Synthetic vs Analytic Geometry for Computers

Computer geometry usually means analytic geometry. Analytic geometry usually means Cartesian coordinates and Euclidean geometry. We consider alternatives: synthetic geometries, such as projective geometry, and coordinate-free analytic geometries. Using classical invariants and Cayley algebra (extended exterior algebra), we describe translations from coordinate analytic geometry to coordinate-free ‘invarant’ analytic geometry and the unsolved problem of translating back to synthetic geometry. The goal is to include more appropriate geometry in computer-aided geometry - and produce ‘better’ proofs from Automated Theorem Provers. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.