The distance transform has found many applications in image analysis. Chamfer distance transforms are a class of discrete algorithms that offer a good approximation to the desired Euclidean distance transform at a lower computational cost. They can also give integer-valued distances that are more suitable for several digital image processing tasks. The local distances used to compute a chamfer distance transform are selected to minimize an approximation error. A new geometric approach is developed to find optimal local distances. This new approach is easier to visualize than the approaches found in previous work, and can be easily extended to chamfer metrics that use large neighborhoods. A new concept of critical local distances is presented which reduces the computational complexity of the chamfer distance transform without increasing the maximum approximation error.