# Thread: Constant circumference of tear drop

1. ## Constant circumference of tear drop

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 $\displaystyle n$?

2. Originally Posted by shawsend
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 $\displaystyle n$?
Your definition of $\displaystyle A(n)$ is independent of $\displaystyle n$. Do you mean find $\displaystyle r$ such that $\displaystyle A=K$?

The way $\displaystyle A$ is defined it is proportional to $\displaystyle r$ so if you know A for the case r=1 it is simple to find A for any other value of r, or find the r corresponding to a desired value of A..

RonL

3. I don't understand then. I thought the arc length is a function of n. When I simplify the radical I get (Mathematica-generated latex):

$\displaystyle \surd \left(r^2 \text{Sin}[t]^2+\left(r \text{Cos}[t] \text{Sin}\left[\frac{t}{2}\right]^{-1+n}+\frac{1}{2} (-1+n) r \text{Cos}\left[\frac{t}{2}\right] \text{Sin}\left[\frac{t}{2}\right]^{-2+n} \text{Sin}[t]\right)^2\right)$

I'll work on it today.

4. . . . wonderful. I get it now:

$\displaystyle r(n)=\frac{K}{\int_0^{2\pi} f(n,t)dt}$