Three Dollar Quill

All right, I did a bunch of work in my head last night, let's sum it up.

I haven't fully put in the legwork to handle non-degenerate projection of conics, but the groundwork should be there. One thing that I need to be careful about is, I didn't take any measures to make sure that the vectors going into the source ellipse are perpendicular, or even linearly independent. Now, I don't know how much emphasis I want on exact perpendicularity (like, there at least needs to be some kind of epsilon), but I believe it is generally true that lower cosine similarity is better, as far as the math goes.

Anyway, working in three dimensions, we cannot guarantee whether the vector from the origin to the center is linearly independent of the "axis" vectors (which we are assuming are independent of each other). If it's linearly dependent, then the "standard" calculations will break down because all coefficients will be zero. Open question as far as I'm concerned: is it possible to have all coefficients zero without this linear dependence? Intuitively, I'd say no, because the linear dependence corresponds to the three vectors lying in a common plane, and if they don't, then the ellipse is going to have to have some kind of non-degenerate structure.

Aside from that, it's for sure possible for the linear dependence to come out exact, in which case, I need fallback calculations, but if it's not exact, then does attempting the standard calculations result in a projection that is "almost right", or does it produce a basically arbitrary or random result? I suppose to answer that, I need to calculate some kind of limit.

For the fallback calculations, I believe I need to use the linear dependence to decompose the vector from the origin to the center in terms of the axis vectors. This is because that represents the ellipse as a unit circle displaced from the origin, and I can calculate the tangents of that, then multiply by the axis vectors to, because linear transformations preserve tangency, find the tangents to the ellipse through the origin. These tangents project to the endpoints of the projection of the ellipse, which is a line segment.

Similar calculations apply to finding the convex hull of the projection of a cone, but the questions of stability differ; we should assume that the axis vectors are basically perpendicular, which should make things basically work out. (Also, if the base projects to a line, congratulations, it's a triangle.)

The one detail I don't think I've gotten down is my thoughts about how to decompose the vector from the center to the origin. It's been a long time, so I don't have an intuitive sense of what calculations to do, but decomposing one axis vector into the parallel and perpendicular components of the other should work; then I can decompose the center vector in the same way, and process those vectors appropriately. I suspect this somehow corresponds to some matrix multiplication or something, which I can figure out how to search for once I sort of understand it.

I've got all of these thoughts, but I need to process them more once I have some rest. (Also, I have worked out a pattern in the "standard" coefficients, but it's not yet at the level of a "formula".)

Good night.