A new class of aesthetic curves based on the self-affinity in equiaffine geometry

TL;DR

This paper introduces a new class of aesthetic planar curves in equiaffine geometry, characterized by extendable self-affinity, expanding the understanding of aesthetic curves like the log-aesthetic curve and logarithmic spiral.

Contribution

It presents a novel class of curves based on self-affinity symmetry, broadening the scope of aesthetic curves in equiaffine geometry and linking them to known curves.

Findings

Includes quadratic curve and logarithmic spiral as special cases

Demonstrates the new class's relation to existing aesthetic curves

Provides a geometric framework for aesthetic curve design

Abstract

In this paper, we consider planar curves in equiaffine geometry and present a family of planar curves characterized by a symmetry called the extendable self-affinity (ESA). The ESA has been recognized through the investigation of the symmetry of the log-aesthetic curve (LAC), which has been studied as a reference for designing aesthetic shapes in CAGD and regarded as an analog of Euler's elastica in similarity geometry. Our new class, characterized by the ESA, includes the quadratic curve and the logarithmic spiral, a special case of the LAC. This implies that the new class can be regarded as an alternate class of ``aesthetic curves" in equiaffine geometry.