AF: Medium: Collaborative Research: Approximate Computational Geometry via Controlled Linear Perturbation

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The investigators will develop an approximate computational geometry that is algorithm independent, accurate, and fast. Geometric predicate evaluation and element construction will be performed approximately using floating point arithmetic. Degeneracy will be handled transparently. The evaluation and construction techniques will be encapsulated in a software library that will be free for nonprofit use.The research challenge is robustness: the output of an approximate algorithm must be correct for a small perturbation of the given input. This definition extends the numerical analysis definition of a stable algorithm to cover combinatorial error. Robustness is a fundamental computer science problem that is a major challenge in computational geometry. The predominant strategy in computational geometry, exact computation using algebraic geometry, has high computational complexity and contradicts the standard scientific and engineering strategy of approximate computation with error bounds. The investigators will adapt approximate computation to the special needs of computational geometry, which is primarily combinatorial. This task involves core research at the interface between computational geometry and numerical computing.Robust approximate computation will transform how computational geometry is taught, how algorithms are developed and implemented, and how the field interacts with the wider scientific and engineering community. Introductory courses will present a rigorous, practical robustness theory, instead of treating robustness in an ad hoc, incomplete way. Programmers will implement real RAM algorithms as stated, using our library to ensure robustness and to handle degeneracy, instead of addressing these problems anew for every algorithm, which is often a major research challenge. Computational geometry will be available to other disciplines in the form of high-quality software libraries, akin to modern applied mathematics libraries.