Attractive Plane Curves in Differential Geometry

Abstract

The purpose of chapter is to discuss plane curves from differential geometric point of view and applications of plane curves to computer aided designs. Plane curves are determined uniquely by curvatures up to Euclidean motions. Thus geometry of plane curves are formulated by the Euclidean motion group \(\mathrm {SE}(2)\). From industrial point of view, other transformation groups are more appropriate for characterizing certain classes of plane curves. For instance, under equiaffine transformation group, conics are characterized as plane curves with constant equiaffine curvatures. Plane curves with monotonous curvature function have been paid much attention in industrial shape design and computer aided geometric design. In this chapter we study plane curves with monotonous curvature function, especially log-aesthetic curves, in terms of similarity transformation group.