The problem of solving triangles on a sphere can be generalized in several ways:
- Higher dimensions: Solving tetrahedra on 4D hyperspheres, then simplices on higher dimensional hyperspheres. As the number of dimensions increases, the number of cases (e.g., SSS, SAS, etc., on 2D manifolds) probably increases exponentially.
- Solving triangles on triaxial ellipsoids, i.e., adjusting the coefficient 1 on the terms in the equation that had defined a sphere in Cartesian coordinates. Different terms can have different coefficients.
- Solving triangles on superellipsoids, i.e., adjusting the exponent 2 that had defined an ellipsoid in Cartesian coordinates. Different terms can have different exponents. We will typically also need absolute value bars.
It would also be nice to compute geodesics -- the actual path, not just their lengths --- on these surfaces and hypersurfaces.
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