Conformal transformations and curvature

In this paper, we discuss the relationship between conformal transformations of [Formula: see text] and the curvature of curves. First, for any non-circular closed curve, there exists a length-preserving inversion such that the maximum pointwise absolute curvature can be made arbitrarily large. In contrast, we show that the total absolute curvatures of a family of curves conformally equivalent to a given simple or simple closed curve are uniformly bounded. Furthermore, we show that the total absolute curvature of an inverted regular [Formula: see text] simple closed curve as a function of inversion center and radius is removably discontinuous along the curve with exactly a [Formula: see text] drop, and continuous elsewhere.