Hello all,

I came across a problem that requires one to transform an upside down cone to the plane. So I need to come up with some kind of transformation that takes me from 3-space to 2-space that is one-to-one and onto.

This problem has nothing to do with my job, but I think it is interesting.

Geometrically it seems that you would just push the height down toward the origin and be left with a circle of radius equal to the hypotenuse of the right triangle formed when slicing the cone.

It would probably be best to parameterize the cone in spherical coordinates with the angle from the z-axis constant somewhere between 0 and pi/2. Then drop this term and just map to cylindrical coordinates keeping only the angle theta around the plane and the distance from the origin to height. I just realized I am thinking of the cone the other way around. Even so, I think it would be similar.

That’s what I am thinking. I wanted to know if there is a standard way of doing this.

It’s been a while since I have taken a math course and my new job doesn’t require much math, so I am pretty rusty. Any help would be great.

On the other hand, the new radius on the plane is probably proportional to the h

It’s been a while since I have taken a math course and my new job doesn’t require much math, so I am pretty rusty. Any help would be great.

thanks