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

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.