Self-affine maps were successfully used for edge detection, image segmentation, and contour extraction. They belong to the general category of patch-based methods. Particularly, each self-affine map is defined by one pair of patches in the image domain. By minimizing the difference between these patches, the optimal translation vector of the self-affine map is obtained. Almost all image processing methods, developed by using self-affine maps, take advantage of either the attracting or repelling behaviors which have been, only, experimentally investigated. In this paper, we analytically study the properties of self-affine maps and prove their attracting and repelling behaviors. Furthermore, the new corner/edge pointing behavior is also proposed for contractive self-affine maps. We show that the conventional cost function of self-affine maps may cause critical uncertainty due to providing multiple equivalent optimal translation vectors. Thus, a new cost function is suggested to effectively tackle this problem. For evaluation, it is used with the self-affine snake (SAS) for contour extraction. Experimental results demonstrated that the enhanced SAS provides better performance compared to a number of different active contour methods in terms of both solution quality and CPU time.