The problems of accuracy and robustness in geometric computation

Practical implementation of geometric operations remains error-prone, and the goal of implementing correct and robust systems for carrying out geometric computation remains elusive. The problem is variously characterized as a matter of achieving sufficient numerical precision, as a fundamental difficulty in dealing with interacting numeric and symbolic data, or as a problem of avoiding degenerate positions. The author examines these problems, surveys some of the approaches proposed, and assesses their potential for devising complete and efficient solutions. He restricts the analysis to objects with linear elements, since substantial problems already arise in this case. Three perturbation-free methods are considered: floating-point computation, limited-precision rational arithmetic, and purely symbolic representations. Some perturbation approaches are also examined, namely, representation and model, altering the symbolic data, and avoiding degeneracies.<>