Polar coordinates of an egg-shaped curve

Question

For a simple 3D-rendering project, I need to get the shape of an egg. (actually, a prism with an egg-shaped based). The idea behind it is to explain how a camshaft in an engine is working.

From Wikipedia, I found the implicit definition: a.(1+k.y).x^2 + b.y^2 = 1

However, the page is fairly simple and has no english translation which seems to point that I'm missing something.

I need that curve in polar coordinates : I tried using x=r(theta).cos(theta) and y=r(theta).sin(theta)but I'm slightly rusted and could not solve it. Looking on the internet, I didn't find an obvious answer, so I'm wondering if there is one.

Any idea ? The reason why I need polar coordinates is that I need to rotate that egg and have the offset for moving my valve. It seems a very common and simple mechanical question, so I think there is an answer.

If there is none, I'll go back to using parts of cylinders.

I'm not comfortable with the usage of tags on this site, so I probably got it wrong (more of a stack overflow person)

Answer 1

I followed a fifferent approach that I described in a blog post: split the shape into parts, then compute the value for each part.

I used the old : split into circles streategy, whbich gives me a C1 surves, which is good enough.

More details in http://www.revelut.ch/2016/02/01/egg-shape-and-camshaft/

Answer 2

I think you could try with something like: $$ r=k(\cos^2\theta+a\cos\theta+b). $$ With suitable choices of parameters $a$ and $b$ the resulting curve is acceptably egg-shaped. See here an interactive demonstration: http://tube.geogebra.org/m/2480891 .