Equation of a curve of a point moving inside a circle

Question

Given circle $C_1$ of radius $r$ moving inside another circle $C_2$ of radius $R$ $(R\gt r)$ and tangent to this, I am in trouble to find the equation of the curve obtained by a fixed point on $C_1$ during the motion of $C_1$ inside $C_2$. Thanks.

Answer 1

I believe you are looking for hypocycloids, as described by http://en.wikipedia.org/wiki/Hypocycloid or http://mathworld.wolfram.com/Hypocycloid.html

An important fact in the derivation of the formulas is that from the time the small circle starts rolling until it returns to its starting position (having rolled around the entire circumference of the larger circle once), it will have rotated about its own center exactly $\frac{R}{r} - 1$ times. The coordinates of the "fixed point" are found by adding the displacement of the fixed point relative to the center of the small circle (which depends only on the small circle's rotation) to the displacement of the center of the small circle relative to the center of the large circle (which depends only on how far the small circle has rolled).

Answer 2

The curve is a hypocycloid.

(picture from Wikipedia)