Polar Curves in the Non-euclidean Geometry

Abstract

We consider the global differential 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 \({\frac{{\pi}}{{2}}}\,{\rm resp.}\,{\frac{{\pi}}{{2}}\,·\,i}\). The evolute of a convex curve may also be represented by its support function in a simple manner.