Equation of a star

Question

About three years back I read an article in wikipedia about equation(it was more like how to draw a curve that remembled a star) of a star. Though I dont remember vividly in that article there was a GIF image that showed how a star was drawn mathematically inside a big circle that had a small circle and the two circles were connected by a straight line. When the small circle was revolved inside the big circle the result was a smooth star(no pointed ends). My problem is that I cannot find anything in the internet about equation of a star or drawing a star mathematically or anything at all of that sort. May be I dont know the exact term for searching. I need this for an android application of mine where I am trying to generate a star programatically. Any hints, links and thoughts are highly appreciated.

Answer 1

You seem to refer to a hypotrochoid, the thing one can draw with a Spirograph toy. It can be drawn with sine and cosine, see the Wikipedia article.

Answer 2

You can get a star in the complex plane.

Let $z_i=e^{2\pi ki/4}$ for $i\in\{0,1,2,3\}$

Let $ |z-z_i|=\frac{\sqrt{2}}{2}$ or $$(z-z_i)(\bar{z}-\bar{z_i})=z\bar{z}-z\bar{z_i}-z_i\bar{z}+z_i\bar{z_i}=\frac{1}{2}$$

$$\frac{1}{2}=x^2+y^2+1-(\sqrt{x^2+y^2}\cos(Tan^{-1}(y/x)+\frac{\pi k}{2}))$$

$$\frac{1}{2}=x^2+y^2+1 -x\cos{\pi k/2}-y\sin{\pi k/2}, k\in\{0,1,2,3\}$$

These will generate tangent circles enclosing a star shaped area.