Re: Curvature for a circle

• To: mathgroup at smc.vnet.net
• Subject: [mg83110] Re: Curvature for a circle
• From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
• Date: Sun, 11 Nov 2007 02:59:46 -0500 (EST)
• Organization: The Open University, Milton Keynes, UK
• References: <fh3qqc\$o9\$1@smc.vnet.net>

```dpdoughe at dialup4less.com wrote:

> It can easily be shown that the curvature for a circle of radius r is
> 1/r.  How can I get Mathematica to show me this?
>
> I have defined a function to calculate the curvature for me assuming a
> vector-valued function.
>
> Curvature =
>   Function[{V, x}, Norm[D[D[V, x]/Norm[D[V, x]], x]/Norm[D[V, x]]]];
>
> S = {2*Cos[theta], 2*Sin[theta]}
>
> ListPlot[Table[{t, Curvature[S, theta] /. theta -> t}, {t, 0, 2*Pi}]]
>
> I don't get anything for a plot.  If I look at what my curvature
> formula calculates for me there are some questions as to if
> Abs'  (Derivative of absolute value??) being left unevaluated.
>
> Any thoughts?  It should be a straight-forward calculation....

Using the built-in function *Norm* may be not a good idea when dealing
with derivatives. As you witnessed, it might generates absolute values
*Abs* and Mathematica does not know how to differentiate such things.

What you should do in such cases is witting your own norm function as
illustrated in the example below.

norm = Function[{V}, Sqrt[V.V]];

Curvature =
Function[{V, x}, norm[D[D[V, x]/norm[D[V, x]], x]/norm[D[V, x]]]];

S = {2*Cos[theta], 2*Sin[theta]}

ListPlot[Table[{t, Curvature[S, theta] /. theta -> t}, {t, 0, 2*Pi}]]

In doing so, you get rid of the *Abs* function (and its unevaluated
derivative) as you can see by evaluating

Table[{t, Curvature[S, theta] /. theta -> t}, {t, 0, 2*Pi}]

HTH,
--
Jean-Marc

```

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