A rational rotation method for robust geometric algorithms

Abstract

Algorithms in computational geometry often use the real-RAM model of computation. This model assumes that exact real numbers can be stored in memory and retreived in constant time, and that field operations (+,-, *, /) and certain other operations, square root, sine, and cosine for instance, are exact, and can be applied in constant time. These assumptions are often difficult to discharge at implementation time. Even well-understood algorithms, like line-sweep for polygon union [PS85], present much trouble. Why? Such algorithms obtain good combinatorial complexity bounds by exploiting geometric orders on the input. These relations are often implicit, so an algorithm can only probe them pointwise with a collection of predicate functions answering, e.g., "Is point p left of edge e?". In implementations, the reply must depend on arithmetic with finitely represented numbers. The trouble originates here.

A Polynomial-Time Algorithm for De Novo Protein Backbone Structure Determination from Nuclear Magnetic Resonance Data

“…While our algorithm is combinatorially precise and uses exact algebraic numbers, to test it in practice we implemented some subroutines exactly (i.e., the closedform exact solutions for internuclear NH and CH bond vectors and backbone (φ, ψ) angles, and used a discrete, combinatorial tree-search over the algebraic cross-product Y of possible solutions) and some numerically (i.e., we used a grid search over SO(3) for the orientation of the first peptide plane and over R 3 to find translations between successive secondary structure elements) for both implementation speed and to avoid some technical issues in approximating rational rotations (Canny et al, 1992;Xavier, 1995a, 1995b;Donald et al, 1992, pages 1-23). In practice, the implementation took about 20 minutes on average on a single-processor Pentium-4 class machine.…”

Boolean Operations on 3D Selective Nef Complexes: Data Structure, Algorithms, and Implementation

“…The current implementation uses inefficient but simple and complete implementations for these substeps. The current implementation supports the construction of Nef polyhedra from manifold solids [Ket99], boolean operations (union, intersection, complement, difference, symmetric difference), topological operations (interior, closure, boundary, regularization), rotations by rational rotation matrices (arbitrary rotation angles are approximated up to a specified tolerance [CDR92]). Our implementation is exact.…”

Flip Distance Between Triangulations of a Simple Polygon is NP-Complete

“…Finally, note that all slopes used in the construction have numerators and denominators that are polynomial in N . Hence, this also holds for the coordinates of the vertices of D. Note that this is, essentially, the parametrization of the unit circle, as discussed in [5].…”