Computing an exact spherical visibility map for meshed polyhedra

Abstract

This paper considers computation of the exact visibility range (or the spherical visibility map) for a closed polyhedron whose boundary is represented as a triangle mesh. For each facet on the mesh, we calculate the set of view directions from which all the points on the facet can be seen from the exterior. The projection of those visible directions onto the unit sphere forms the visibility map for the facet. We show that the exact visibility map is a spherical arrangement of closed 0-cells, 1-cells, and 2-cells embedded on the surface of the unit sphere. Based on a provable method for calculating the potential occlusion regions of a facet, a vector visibility algorithm is developed for computing the exact solution of the spherical visibility map for a facet. Examples are given to illustrate our algorithm.

On a Novel Geometric Representation of Rotation

“…That is nearest plane parallel to x-plane is defined by the x-coordinate of V, and so on. 4. A point is contained in a parallelepiped if it is contained in each of the half-space defined by its faces.…”

“…Representing directions in space is important for diverse applications; it occurs naturally in geography [1], astronomy and in the problems of Gauss maps [2], visibility map [3] [4]. In these applications a geometric rather than an algebraic representation of direction is more relevant.…”

A Polyhedral-based Approach of Accessibility Computation for a NURBS Model in Tool-based Manufacturing

“…First, accessibility can be computed from a polyhedral model (i.e. a tessellated NURBS model), called facet accessibility in [30,31,32,33] and extended to be region accessibility in [34]. For this kind of computation, every facet of the model (or in the concave region) must be analyzed with a fixed resolution without mentioning a freeform NURBS model.…”

Computing axes of rotation for 4-axis CNC milling machine by calculating global visibility map from slice geometry

“…[29] proposed an algorithm that calculated global a visibility map for a triangular mesh by an occlusion calculation between a pair of triangle facets. [13] extends the work to the occlusion calculation between a pair of convex facets. A similar approach also calculates the occlusion between a pair of convex facets by a boundary tracing method [30].…”

“…There are also a few approaches that aim at obtaining an exact solution for a global visibility map [13,[27][28][29][30]. [29] proposed an algorithm that calculated global a visibility map for a triangular mesh by an occlusion calculation between a pair of triangle facets.…”