Efficient Delaunay triangulation using rational arithmetic

Abstract

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 es floating point, they can produce incorrect answers; when they are implemented using arbitraryprecision arithmetic, they are expensive to compute. We present adaptive-precision algorithms for finding the signs of determinant 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.

Reliable Geometric Computations with Algebraic Primitives and Predicates

“…Unfortunately, exact implementations are often far too slow, especially when we are dealing with nonlinear primitives. Karasick et al [22] noted that naive implementations can take several orders of magnitude longer than an equivalent floating-point implementation, an observation that is consis-tent with our experience. The goal has been to find techniques that reduce the performance penalty to an acceptable level.…”

Degeneracy and Instability in Geometric Computation

“…This principle has been used widely in numerically robust geometric computation (Ottmann et al, 1987;Sugihara and Iri, 1989;Karasick et al, 1991;Sugihara, 1992;Schorn, 1991;Benouamer et al, 1993;Fortune and von Wyk, 1993).…”

“…The first group is the "exact-arithmetic approach", which relies on numerical computation absolutely (Sugihara and Iri, 1989;Ottmann et al, 1987;Karasick et al, 1991;Sugihara, 1992;Schorn, 1991;Benouamer et al, 1993;Fortune and von Wyk, 1993;Sugihara, 1997;Yap, 1995). The topological structure of a geometric object can be decided by the signs of the results of numerical computations.…”

Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates

“…Karasick et al [16] report their experiences optimizing a method for determinant evaluation using rational inputs. Their approach reduces the bit complexity of the inputs by performing arithmetic on intervals (with low precision bounds) rather than exact values.…”