In this thesis based on a paper of leichtwess in polar curve in the Non-euclidean Geometry we consider the global di?erential geometry of polar closed convex curves in the spherical resp. hyperbolic plane (in the last case after restriction to horocyclic convex curves) by means of a curve representation by a suitable support function. In a certain sense the results are simpler than in the euclidean case. There exists a connection between the polar curve and the parallel curve at distance ?/2 resp. ?/2.i. The evolute of a convex curve may also be represented by its support function in a simple manner. We will do so globally for polar closed convex curves in the two self-dual and nondegenerated Cayley–Klein planes, namely the spherical and the hyperbolic plane (in this case we must restrict to horocyclic convex curves), by means of a curve representation by a suitable support function.