Implicit arc length differentiation and perimeter computation?

• To: mathgroup at smc.vnet.net
• Subject: [mg33213] Implicit arc length differentiation and perimeter computation?
• From: jflanigan at netzero.net (jose flanigan)
• Date: Sat, 9 Mar 2002 03:19:35 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Hi,

Suppose I have the following equation:

f(x,y) = const.

Now, as I understand it, this implicitly defines y as a function of x,
y(x). Suppose that I am able to solve for this explicitly. Next, to
compute the perimeter defined by this equation:

ArcLength = integrate[ sqrt(1+y'(x)^2) dx]

Now, we are up to my question: If I derive a unit vector tangent to
the curve defined by f(x,y)=const. called v such that <v,del f(x,y)> =
0 at every point on the contour, how can I use this to compute the
same perimeter using polar coordinates. Note that del is the usual

I am thinking of something like

ArcLength = integrate[F,dphi]

where  F=<v,del some_other_function> which would evaluate to the
correct perimeter.

The question is how  can I get at this mysterious some_other_function

Any help appreciated. Thanks in advance.

By the way, I have tried a number of things like some_other_function =
ArcTan[x,y].

```

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