Efficient Delaunay triangulation using rational arithmetic

Many fundamental tests performed by geometric algorithms can be formulated in terms of finding the sign of a determinant. When these tests are implemented using fixed precision arithmetic such as floating point, they can produce incorrect answers; when they are implemented using arbitrary-precision arithmetic, they are expensive to compute. We present adaptive-precision algorithms for finding the signs of determinants of matrices with integer and rational elements. These algorithms were developed and tested by integrating them into the Guibas-Stolfi Delaunay triangulation algorithm. Through a combination of algorithm design and careful engineering of the implementation, the resulting program can triangulate a set of random rational points in the unit circle only four to five times slower than can a floating-point implementation of the algorithm. The algorithms, engineering process, and software tools developed are described.

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