Three-dimensional strong convexity and visibility

Eugene Fink and Derick Wood

In Proceedings of the Vision Geometry IV Conference, pages 61-72, 1995. Also available as Technical Report HKUST-CS95-33, Hong Kong University of Science and Technology.

Abstract

We define the notions of strong convexity and strong visibility. These notions generalize standard convexity and visibility, as well as several types of nontraditional convexity, such as iso-oriented rectangles and C-oriented polygons. We explore the properties of strong convexity and strong visibility in two and three dimensions. In particular, we establish analogs of the following properties of standard convex sets:

• Every two points of a convex set are visible to each other.
• The intersection of convex sets is a convex set.
• For every point in the boundary of a convex set, there exists a supporting plane through this point.
• A closed convex set in three dimensions is the intersection of all halfspaces that contain it.