Rectilinear convex hull of points in 3D and applications

Abstract

AbstractLet P be a set of n points in $$\mathbb {R}^3$$ R 3 in general position, and let RCH(P) be the rectilinear convex hull of P. In this paper we obtain an optimal $$O(n\log n)$$ O ( n log n ) time and O(n) space algorithm to compute RCH(P). We also obtain an efficient $$O(n\log ^2 n)$$ O ( n log 2 n ) time and $$O(n\log n)$$ O ( n log n ) space algorithm to compute and maintain the set of vertices of the rectilinear convex hull of P as we rotate $${\mathbb {R}}^3$$ R 3 around the Z-axis. We study some combinatorial properties of the rectilinear convex hulls of point sets in $$\mathbb {R}^3$$ R 3 . Finally, as an application of the obtained results, we show an approximation algorithm to an optimization fitting problem in $$\mathbb {R}^3$$ R 3 .