Conformal mapping and ellipses

Abstract

Abstract. We answer a question raised by M. Chuaqui, P. Duren, and B. Osgood by showing that a conformal mapping of a simply connected domain cannot take two circles onto two proper ellipses.The goal of this note is to prove the following. Proposition 1.Let Ω be a simply connected domain on C = C ∪ {∞} containing circles (on C) C 1 and C 2 , C 1 = C 2 . Let f be a conformal mapping from Ω into C. If f maps each of the circles C 1 and C 2 onto an ellipse, then f is a Möbius transformation and therefore the ellipses are actually circles.Thus this proposition answers affirmatively a question raised by M. Chuaqui, P. Duren, and B. Osgood in [1] that a conformal mapping of a simply connected domain cannot take two circles onto two ellipses.Precomposing f with a Möbius transformation and postcomposing it with a linear transformation, we may assume that C 1 = {z : |z| = 1} and that f maps the unit disc D = {z : |z| < 1} onto the interior of an ellipse L 1 with foci ±1 such that f (r) = 1, f (−r) = −1 for some 0 < r < 1. Then it is a standard exercise in complex analysis to verify that w = f (z) is defined bywhere K(·) denotes the complete elliptic integral of the first kind. Since the image f (D) is symmetric with respect to the coordinate axes, it follows that f is an odd function having real coefficients, i.e.,We want to stress that our proof below does not require the explicit form (1) of the mapping function, which is included here for completeness. Instead, we will use the fact that every ellipse L having foci w 1 and w 2 is a trajectory of the quadratic differential (2) Q 1 (w) dw 2 = − dw 2 (w − w 1 )(w − w 2 ) ;

Ellipses and similarity transformations with norm functions

“…Images of circles and ellipses (corresponding to the Euclidean norm or to another norm function on the complex plane C) have been studied extensively under some special transformations such as Möbius transformations or harmonic Möbius transformations (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and the references therein). Let us consider the real linear space structure of the complex plane C. In [14], the present author proved that the image of any ellipse E r (F 1 , F 2 ) = {z ∈ C : ∥z − F 1 ∥ + ∥z − F 2 ∥ = r} corresponding to any norm function ∥.∥ (except in the Euclidean case) on C under the similarity transformation w = f (z) = αz + β ; α ̸ = 0 , α, β ∈ C (which is a special Möbius transformation) is an ellipse corresponding to the same norm function or corresponding to the norm function ∥z∥ ϕ = e −iϕ z , (1.1) where ϕ = arg(α).…”