I'm trying to solve a geometric wrapping problem which is as such;
If I have an ellipsoid of which I know all dimensions, given two points and a quadrant which is considered to be the "wrapping quadrant" (indicating for example +x requiring wrapping around the +x side of the ellipsoid), should the line wrap around the ellipsoid or can it stay as is.
The information I have is:
- point 1: $p1 = [x_1,y_1,z_1]'$
- point 2: $p2 = [x_2,y_2,z_2]'$
- Center of ellipsoid: $m \equiv [0,0,0]'$
- Radii of ellipsoid: $R = [r_x,r_y,r_z]$
- Wrapping quadrant: $q = +x$ or for example $q=-z$
That's all quite vague I suppose so here are some examples: here, on the top image, the green dotted line indicates what a wrapping should look like. I know there is some sort of fundamendal difference between duo's of points that would either wrap around quadrant 1 or quadrant 2 but I've yet to figure it out....
So the issue is in 3D but If someone can give me some guidance on the 2D example I should be able to figure it out for the 3D case (hopefully). Some of the things I considered are
- Slopes, in 2D this might work because you have two tangents on an ellipse equal to the line ellipse so you could work from that but that does not work in 3D
- Middle point of the line, or projection on an axis of a line but this all seems kinda hacky.
EDIT: I will elaborate a bit; two points, $p1$ and $p2$ should theoretically be connected by a simple line. However, if given a quadrant which can be $+x$, $+y$, $+z$, $-x$, $-y$, and $-z$, should the line be wrapped around the ellipsoid if the predominant part of the wrapping is able to wrap over that specific quadrant. For example, in the picture (the top one), the lowest red-colored line would predominantly wrap over the $-x$ quadrant so if the $+x$ quadrant would be given as 'question' quadrant, it should say that the points should just be connected by a line.
Two points which are inside the diameter of the ellipse in the top picture should wrap partly over $-x$ as well as $+x$ but as the second green line from left to right (the first short one) would indicate, the predominant wrapping would occur over the $+x$ quadrant and only slight parts would wrap over the $-x$ quadrant.