rotational alignment

[YoungRainy]

That's really a great job, Thank you.
I was reading your code now, and try to figure out how you implement the process of "Tessellating the conformal parametrization to a regular equirectangular grid".
Your code is great, but it's a little difficult for me because I am using python.
Can you explain more about how you implement this progress?

I also read the paper "Mobius Registration", In chapter 6.2, it is also described very roughly,

Thank you so much.

[mkazhdan]

The idea is to use the equitrectangular (theta/phi) parameterization of the sphere.
Specifically, store a vector of polygons with each cells of the spherical grid and then for each triangle in the mesh, compute its intersection with the grid cells and add the intersection to the vector associated with the cell. This can be done efficiently as follows.

For each triangle:

1. Compute it's theta extent.
2. Successively clip the triangle with the planes corresponding to the discrete values of the theta in its extent.
3. For each clipped polygon.
   3a. Compute its phi extent.
   3b. Successively clip the polygon with the planes corresponding to the discrete values of the phi in its its extent.
   4b. Add the theta/phi-clipped polygon into the spherical cell.

There are two caveats to doing this:

1. Since the theta is in [0,2\pi), the notion of "theta extent" isn't well defined. We get around this by fist clipping each triangle with the yz plane.
2. If a triangle overlaps one of the poles, you can run into numerical issues when clipping against the theta planes. This needs to be handled carefully.

[YoungRainy]

Thank you so much.
Your explanation helped me a lot.

I am sorry I still feel a little confused about the cell of the spherical grid.
In Figure 6 in the paper, the cell of the grid is a rectangle, except cells around the poles.
I run your program, it seems like all cells of the spherical grid are triangles.
Does it affect the final result?
(The polygon is more like a rectangle or a triangle?)

Thanks again.

[mkazhdan]

As you said, the spherical cells are either quads (away from the poles) or triangles (at the poles). Their intersection with the the triangles of the mesh will be convex spherical polygons, but not necessarily either triangles or quads. (These are the intersections of the light gray triangles with the thick black quads/triangles.)

[YoungRainy]

Thank you, I got it.