Indirect predicates for geometric constructions

by   Marco Attene, et al.

Geometric predicates are a basic ingredient to implement a vast range of algorithms in computational geometry. Modern implementations employ floating point filtering techniques to combine efficiency and robustness, and state-of-the-art predicates are guaranteed to be always exact while being only slightly slower than corresponding (inexact) floating point implementations. Unfortunately, if the input to these predicates is an intermediate construction of an algorithm, its floating point representation may be affected by an approximation error, and correctness is no longer guaranteed. This paper introduces the concept of indirect geometric predicate: instead of taking the intermediate construction as an explicit input, an indirect predicate considers the primitive geometric elements which are combined to produce such a construction. This makes it possible to keep track of the floating point approximation, and thus to exploit efficient filters and expansion arithmetic to exactly resolve the predicate with minimal overhead with respect to a naive floating point implementation. As a representative example, we show how to extend standard predicates to the case of points of intersection of linear elements (i.e. lines and planes) and show that, on classical problems, this approach outperforms state-of-the-art solutions based on lazy exact intermediate representations.


page 1

page 2

page 3

page 4


Computing the exact sign of sums of products with floating point arithmetic

IIn computational geometry, the construction of essential primitives lik...

Capturing the Future by Replaying the Past

Delimited continuations are the mother of all monads! So goes the slogan...

Constrained Delaunay Tetrahedrization: A Robust and Practical Approach

We present a numerically robust algorithm for computing the constrained ...

Approximate Vertex Enumeration

The problem to compute a V-polytope which is close to a given H-polytope...

FFT Convolutions are Faster than Winograd on Modern CPUs, Here is Why

Winograd-based convolution has quickly gained traction as a preferred ap...

Exact arithmetic as a tool for convergence assessment of the IRM-CG method

Using exact computer arithmetic, it is possible to determine the (exact)...

A Novel Approach to Generate Correctly Rounded Math Libraries for New Floating Point Representations

Given the importance of floating-point (FP) performance in numerous doma...

Please sign up or login with your details

Forgot password? Click here to reset