Basic Contacts and Applicability Conditions

There are two types of basic contacts, vertex - face and edge - edge, in terms of which any other contact between the features of polyhedra can be expressed. For example, the contact between an edge and a face can be expressed as two vertex (endpoints of the edge) - face contacts. For each basic contact there is a mathematical relationship that expresses whether the contact is actually possible, for a given relative orientation between two convex polyhedra, if the necessary translations are performed, or not.

Vertex - Face

The contact between vertex v and face f is applicable iff
< v_i - v, f > >= 0, for all v_i adjacent to v.

Edge - edge

The contact between edge e₀ and edge e₁ is applicable iff
k_a < > k_b,
where k_a = sign(< T₁ , f_p >) = sign(< T₂ , f_p >),
and k_b = sign(< T₃ , f_p >) = sign(< T₄ , f_p >),
with T_i = s_i·(f_i X e₀), f_i adjacent to e₀,
and T_j = s_j·(f_j X e₁), f_j adjacent to e₁;
s_i, s_j in {+1, -1} such that T_l is oriented towards the interior of face f_l;
and f_p = e₀ X e₁ (or the opposite direction, the choice is arbitrary).

Pablo Jimenez Schlegl

Last modified: Mon Apr 8 12:19:00 MET DST 2002