Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Form Convex Hull to Concavity: Surface Contraction Around a Point Set

Netzer Moriya

TL;DR

This paper investigates the transformation of a convex hull, derived from a d-dimensional point cloud, into a concave surface and employs an iterative process of facet replacement and expansion to evolve the surface into Scc, which accurately conforms to the complex geometry of the point cloud.

Abstract

This paper investigates the transformation of a convex hull, derived from a d-dimensional point cloud, into a concave surface. Our primary focus is on the development of a methodology that ensures all points in the point cloud are encapsulated within a closed, non-intersecting concave surface. The study begins with the initial convex hull and employs an iterative process of facet replacement and expansion to evolve the surface into Scc, which accurately conforms to the complex geometry of the point cloud.

Form Convex Hull to Concavity: Surface Contraction Around a Point Set

TL;DR

This paper investigates the transformation of a convex hull, derived from a d-dimensional point cloud, into a concave surface and employs an iterative process of facet replacement and expansion to evolve the surface into Scc, which accurately conforms to the complex geometry of the point cloud.

Abstract

This paper investigates the transformation of a convex hull, derived from a d-dimensional point cloud, into a concave surface. Our primary focus is on the development of a methodology that ensures all points in the point cloud are encapsulated within a closed, non-intersecting concave surface. The study begins with the initial convex hull and employs an iterative process of facet replacement and expansion to evolve the surface into Scc, which accurately conforms to the complex geometry of the point cloud.
Paper Structure (7 sections, 3 theorems, 3 equations, 3 figures)

This paper contains 7 sections, 3 theorems, 3 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Mathematical Approach
  4. Proofs for Lemmas
  5. Methodology
  6. Complexity Analysis
  7. Results

Key Result

Theorem 3.1

Let $P = \{p_1, p_2, \ldots, p_n\}$ be a finite set of distinct points in d-dimensional space $\mathbb{R}^d$, and let $S$ be a surface that initially coincides with the convex hull of $P$, denoted as $\text{conv}(P)$. Suppose $S$ undergoes a continuous contraction process defined by the following pr Then, the final structure of $S$ after the completion of the contraction process is such that every

Figures (3)

  • Figure 1: The 3D point distributions of a 50-point (left) and a 500-point clouds sphered shape (right). Red points refer to on-surface points.
  • Figure 2: Replacement of a 3-point facet $F_{ABC}$ and an external point $P$ into the tetrahedron represented by $\{F_{ABP}$, $F_{BCP}$ and $F_{CAP}\}$
  • Figure 3: The final surfaces encapsulating all 50 points (left) and 500 points (right). Red points refer to on-surface points.

Theorems & Definitions (6)

  • Theorem 3.1: Point Cloud Contraction Theorem
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof