A new class of aesthetic curves based on the self-affinity in equiaffine geometry
Shun Kumagai, Kenji Kajiwara
TL;DR
The paper develops a new class of planar aesthetic curves in equiaffine geometry characterized by extendable self-affinity (ESA). It shows that curves with ESA in this setting have equiaffine curvature $\kappa^{SA}(u)=\pm(\xi u+\eta)^{-2}$, which, up to affine changes of coordinates, correspond to graphs of power functions, $\log x$, $x\log x$, a logarithmic spiral, and quadratic curves, thereby unifying several familiar shapes under a single symmetry framework. The work provides both the theoretical derivation of the curvature condition and explicit parametrizations of curves realizing the ESA, including detailed treatment of the ESA-parameter dynamics and its consequences. It positions the new ESA-based class as an alternative, geometry-centered extension of aesthetic curves, complementary to the LAC in similarity geometry, and suggests avenues for unification within broader geometric frameworks such as Möbius geometry with potential CAD applications. Overall, the paper advances the understanding of symmetry-driven design curves and expands the repertoire of aesthetically useful shapes in CAD systems.
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.