Recently we have proposed a range image segmentation method based on the equidistance contour map extracted from a range image. An equidistance contour of a range image is formed by pixels on the range image whose corresponding scene points are all at a same specified distance from the sensor. We have observed that in different ways the contours reflect the existence of object surface edges, the geometries of object surfaces, and the orientations of these surfaces in the 3-D space. In this paper we present a method which uses the equidistance contours to further characterize the regions obtained from segmentation. This is an integral part of an object recognition process. The meaning of characterization is to find the geometries of the corresponding 3-D object surfaces of the regions from the contours. If a surface has nice analytic form such as planar, spherical, cylindrical, or conical, we determine not only its type but also the values of the parameters which describe its geometry.