A tear drop can be represented by the parametric equations:

$\displaystyle x(t)=r \cos(t)$

$\displaystyle y(t)=r \sin(t)\sin^{n-1}(t/2)$

The arc length is:

$\displaystyle A(n)=\int_0^{2\pi}\sqrt{(x'(t))^2+(y'(t))^2}dt$

Find $\displaystyle r(n)$ such that $\displaystyle A(n)=K$. That is, how must I adjust the value of $\displaystyle r$ so that the arc length remains constant as I vary the parameter $\displaystyle n$?