Kernekoncepter
The author presents an efficient algorithm for eight-partitioning points in 3D space, demonstrating the existence of specific configurations that satisfy key properties.
Resumé
The content discusses the concept of eight-partitioning points in 3D space. It introduces an algorithm to efficiently calculate an eight-partition of a set of points with prescribed normal directions for planes. The article explores the theoretical foundations and practical implications of this algorithm, providing insights into geometric methods and computational geometry.
An eight-partition divides a set of points or mass distribution into octants using three planes. Hadwiger's theorem is discussed, showing the existence of such partitions. Various results and proofs related to equipartitioning problems are presented, including new variants like Theorem 1.2 and Theorem 1.3.
The article delves into topological combinatorics, discussing arrangements of planes and their intersections. It highlights the complexity analysis of arranging planes and computing intersection curves efficiently.
Overall, the content provides a comprehensive overview of partitioning algorithms in 3D space, offering valuable insights into computational geometry concepts.
Statistik
Any mass distribution (or point set) in R3 admits an eight-partition for which the intersection of two planes is a line with a prescribed direction.
An efficient algorithm for calculating an eight-partition of a set of n points in R3 (with prescribed normal direction of one plane) runs in time O∗(n5/2).
Let µ be a mass distribution on R3, then there exists a triple of planes that form an eight-partition for µ with specific properties.
The maximum number h3(n) halving planes for an n-point set R3 is related to the complexity g3(n) levels in arrangements.