Re: Natural Eq'm of Cardioid

In article <gerry-9E8F09.11223702052007@xxxxxxxxxxxxxxxxxx>,
Gerry Myerson <gerry@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

In article <1178063837.566281.103700@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Rodo <mattrodo@xxxxxxxxx> wrote:

I'm having a difficult time with what is probably an easy problem.
The problem states:<br> "Consider the cardioid
r = 1 - cos(theta), 0 less than or equal to (theta) less than or
equal
to 2pi. Let s(theta) be the arc length from the point (2, pi) on the
cardioid to the point (r, theta) and let "rho" (theta) = 1 / K be the
radius of the curvature at the point (r, theta). Show that s and
"rho" (theta) are related by the equation s^2 + 9("rho")^2 = 16.


I haven't even the slighest clue where to begin with this. Any
insight is greatly appreciated!

Well, there's probably a formula somewhere for calculating
the arc length along a curve given in polar co-ordinates,

http://en.wikipedia.org/wiki/Arc_length

and there's probably a formula somewhere for calculating
the radius of curvature along a curve given in polar co-ords.

http://en.wikipedia.org/wiki/Curvature

I'd start by seeing what those formulas have to say about
your cardioid.
.

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