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Conformal invariants of curves via those for inscribed polygons with circular edges

Harald Dorn

Abstract

The conformal nature of smooth curves in $\mathbb{R}^3$ is characterised by conformal length, curvature and torsion. We present a derivation of these conformal parameters via a limiting process using inscribed polygons with circular edges . The procedure is based on elementary geometry in $\mathbb{R}^3$ only and similar to the rectification of curves in the metrical case. It seems to be not available in the literature so far.

Conformal invariants of curves via those for inscribed polygons with circular edges

Abstract

The conformal nature of smooth curves in is characterised by conformal length, curvature and torsion. We present a derivation of these conformal parameters via a limiting process using inscribed polygons with circular edges . The procedure is based on elementary geometry in only and similar to the rectification of curves in the metrical case. It seems to be not available in the literature so far.
Paper Structure (4 sections, 38 equations, 2 figures)

This paper contains 4 sections, 38 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A short recap of the metrical case
  3. The conformal case
  4. Cross ratios and circumcircles

Figures (2)

  • Figure 1: In blue is shown a certain smooth curve with five randomly chosen points $x_1,\dots ,x_5$. Each set of three consecutive points $(x_{j-1},x_j,x_{j+1})$ fixes a circle $cc_j$. The arcs from $x_{j-1}$ to $x_j$ are depicted in red, those from $x_j$ to $x_{j+1}$ in black. Both the black and red polygon is a conformal covariant approximation (rectification) of the respective piece of the blue smooth curve.
  • Figure 2: In yellow the boundary of the rounded tetrahedron \ref{['sym-constraint']} and in yellow/blue that part, for which $(p,q,r)$ is allowed by \ref{['abc,uv']} and \ref{['uv-constraint']}.