A uniform color space has, according to various literature, two definitions: (1) a global uniform color space is a space in which perceptional color difference agrees with the Euclidean distance; (2) a local uniform color space is a space in which discrimination elliptics/ellipsoids are unit circles/spheres everywhere. Unfortunately, it seemed that the relationship between them was not well understood and uniform color spaces have been constructed following these two different definitions independently. In this paper, we discuss the issue from a point of view of global Riemannian geometry and show that these two uniform color spaces are actually equivalent.Giving perceptive metric in a color space, an efficient algorithm is shown to construct a “pollar coordinate system” for the color space, which is the image of the pollar coordinate system in its uniform space.