Polar curves in the non-Euclidean geometry
The author studies the geometry of polar closed convex curves in the spherical and in the hyperbolic plane (in the last case, the curves are also supposed horocyclic) by means of a representation of these curves by suitable support functions. He proves the following: (1) The differential quotients of the arclengths of a curve \(C\) and the polar curve \(C^*\) coincide with the curvature of \(C\). (2) The (suitably defined) curvature of \(C^*\) is inverse to the curvature of \(C\). The author notes that these properties are simpler than the corresponding properties in Euclidean geometry. He also notes that the evolute of a convex curve can be represented by support functions in a simple manner.
support function of spherical convex curve
support function of horocyclic convex curve
scientific article; zbMATH DE number 5320909