Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles become convex, and no bars cross while preserving the bar lengths. Furthermore, our motion is piecewise-differentiable, does not decrease the distance between any pair of vertices, and preserves any symmetry present in the initial configuration. In particular, this result settles the well-studied carpenter’s rule conjecture.

We analyze the behavior of two line arrangement algorithms, a sweepline algorithm and an incremental algorithm, in approximate arithmetic. The algorithms have running times O(n 2 log n) and O(n 2 ) respectively. We show that each of these algorithms can be implemented to have O(nffl) relative error. This means that each algorithm produces an arrangement realized by a set of pseudolines so that each pseudoline differs from the corresponding line relatively by at most O(nffl). We also show that there is a line arrangement algorithm with O(n 2 log n) running time and O(ffl) relative error. 1 Introduction We analyze the behavior of line arrangement algorithms in approximate arithmetic. Approximate arithmetic is a set of arithmetic operations defined on the real numbers that make relative error ffl; this models floating point arithmetic. The input to a line arrangement algorithm is a set of n lines specified by real number coefficients. The output is a "combinatorial arrangement", ...

There is much current interest among researchers to find algorithms that will draw graphs in three dimensions. It is well known that every 3-connected planar graph can be represented as a strictly convex polyhedron. However, no practical algorithms exist to draw a general 3-connected planar graph as a convex polyhedron. In this paper we review the concept of a stressed graph and how it relates to convex polyhedra; we present a practical algorithm that uses stressed graphs to draw 3-connected planar graphs as strictly convex polyhedra; and show some examples. Key words: graph, stressed graph, convex polyhedron, reciprocal polyhedron 1 Introduction It is well known that 3-connected planar graphs can be drawn as convex polyhedra. However, no practical algorithms exist to draw general 3-connected planar graphs as convex polyhedra. The two-dimensional (2D) drawing in Figure 1 is 3-connected and planar, and the corresponding polyhedron is drawn in Figure 2 as three different views. The 2D ...

The article provides a set of algorithms for performing set operations on polygonal regions in the plane using standard floating point arithmetic. The algorithms are robust, guaranteeing both topological consistency and numerical accuracy. Each polygon edge is modeled as an implicit or explicit polygonal curve which stays within some distance fi of the original line segment. If the curve is implicit, fi is bounded by a small multiple of the rounding unit. If the curves are explicit, the bound on fi may grow with the number of curves. One can mix implicit and explicit representations to suit the application.

Geometric modelers are becoming faster and more powerful, but they still suffer from reliability problems because of floating point errors. Previous work in the field of robust geometric modeling tends to be problem specific and has proven hard to generalize. The approach described here is a general paradigm for handling the accuracy problem for a large set of geometric algorithms. This approach brings together ideas and techniques from interval arithmetic, constraint management, randomization, and algebraic geometry. It acknowledges that input values have tolerances, that objects within tolerance are equivalent, and that certain geometric singularities must be maintained because they reflect design intent or the laws of geometry. Our approach is systematic, and can be applied almost mechanically to the large domain of problems, that can be solved by algorithms using the operations +, -, * and /. The required theory and algorithms have been developed, and the viability of the concepts ...

Introduction We present a locality-based algorithm to solve the problem of splitting a complex of convex polytopes with a hyperplane or a convex subset of it. The solution to this problem has several applications. One goal is to perform boolean set operations. The solution can also be used to decompose a polyhedron into convex polytopes [3] and to generate good meshes [4]. In higher dimensional spaces it can be used to efficiently compute isocontours of linear approximations of scalar fields (a basic technique of Scientific Visualization) [17, 19]. The approach taken here can also be included in a set of robust algorithms [11, 13, 15, 20, 27, 28] based on finite precision arithmetic. It is also defined in a dimension independent framework [5, 16, 24, 25]. The main contributions of this approach are: (i) it can be applied to polyhedral complexes of any dimension d; (ii) the algorithm is robust (it always produces valid output) and consistent (the topological structure of the resu

This paper is concerned with applications of Tutte’s barycentric embedding theorem (Proc. London Math. Soc. 13 (1963), 743–768). It presents a method for building isotopies of triangulations in the plane, based on Tutte’s theorem and the computation of equilibrium stresses of graphs by Maxwell–Cremona’s theorem; it also provides a counterexample showing that the analogue of Tutte’s theorem in dimension 3 is false.

The goal of this paper is to prove the applicability of algorithm engineering and software design concepts to geometric computing through a vertical case study on the implementation of planar point location algorithms. The work is presented within the framework of the GeomLib project, aimed at developing an easy to use, reliable, and flexible library of robust and e#cient geometric algorithms. We present the criteria that have inspired the preliminary design of GeomLib and discuss the guidelines that we have followed in the initial implementation. Keywords: Algorithm engineering, geometric computing, software libraries, point location. # Research supported in part by the U.S. Army Research O#ce under grant DAAH04--96--1--0013 and by the National Science Foundation under grants CCR--9423847, CCR--9732327, and CDA-9703080. Portions of the results described in this paper were presented at the Workshop on Geometric Computing, Sophia Antipolis, France, 1997, and at the 1st Workshop o...

Any 3-dimensional convex polytope with n vertices can be realized in Euclidean 3-space with all coordinates of all vertices being integers of absolute value not exceeding n^169n³.

Abstract. We study the problem how to obtain a small drawing of a 3-polytope with Euclidean distance between any two points at least 1. The problem can be reduced to a one-dimensional problem, since it is sufficient to guarantee distinct integer x-coordinates. We develop an algorithm that yields an embedding with the desired property such that the polytope is contained in a 2(n−2)×2×1 box. The constructed embedding can be scaled to a grid embedding whose x-coordinates are contained in [0, 2(n − 2)]. Furthermore, the point set of the embedding has a small spread, which differs from the best possible spread only by a multiplicative constant. 1