# Hypocycloid Problem

• April 14th 2009, 02:29 PM
coolguy99
Hypocycloid Problem
"A hypocycloid is a curve traced by a fixed point P on a circle C with radius b as C rolls on the inside of a circle with the center on the origin and radius a. If the initial point P is (a,0) and the parameter is theta, then the parametric equation is

x = (a-b)cos(theta) + bcos((theta)(a-b)/b))
y = (a-b)sin(theta) + bsin((theta)(a-b)/b))

Question 1: Show that the parametric equation is correct"

This is the picture I drew out to clarify..

http://i155.photobucket.com/albums/s...on9/what-1.jpg

But I don't even know how to start explaining why it is correct... any suggestions?
• April 14th 2009, 10:38 PM
earboth
Quote:

Originally Posted by coolguy99
"A hypocycloid is a curve traced by a fixed point P on a circle C with radius b as C rolls on the inside of a circle with the center on the origin and radius a. If the initial point P is (a,0) and the parameter is theta, then the parametric equation is

x = (a-b)cos(theta) + bcos((theta)(a-b)/b))
y = (a-b)sin(theta) + bsin((theta)(a-b)/b))

Question 1: Show that the parametric equation is correct"

...

1. In my opinion the given equation describes an epicycloid. If you want to get a hypocycloid the equation should read:
$
\left|\begin{array}{l}x = (a-b) \cos(\theta) + b \cos\left(\dfrac{\theta(a-b)}{b} \right) \\ y = (a-b) \sin(\theta) {\color{red}\bold{-}} b \sin\left(\dfrac{\theta(a-b)}{b} \right) \end{array} \right.$

2. The distance in your sketch which is labeled a - b is actually $a - 2b$

3. To start:
- Determine where you can find $\theta$ in your sketch.
- Determine where you can find $\left(\dfrac{\theta(a-b)}{b} \right)$ in your sketch.
- Calculate the corresponding distances which form the coordinates of the points placed on the cycloid.