biVector forum

GA for Spherical Geometry

Hello everyone,

I am curious about using GA for doing spherical geometry. From this post I know that P(\mathbb{R}_{ 3,0,0}^*) should describe the problem nicely. Are there some explicit examples of using this GA somewhere?

I have a very particular problem in mind. It is Stratified Sampling of Spherical Triangle (paper) where two random numbers \xi_1 and \xi_2 are mapped onto triangle ABC. The idea is to first find a sub triangle AB\hat C with area \hat A=A\xi_1 and then \xi_2 defines a point P on the arc B\hat C.

The problem is numerical stability of the calculations described in the paper. If the triangle starts to cover almost full hemisphere an angle approaches \pi and we might have devision by zero at some point. So I wonder whether GA approach can help with finding an alternative formulation of the solution which is numerically stable.

Update: I have realised that the topic probably belongs to PGA section.

The easiest way of dealing with the triangle-approaches-a-whole-hemisphere problem without reworking the rest of the method would be to choose the longest triangle edge and bisect it, breaking the original triangle in two. Then the original method can be used over each of the sub-triangles. You can treat the two sides as corresponding to two rectangular pieces of the square, with widths based on their relative areas.

not an answer to your question, but i have been using GA for spherical geometry and i find it useful, since you have rotors. There is a section in the appendices of ‘new foundations for classical mechanics’ by david hestenes on spherical geometry.
im not sure it would be helpful for that specific problem you are asking about.

Here is an older book that used quaternions to solve spherical geometry problems. It does this by mapping spherical triangles to vectors from the sphere’s center to the surface and back in a neat fashion. Branets, V.N., Shmyglevskiy, I.P.: Primeneniye Kvaternionov v Zadachakh Orientatsii Tverdogn Tela (Application of Quaternions to Rigid Body Rotation Problems), Nauka Press, 1973. This may be useful, especially given quaternions as sub-field of PGA.

For anyone curious about the book mentioned, NASA made a translation: Application of quarternions to rigid body rotation problems : Branets, V. N. : Free Download, Borrow, and Streaming : Internet Archive