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Elastic Analysis of Augmented Curves and Constrained Surfaces

Esfandiar Nava-Yazdani

TL;DR

This work addresses intrinsic comparison of curves and augmented curve-driven surfaces within a Riemannian elastic framework using the Square Root Velocity (SRV) transform, with $q(c)=\dot{c}/\sqrt{|\dot{c}|}$. It derives explicit SRV-based relations for plane curves, including $\tilde{\omega}$ and $\tilde{\kappa}$, and identifies invariants such as total curvature under SRV. The framework is extended to augmented curves to generate tubes, ruled surfaces, and spherical strips, as well as to hurricane tracks, by mapping to homogeneous spaces and computing geodesics and distances $d^S$. The approach enables efficient computation of means, principal geodesics, splines, and other statistical summaries, with practical implementation in the Morphomatics library.

Abstract

The square root velocity transformation provides a convenient and numerically efficient approach to functional and shape data analysis of curves. We study fundamental geometric properties of curves under this transformation. Moreover, utilizing natural geometric constructions, we employ the approach for intrinsic comparison within several classes of surfaces and augmented curves, which arise in the real world applications such as tubes, ruled surfaces, spherical strips, protein molecules and hurricane tracks.

Elastic Analysis of Augmented Curves and Constrained Surfaces

TL;DR

This work addresses intrinsic comparison of curves and augmented curve-driven surfaces within a Riemannian elastic framework using the Square Root Velocity (SRV) transform, with . It derives explicit SRV-based relations for plane curves, including and , and identifies invariants such as total curvature under SRV. The framework is extended to augmented curves to generate tubes, ruled surfaces, and spherical strips, as well as to hurricane tracks, by mapping to homogeneous spaces and computing geodesics and distances . The approach enables efficient computation of means, principal geodesics, splines, and other statistical summaries, with practical implementation in the Morphomatics library.

Abstract

The square root velocity transformation provides a convenient and numerically efficient approach to functional and shape data analysis of curves. We study fundamental geometric properties of curves under this transformation. Moreover, utilizing natural geometric constructions, we employ the approach for intrinsic comparison within several classes of surfaces and augmented curves, which arise in the real world applications such as tubes, ruled surfaces, spherical strips, protein molecules and hurricane tracks.
Paper Structure (4 sections, 1 theorem, 18 equations)

This paper contains 4 sections, 1 theorem, 18 equations.

Table of Contents

  1. Introduction
  2. Riemannian Framework
  3. Preliminaries
  4. Plane Curves

Key Result

proposition thmcounterproposition

Denoting the speed of $\tilde{c}$ by $\tilde{\omega}$, we have Moreover, $\tilde{c}$ is an immersion if and only if $\kappa$ and $\dot{\omega}$ have no common zeros. In this case, where $\tilde{\kappa}$ denotes the curvature of $\tilde{c}$ and

Theorems & Definitions (2)

  • proposition thmcounterproposition
  • proof