Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates

Abstract

Abstract. Exact computer arithmetic has a variety of uses, including the robust implementation of geometric algorithms. This article has three purposes. The first is to offer fast software-level algorithms for exact addition and multiplication of arbitrary precision floating-point values. The second is to propose a technique for adaptive precision arithmetic that can often speed these algorithms when they are used to perform multiprecision calculations that do not always require exact arithmetic, but must satisfy some error bound. The third is to use these techniques to develop implementations of several common geometric calculations whose required degree of accuracy depends on their inputs. These robust geometric predicates are adaptive; their running time depends on the degree of uncertainty of the result, and is usually small. These algorithms work on computers whose floating-point arithmetic uses radix two and exact rounding, including machines complying with the IEEE 754 standard. The inputs to the predicates may be arbitrary single or double precision floating-point numbers. C code is publicly available for the two-dimensional and three-dimensional orientation and incircle tests, and robust Delaunay triangulation using these tests. Timings of the implementations demonstrate their effectiveness.

Multimesh finite element methods: Solving PDEs on multiple intersecting meshes

“…However, the collision detection and intersection construction depend on the geometrical (and topological) dimension. The geometric predicate library [49] provides routines in both 2D and 3D. The intersection construction is currently done in a case-by-case basis and is work in progress.…”

“…The collisions are computed by first constructing and then colliding axis-aligned bounding box trees (AABB trees) for the two meshes [47,48] to find candidate element intersections. The candidate elements are then checked for collision using robust geometric predicates of adaptive precision [49]. The mesh-mesh collision is carried out for each (unordered) pair of premeshes among the 1 + N meshes.…”

Load-Balancing for Parallel Delaunay Triangulations

“…For a given simplex s of partition i we use the AABB intersection test from the previous section to determine the set C j ⊆ C j of cells intersected by the circumhypersphere of s in partition j. For all points contained in these cells an adaptive precision inSphere-test [29] is performed to determine whether s violates the Delaunay property and thus its vertices need to be added to the border triangulation. Table 2: Input point sets and their resulting triangulations.…”

The moving mesh code Shadowfax

“…As Richard Shewchuk (1997) pointed out, incremental Delaunay construction algorithms can become unstable due to numerical round-off error. To prevent this from happening, we employ arbitrary exact arithmetics for all geometrical tests involved.…”

“…To prevent this from happening, we employ arbitrary exact arithmetics for all geometrical tests involved. Since the predicates of Richard Shewchuk (1997) depend on a number of assumptions on the internal CPU precision that are not met on all hardware architectures, we use the technique outlined by Springel (2010) : we map the floating point coordinates of the mesh generators to the interval [1, 2] and use an integer representation of the mantissa to exactly calculate the result of a geometrical test if the numerical error could lead to a wrong result. We pre-calculated the maximal size of an integer necessary to store the exact result and use the Boost Multiprecision library 11 to perform the calculations using long integer arithmetics.…”