Application of the Fourier Series to Convex Curves With Axes of Symmetry

Abstract

Abstract. In this paper we consider a class O of all ovals. A sufficient and necessary condition for existence of axes of symmetry in the class O is given. Moreover, a form of Fourier series expansion of a support function of a curve K £ O is also given. PreliminariesOur geometric results complete and deepen the properties obtained in This article is concerned with ovals with axes of symmetry. The class of all ovals we denote by O. Obviously each oval K: z = z(t), t € R of the class O belongs to the class C 2 and R means real numbers.We write down some sentences about a special parametrization by an oriented support function. This parametrization is a natural generalization of the ordinary one in [1], [3], [5] or [6], Let us consider an oval K: z = z(s) parametrized by arc lenght. Let a point O be the origin of our coordinate system and suppose that the curve K is considered in this system. Let us fix a point z0 = z(s0) and consider the tangent line at za. We can assume that zQ is chosen in such a way, that the tangent line is perpedicular to the x-axis. For an arbitrary point z(s) we define a vector e lt = cos t + i sin t, where t is an oriented angle between the positive direction of the x-axis and the vector e lt . Now we consider an oriented distans p(t) from the origin O of the coordinate system to the tangent line to K. 1. it is a periodic function ( the period T = 2-rr); 2. it is at least at the class C 1 ; 3. it is a positive one if only 0€ intV, where dV = K.Using p(t) we obtain the special parametrization of K, given by (1.1) z{t) = pity* +p'(t)ie a , for t 6 R.