Understanding the Dual Polytope for Hull Simplification
16 hours ago
- The dual polytope of a convex hull can be used for hull simplification by converting face planes into points.
- The support function h(y) of a polytope P is piecewise linear, with wedges corresponding to vertices.
- The level set {h <= c} equals c times the polar dual P°, and P° is convex and bounded.
- Faces of P° correspond to vertices of P, and vertices of P° correspond to faces of P.
- For a face of P with plane n·x=d, its dual point in P° is n/d.
- Projecting edges of P° onto the sphere yields the Gauss map of P.
- In hull simplification, merging or dropping face normals edits vertices of P°, and the convex hull of these points rebuilds the simplified polytope.
- Each face of the new dual polytope gives a vertex of the simplified polytope.