Line accessibility of free form surfaces

Abstract

Given a three dimensional object B⊂R3, the visual hull of B (a concept introduced by Laurentini in 1992) is the smallest set VB⊂R3 containing B with the following property: for each point p on the boundary of VB there exists a direction from which p is on a silhouette of VB. The notion of the visual hull has applications in computer graphics (visibility and silhouette based algorithms), manufacturing (accessibility, wire EDM and hot wire cutting), and more.We present a tractable algorithm for computing a decomposition of a C1 smooth, closed parametric surface S⊂R3, positioned in an arrangement with other occluding surfaces, into those regions that belong to the visual hull and those that do not (later referred to as the “line-accessible” or “line-inaccessible” regions). For efficiency, our method introduces two early domain pruning criteria, using a recursive subdivision of the parameter domain, along with a tangent plane bounding cone: one criterion detects regions that are guaranteed to belong to the visual hull, while the other detects regions that cannot belong to it (both use sufficient and not necessary conditions). Only to those sub-domains that remain after this significant domain reduction, an algebraic (subdivision based) solver is applied, to solve two sets of algebraic equations, already presented by Laurentini in 1999, gaining one to two orders of magnitude improvement in the computation times. We provide a derivation of these equations via a different approach, compared to Laurentini in 1999, using the model we refer to as the dynamic SSI solution of the surface and the moving tangent plane. Concrete computational examples are provided as well.