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The strength of a geometric simplex is a key ingredient in a polynomial-time classification of unordered point clouds by Lipschitz continuous invariants

Olga Anosova, Vitaliy Kurlin

TL;DR

The strength of any geometric simplex is defined and its continuity under perturbations under perturbations is proved with explicit bounds for Lipschitz constants.

Abstract

The basic input for many real shapes is a finite cloud of unordered points. The strongest equivalence between shapes in practice is Euclidean motion. The recent polynomial-time classification of point clouds required a Lipschitz continuous function that vanishes on degenerate simplices, while the usual volume is not Lipschitz. We define the strength of any geometric simplex and prove its continuity under perturbations with explicit bounds for Lipschitz constants.

The strength of a geometric simplex is a key ingredient in a polynomial-time classification of unordered point clouds by Lipschitz continuous invariants

TL;DR

The strength of any geometric simplex is defined and its continuity under perturbations under perturbations is proved with explicit bounds for Lipschitz constants.

Abstract

The basic input for many real shapes is a finite cloud of unordered points. The strongest equivalence between shapes in practice is Euclidean motion. The recent polynomial-time classification of point clouds required a Lipschitz continuous function that vanishes on degenerate simplices, while the usual volume is not Lipschitz. We define the strength of any geometric simplex and prove its continuity under perturbations with explicit bounds for Lipschitz constants.
Paper Structure (3 sections, 4 theorems, 2 figures)

This paper contains 3 sections, 4 theorems, 2 figures.

Table of Contents

  1. The importance of Lipschitz continuity for distinguishing mirror images under noise
  2. The strength is a Lipschitz continuous invariant
  3. Proofs of strength properties in all dimensions

Key Result

Theorem 2.4

(a) The strength $\sigma(A)$ and signed strength $s(A)$ of any simplex $A\subset\mathbb R^n$ are invariant under isometry and rigid motion in $\mathbb R^n$, respectively, and can be computed in time $O(n^3)$. The uniform scaling of $\mathbb R^n$ by any factor $c>0$ multiplies both $\sigma(A)$ and $s

Figures (2)

  • Figure 1: Left: Example \ref{['exa:strength_triangle']}(b) parametrizes the space of (normalized) triangles with sides $0<a\leq b\leq c$ by $x=\dfrac{a}{c}$ and $y=1-\dfrac{b}{c}$ over the region $\Delta=\{(x,y)\in\mathbb R^2 \mid x\in[0,1],\, x\geq y,\, x+y\leq 1\}$. Right: the strength $\sigma(x,y)=\dfrac{(2-x-y)(x^2-y^2)}{2(2+x-y)^2}$ of a normalized triangle over the region $\Delta$.
  • Figure 2: Vertical sections of the strength $\sigma$ of normalized triangles parametrized by $x,y$ in Fig. \ref{['fig:strength_plots']}. Left: isosceles triangles with $\tilde{a}=\tilde{b}\leq \tilde{c}=1$ have $y=0$ and $\sigma=\dfrac{(2-x)x^2}{2(2+x)^2}$ for $x\in[0,1]$. Right: isosceles triangles with sides $\tilde{a}\leq \tilde{b}=\tilde{c}$ have $y=1-x$ and $\sigma=\dfrac{2x-1}{2(2x+1)^2}$ for $x\in[\frac{1}{2},1]$.

Theorems & Definitions (18)

  • Definition 1.1: Lipschitz continuity
  • Definition 1.2: a geometric simplex $A$ on $n+1$ points in $\mathbb R^n$
  • Example 1.3: the area of a triangle is not Lipschitz continuous
  • Definition 2.1: the strength $\sigma(A)$ and signed strength $s(A)$ of a simplex $A\subset\mathbb R^n$
  • Example 2.2: the strength of a line segment
  • Example 2.3: the strength of a triangle
  • Theorem 2.4: strength properties: invariance, computational time, and Lipschitz continuity
  • proof : Proof of Theorem \ref{['thm:strength_properties']}(b) for dimension $n=1$ and $\lambda_1=2$
  • proof : Proof of Theorem \ref{['thm:strength_properties']}(b) for dimension $n=2$ and $\lambda_2=\sqrt{3}$
  • Lemma 3.1: edge ratios
  • ...and 8 more