Understanding the Dual Polytope for Hull Simplification

Support function h(y) = max_i v_i · y of a convex polytope with the origin inside is piecewise-linear on the sphere, with one linear wedge per vertex. Level sets {h ≤ c} are flat inside each wedge and bend only along seams between wedges; because h(ty)=t h(y) they are scaled copies, and {h ≤ 1} equals the polar dual P° = {y : x · y ≤ 1 for all x ∈ P}. Concretely P° is the finite intersection of halfspaces v_i · y ≤ 1, so each wedge i induces a face lying in the plane v_i · y = 1 and different vertices of the original produce different faces of P°. Rays from the origin hit the boundary at y/h(y), so projecting the boundary of P° onto the sphere recovers the Gauss map; vertices of P° occur along face normals n of the original, at n/d when that face has equation n · x = d.

That geometric identification gives a clean hull-simplification recipe: editing, merging or dropping face normals changes the vertex set of P°, and rebuilding by taking the convex hull of the edited dual points reconstructs a dual polytope whose faces v · y = 1 correspond to vertices v of the simplified primal hull. In practice this is why converting face planes n · x = d to dual points n/d and taking their convex hull yields the simplified hull, an alternative to clipping large plane quads.