Computations that require higher than double precision for robust and exact decision making

Abstract

Consider the computation of deciding relative orientations of objects undergoing multiple translations and rotations. Such an orientation test involves the computation of expressions based on arithmetic operations, square roots and trigonometric functions. The computation of signs of such expressions using double precision floating-point arithmetic in modern computers may result in errors. In this article we demonstrate the existence of examples where double precision is not sufficient to compute the correct sign of an expression. We consider (i) simple expressions involving only the four basic arithmetic operations, (ii) expressions involving the square-root function and (iii) expressions representing orientation tests in two- and three-dimensions involving objects undergoing arbitrary rotations by angles given in radians, thereby requiring the computation of trigonometric functions. We develop a system that uses requisite high precision for computing the correct sign of such expressions. The system uses our floating-point filter called L-filter and the bigfloat extended precision package in LEDA (Library of Efficient Data Types and Algorithms).

Keywords:

• Floating-point filter
• Exact computation
• Geometric transformations
• Collision detection

Acknowledgement

The author S. P. Pal acknowledges the support of a research grant from All India Council for Technical Education, New Delhi, India, during the period 1997–2000.