Exact Geometric Computation: Theory and Applications

This dissertation explores the theory and applications of Exact Geometric Computation (EGC), a general approach to robust geometric computing. The contributions of this thesis are organized into three parts. A fundamental task in EGC is to support exact comparison of algebraic expressions. This leads to the problem of constructive root bounds for algebraic expressions. Such root bounds determine the worst-case complexity of exact comparisons. In the first part, we present a new constructive root bound which, compared to previous bounds, can give dramatically better performance in many common computations involving divisions and radical roots. We also improve the well-known degree-measure bound by exploiting the sharing of common sub-expressions. In the second part, we discuss the design and implementation of the , a C++ library which embraces the EGC approach to robust numerical and geometric computation. Our design emphasizes ease of use and facilitates the rapid development of robust geometric applications. It allows non-specialist programmers to add robustness into new or existing applications with little extra effort. A number of efficiency and implementation issues are investigated. Although focused on geometric computation, the EGC techniques and software we developed can be applied to other areas where it is critical to guarantee numerical precision. In the third part, we introduce a new randomized test for the vanishing of multivariate radical expressions. With this test, we develop a probabilistic approach to proving elementary geometry theorems about ruler-and-compass constructions. A probabilistic theorem prover based on this approach has been implemented using the . We present some empirical data.