does anyone know if there is a generalized approach to generating opposite points within n-shapes such as an n-cube or n-sphere. I am trying to find out if a uniform distribution will be better at calculating CoG rather than an opposite points approach. Any further resources regarding opposite points would be appreciated.
Just for clarification by opposite point I mean exactly that for example on a unit sphere the opposite point to (1,1,1) would be (-1,-1,-1). I am not sure how to identify opposite points within dimensions higher than 3.
I would have thought points on a unit sphere would have $3$ coordinates, not $2$.
For a point on the $(n-1)$-boundary of a regular $n$-cube (measure polytope), $n$-orthoplex (cross polytope, analogue of the octahedron) or $n$-sphere, you just negate all the Cartesian coordinates to get the opposite point on the boundary, taking 'opposite point' to mean 'line between opposite point and original point passes through the origin'.