A theorem for the determination of vertical and horizontal tangents to the hyperpolar image of a circle

The aim of this article is to complete the basic theory of the hyperpolar images of Cartesian circles as developed in a previous article (Glaser, 1992). These images consist of a certain number of loops that depends on the radius of the circle that has been mapped into the hyperpolar plane. In order to have a complete theory of these curves and to be able to draw their graphs with precision, this theorem is of great importance. Two examples showing its application will also be considered.