Abstract
It is known that a color space is a Riemann space rather than a Euclidena space. For such a color space, one can still construct a generalized polar coordinates in the color space using Riemann geometry. This so-called Riemann normal coordinate system is consisted of geodesies and equa-geodesic-distance lines and provide us a generalization of the Munsell coordinates. In this paper we show that the curvature of a color space is an essentially invariant of color spaces which play an important role in construction of the Riemann normal coordinates. In particular, only for color spaces with negative curvature everywhere it is possible to build a global Riemann normal coordinates with geodesies. We calculate the curvatures in CIELUV and CIELAB and show possitive curvature is the reason for intersection of geodesies. A multipatch algorithm is proposed also to build a Riemann normal coordinates in a color space with possitive curvature.
