Improved Binary Space Partition merging

Abstract

This paper presents a new method for evaluating boolean set operations between Binary Space Partition (BSP) trees. Our algorithm has many desirable features including both numerical robustness and O(n) output sensitive time complexity, while simultaneously admitting a straightforward implementation. To achieve these properties, we present two key algorithmic improvements. The first is a method for eliminating null regions within a BSP tree using linear programming. This replaces previous techniques based on polygon cutting and tree splitting. The second is an improved method for compressing BSP trees based on a similar approach within binary decision diagrams. The performance of the new method is analyzed both theoretically and experimentally. Given the importance of boolean set operations, our algorithms can be directly applied to many problems in graphics, CAD and computational geometry.

Fast Exact Booleans for Iterated CSG using Octree-Embedded BSPs

“…Additionally, the conversion from and to a mesh requires the same polygon cutting algorithm. Unfortunately, polygon cutting tends to be highly unstable when using fixed length floating point arithmetics [10]. Lysenko et al [10] present an improved BSP CSG algorithm by eliminating the polygon cutting from the merge procedure and replacing it by a satisfiability check for a linear program.…”

“…Unfortunately, polygon cutting tends to be highly unstable when using fixed length floating point arithmetics [10]. Lysenko et al [10] present an improved BSP CSG algorithm by eliminating the polygon cutting from the merge procedure and replacing it by a satisfiability check for a linear program. The constraints of the program are the same as the half-spaces from the current subtree to the root of the BSP.…”

“…Vertices are defined as the intersection of three non-coplanar planes, convex polygons consist of a supporting plane and a list of planes determining the boundary of the polygon. Unfortunately the BSP merging procedure scales poorly with the size of the input BSP [10]. To mitigate the scaling problems Campen and Kobbelt [2] embed Bernstein's BSP merging algorithm into an octree that localizes CSG operations.…”

“…However, they require conversion from mesh to BSP (import) and BSP to mesh (extraction), both with their own robustness issues. While fast for small trees, BSP-based Booleans, in the worst-case, scale at least with O(n 2 ) for n nodes, even if the output complexity is only O(n) [10]. Extracting a polygonal mesh from a BSP is conceptually simple: start with a cube large enough to contain the result and recursively perform mesh-plane cuts until the leaves of the BSP are reached.…”

Fast, Exact, Linear Booleans

“…BSP trees afford an alternative to B‐rep algorithms that avoid their concomitant case explosion by explicitly handling all degenerate configurations of geometry [TN87, NAT90]. These have been demonstrated to have suitable performance for interactive volumetric sculpting [Nay90], and recent work [LDS09] suggests further possible performance improvements. Unfortunately, all of these approaches are fragile and therefore subject to failures and user workarounds.…”