# Fast Planar Projection Onto An Ellipsoid

In a previous post I talk about fast planar projections onto spheres. After I put that out I wondered if it were possible to do the same thing with other shapes. If you haven't read that original post, I would recommend it because in this post I am only going to cover the part that is unique to the ellipsoids.

The formula that defines an ellipsoid is:

It isn't necessary to understand how this formula is derived but if you are interested then here is an explanation of the formula for an ellipse which can be generalized and used for an ellipsoid.

a, b, and c are known constants, and we can find find x,y just like we did in the previous tutorial using a point-matrix product. So we just need to solve for Z. Once we do that we get the formula:

See the previous tutorial if you are confused about how to deal with the plus or minus ambiguity.

That's everything, sorry for not going into detail in this post, but the previous post should give you all the info you need. I've got several other posts on the way so I didn't want to take the time to re-record a tutorial that wasn't necessary. If you have any questions, feel free to email me.