We propose an efficient and exact method for the adaptive sign detection of 4x4 determinants using a standard arithmetic unit. The entities of determinants are variable length integers (integers of arbitrary bit lengt...
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We propose an efficient and exact method for the adaptive sign detection of 4x4 determinants using a standard arithmetic unit. The entities of determinants are variable length integers (integers of arbitrary bit length). The integers are expressed in 16-bit data units, and the sign detection is reduced to the computation of 4x4 determinants of 16-bit integers. To accelerate the computation, the calculation is performed by using a standard arithmetic unit. We have implemented our method and confirmed that it significantly improves the computation time of 4x4 determinants. The method can be applicable to many geometric algorithms that need the exact sign evaluation of 4x4 determinants, especially to construct robust geometric algorithms.
Geometric algorithms are usually described assuming that arithmetic operations are performed exactly on real numbers. A program implemented using a naive substitution of floating-point arithmetic for real arithmetic c...
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Geometric algorithms are usually described assuming that arithmetic operations are performed exactly on real numbers. A program implemented using a naive substitution of floating-point arithmetic for real arithmetic can fail, since geometric primitives depend upon sign-evaluation and may not be reliable if evaluated approximately. Geometric primitives are reliable if evaluated exactly with integerarithmetic, but this degrades performance since software extended-precision arithmetic is required. We describe static-analysis techniques that reduce the performance cost of exact integer arithmetic used to implement geometric algorithms. We have used the techniques for a number of examples, including line-segment intersection in two dimensions, Delaunay triangulations, and a three-dimensional boundary-based polyhedral modeller. In general, the techniques are appropriate for algorithms that use primitives of relatively low algebraic total degree, e.g., those involving flat objects (points, lines, planes) in two or three dimensions. The techniques have been packaged in a preprocessor for reasonably convenient use.
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