The representation of images is an active and very important area in image processing and pattern recognition. Therefore, in the literature, different contour codes for binary images have been proposed, such as F4,F8,VCC,3OT , and AF8 . These codes have been used in many papers since the first chain code, F8 , was introduced by Freeman in 1961. All the codes have been tried here as vector representations, including vertex chain code (VCC ). To know their properties, this paper provides an analysis of comparisons of each code, and as an important contribution, we investigated the relationship between them and found a series of transformations that allow simple and efficient sets to go from one code to another. We found the equivalences between F4,VCC,3OT,F8 , and AF8 . As an important consequence of the transition matrix concept, we proposed a new code for eight connectivity by observing a missing code in the state of the art and in the inspiration of the 3OT code.