Performing geometric transformations by program transformation

Problems in geometry often possess symmetry properties that may be exploited in order to develop solutions. Algorithms based on these symmetry properties will frequently use geometric transformations to transform one case into another (symmetric) case. One advantage of this approach is that the algorithm avoids enumeration of cases and thus is shorter and generally easier to read. One disadvantage is that some additional execution time is required to perform these transformations. We describe how simple program equivalences may be used as program transformations to eliminate this additional execution time from programs that use geometric transformations. This approach has been used to develop an efficient implementation of a new algorithm for the two-dimensional line-clipping problem.