I have rather embarrassing request to make. See, I visualize badly in anything above 2 dimensions, so I have to work hard to figure out equations for 3D shapes. Additionally I am slightly "technologically challenged" as to 3D graphing software: the only one I've ever understood was Mathematica, which I no longer have a license for.

So it comes down to this: I would like someone to graph a surface for me so I can double check that I have something of the correct form.

What I am going for is this: The "base" of this shape is, in Physics, called the "sombrero" potential. (I've attached a cross-section of the graph below. It is named so for the rough similarity the shape has to the Mexican hat.) The basic form of this equation is $\displaystyle z = -a \rho ^2 + b \rho ^4$ where a and b are positive. (I'm using cylindrical coordinates.)

Now, imagine a ball sitting at $\displaystyle \rho = 0$. It is in an unstable equilibrium. It will roll down the "sombrero" and eventually settle at the lowest point, of which there are an infinite number, due to the rotational symmetry.

What I want to do is alter this function so that the ball can only roll down certain "channels": instead of having the same probability for ending up at any point on the minimum, I want it to have to choose one of several locations I have predetermined. So we must introduce some sort of sinusoidal variation into the sombrero.

What I have come up with is this:

$\displaystyle z = (-a \rho ^2 + b \rho ^4) + c \rho ~ sin \left ( \frac{ 2\pi \theta}{k} \right )$

The value of k will be integral and will determine the number of "valleys" (absolute minima) the surface will contain.

Since I need someone to graph this, let me give a sample graph to look at:

$\displaystyle z = (-6 \rho ^2 + \rho ^4) + \frac{1}{5} \rho ~ sin \left ( \frac{ \pi \theta}{2} \right )$

I would like to verify that this surface has the properties I am looking for (and that my poor deficient mind is coming up with as a solution.)

Thank you for your trouble.

-Dan

PS If anyone has a "correct" Mathematical name for this surface I would be interested in knowing it. For the moment I'm calling it the "crumpled sombrero."