Of Spherical Evolute And Spherical Involute

Bruce, J.W. and Giblin, P. J. apply the unfolding theory to the study of the plane and space curves by means of the distance-squared function and the height function. Pei, Donghe studied time-like curves and space-like curves in Minkowski space. Porteous, I. R. introduced the notion of the spherical evolute. In this paper we'll define the spherical evolute using the geodesic distance on the unit sphere and investigate some properties of the spherical evolute and the spherical involute, and their relationship. Finally we'll use the method of Bruce, J.W. and Giblin, P. J. to determine the local diffeomorphic image of the spherical evolute.This paper consists of four parts studying the spherical evolute and the spherical involute. First we give the characteristic property of a spherical curve, then we discuss the contact between the geodesic circle and the spherical evolute, and the properties of the spherical evolute. In the last part we mainly deal with the relationship between the spherical evolute and the spherical involute and give the local diffeomorphic image of the spherical evolute.