
wrote:

[From Rick Marken (2017.11.13.2200)]

Martin Taylor (2017.11.13.09.39)-

``````            MT: Rick, you say you are going to write a rebuttal if
``````

(and by implication only if) my comment gets published.

``````          RM: No I'll write a rebuttal no matter what. If your
``````

comment gets published I’ll be astonished and disappointed
in the judgement of the editors.

``````          But I'll write a rebuttal to your comment as well if it
``````

comment sickening, unprofessional and wrong. If your
“proof” that log(V) = 1/3 log® + log (V) + log (S) were
correct (which it’s not)

``````          then it would be power law researchers who have been
``````

doing things the wrong way for the last 20+ years, not
me.

``````          RM: The analyses in the Marken/Shaffer paper was done
``````

using the formulas for calculating velocity and curvature
that have used by power law researchers for decades; the
formulas published in their articles. The clearest
formulas are those in Gribble and Ostry:

``````          These are the ones you don't like because you say that
``````

the derivatives for R are with respect to spatial rather
than time differentials.

``````          But, in fact, the differentials in (8) and (9) are all
``````

with respect to time, just as the equations say.

Contrary to your contention these equations are correct

``````          (Gribble and Ostry are much better mathematicians than
``````

you think you are). They are derived from the following
formulas from Viviani (1992):

``````          RM: It can be proved (if you could do the math) that
``````

(8) and (9), using time differentials, are equivalent to
(A5) and (A6), using spatial and time differentials. It’s
easier to calculate V and R using (8) and (9) rather than
(A5) and (A6).

RM: I will now go wash my mind out with Bach.

[From Richard Kennaway 2017.11.15 1352 GMT]

R, the radius of the curvature, is a property of each point of the curve and is independent of the velocity with which it is traversed, or whether it is traversed at all.

That G&O's formula for R involves derivatives with respect to time does not change that. Time can be reparameterised in any way whatever and R will be exactly the same function of position.

Martin Taylor's formula in [Martin Taylor 2017.11.14.11.16] expresses R in terms of the parameterisation of the curve by arc length: 1/R = (dx/ds)(d2y/ds2) - (dy/ds)(d2x/ds2). (In words, that formula expresses the definition of curvature as the rate of change of the direction of the tangent vector with respect to arc length.) This definition expresses R without reference to any velocity profile.

G&O describe R in terms of the time parameterisation: R = V^3 / ((dx/dt)(d2y/dt2) - (dy/dt)(d2x/dt2)). For a given shape of path, both formulas give exactly the same value for R at the same point of the path. They are mathematically equivalent. The latter formula will continue to give the same value of R at the same point of the path whatever V looks like. V can be eliminated to give the previous formula, expressing R in terms of nothing but the shape of the curve.

No matter how much you look at the equation V^3 = DR, or equivalent formulations in terms of logarithms, V and R remain independent quantities. I can take an arbitrary path, and define velocity as an arbitrary function of time, and traverse that path with that velocity profile. The equation will still hold. However I change V, D = (dx/dt)(d2y/dt2) - (dy/dt)(d2x/dt2) = | cross(V,dV/dt) | will also change and R will remain the same.

The equation V^3 = DR is not an equation relating V and R, or V and D, or D and R. It is an equation relating all three. No relationship between any two of them exists without assuming some constraint on the third.

···

--
Richard Kennaway
School of Computing Sciences, University of East Anglia, Norwich, UK
Cell and Developmental Biology, John Innes Centre, Norwich, UK