[Martin Taylor 2016.07.18.21.49]

```
So why has time mysteriously disappeared from equation 1? V =
```

sqrt((dx/dt)^2 + (dy/dt^2)) is indeed a proper velocity. But Why should you expect to compute V from R or vice versa, since the

essence of the problem is to ask why control systems tend to slow

down around sharp curves, but less so if the organism is a larva or

a human moving in a viscous medium. Then the power-law relation

between linear velocity and R has an exponent nearer 0.25 than 0.33.

The geometry is the same. Why is the relation between V and R

different?

Martin

## ···

On 2016/07/18 6:17 PM, Richard Marken

wrote:

[From Rick Marken (2016.07.18.1515)]

Martin Taylor (2017.07.18.14.13)

`Alex:`

you have simply numerically checked

a mathematical equality by writing

curvature as a function of speed.

Every single scientist working on

the power law has done that. But

they don’t believe it is such a big

achievement.

`RM: If they know this then I don't`

understand why they still think that any

observed relationship between R and V would

tell them anything about how the curves are

produced. There is simply no way for them to

find any relationship between R and V other

than

`V`

= |dX

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

(1)

`MT: The question`

isn’t about geometry, it’s about velocity. V =

d(distance)/d(time). Your V is just a measure of local

curvature. There’s no time in it at all. Alex keeps

telling you as much. His original question was about why

people slow down at sharp curves, not about how you

describe curves in a Cartesian space.`V(angular velocity) = (d(distance travelled)/d(time))/R`

for a portion of a circular arc of radius R.

`RM: In determining the power relationship between V and R,`

V (velocity) is measured as:

`where X.dot and Y.dot are Newton's notation for the`

time derivatives of X (dX/dt) and Y(dY/dt) respectively.

This measure of V is the variable that is actually

computed from the curved paths that are observed in the

velocity/curvature power law studies. This measure of

velocity is then related to the R measure of curvature,

which is computed as follows:

`RM: Equation 1 is found by solving the two equations`

above, for V and R, simultaneously. X and Y are, of

course, the coordinates of the curve (or path) that was

observed to have been produced by the organism under

study.

```
V
```

= |dX*d ^{2}Y-d^{2}X*dY|

^{1/3 }*R

^{1/3}is not.

Something (a bunch of dt’s) got lost in the translation.

`RM: So whatever the merits of your equation for V might`

be (and one of its demerits is that it doesn’t specify how

to compute the value of the radius, which you call R) it

is not the V that they are talking about in the power law

research.

Best

Rick

–

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.

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