2024/07/24 by Kumagai, Shun, Kajiwara, Kenji
#53A15 #65D18 #93B51 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2407.17008
In this paper, we consider the self-affinity of planar curves. It is regarded as an important property to characterize the log-aesthetic curves which have been studied as reference curves or guidelines for designing aesthetic shapes in CAD systems. We reformulate the two different self-affinities proposed in the development of log-aesthetic curves. We give rigorous proof that one self-affinity actually characterizes log-aesthetic curves, while another one characterizes parabolas. We then propose a new self-affinity which, in equiaffine geometry, characterizes the constant curvature curves (the quadratic curves). It integrates the two self-affinities, by which constant curvature curves in similarity and equiaffine geometries are characterized in a unified manner.