In image analysis, Euclidean distance transform is often an useful tool for a variety of applications like image compression, skeletonization etc. A distance transform based on octagonal digital distances are often close approximations to the Euclidean distance transform. There has been a continual search for simple yet good octagonal distances. In this paper, we follow a geometric approach to show that octagonal distances based on the neighborhood sequence {1,1,2}, {1,2} and {1,1,1,2} in 2D and {1,1,3}, {1,2} and {1,2,2} in 3D are indeed good approximants. We use various features of digital circles and spheres for arriving at these conclusions.