4 Preliminaries
TODO: This whole chapter needs organization, it’s just a grab bag of miscellaneous results for now.
4.1 Rupert Sets
The following are equivalent:
The convex polyhedron with vertex set \(v\) is Rupert.
The convex closure of \(v\) is a Rupert set.
Proof
TODO: import this from the other repo
4.2 Poses
TODO
Given a pose with zero offset, there exists a 5-parameter pose that is equivalent to it.
Proof
By putting the pose into a canonical form as a Z rotation followed by a Y followed by a Z.
4.3 Pointsymmetry and Rupertness
If a set is point symmetric and convex, then it being Rupert implies it being purely rotationally Rupert.
Proof
TODO: informalize proof