Triangle is a robust implementation of two-dimensional constrained Delaunay triangulation and Ruppert's Delaunay refinement algorithm for quality mesh generation. Several implementation issues are discussed, including the choice of triangulation algorithms and data structures, the effect of several variants of the Delaunay refinement algorithm on mesh quality, and the use of adaptive exact arithmetic to ensure robustness with minimal sacrifice of speed. The problem of triangulating a planar straight line graph (PSLG) without introducing new small angles is shown to be impossible for some PSLGs, contradicting the claim that a variant of the Delaunay refinement algorithm solves this problem.

/pdf/triangle-engineering-a-2d-quality-mesh-generator-and-34ox9m7czj.pdf

Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator

TetGen is a Cpp program for generating good quality tetrahedral meshes aimed to support numerical methods and scientific computing. The problem of quality tetrahedral mesh generation is challenged by many theoretical and practical issues. TetGen uses Delaunay-based algorithms which have theoretical guarantee of correctness. It can robustly handle arbitrary complex 3D geometries and is fast in practice. The source code of TetGen is freely available.This article presents the essential algorithms and techniques used to develop TetGen. The intended audience are researchers or developers in mesh generation or other related areas. It describes the key software components of TetGen, including an efficient tetrahedral mesh data structure, a set of enhanced local mesh operations (combination of flips and edge removal), and filtered exact geometric predicates. The essential algorithms include incremental Delaunay algorithms for inserting vertices, constrained Delaunay algorithms for inserting constraints (edges and triangles), a new edge recovery algorithm for recovering constraints, and a new constrained Delaunay refinement algorithm for adaptive quality tetrahedral mesh generation. Experimental examples as well as comparisons with other softwares are presented.

TetGen, a Delaunay-Based Quality Tetrahedral Mesh Generator

Random sampling is used for several new geometric algorithms. The algorithms are “Las Vegas,” and their expected bounds are with respect to the random behavior of the algorithms. One algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the size of the answer, the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of a point set in E3 in O(n log A) expected time, where n is the number of points and A is the number of points on the surface of the hull. A simple Las Vegas algorithm triangulates simple polygons in O(n log log n) expected time. Algorithms for half-space range reporting are also given. In addition, this paper gives asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets.

/pdf/applications-of-random-sampling-in-computational-geometry-ii-3smdvp6m82.pdf

Applications of random sampling in computational geometry, II

This survey describes the state of the art of online handwriting recognition during a period of renewed activity in the field. It is based on an extensive review of the literature, including journal articles, conference proceedings, and patents. Online versus offline recognition, digitizer technology, and handwriting properties and recognition problems are discussed. Shape recognition algorithms, preprocessing and postprocessing techniques, experimental systems, and commercial products are examined. >

http://www.csis.pace.edu/~ctappert/dps/pdf/pen-tappert.pdf

The state of the art in online handwriting recognition

http://compalg.inf.elte.hu/~tony/Kutatas/BinPacking/Johnson-BinPacking-NP-Column-JAlg-2002.pdf

The NP-completeness column: An ongoing guide

From the Publisher: \nThis is the newly revised and expanded edition of a popular introduction to the design and implementation of geometry algorithms arising in areas such as computer graphics, robotics, and engineering design. The basic techniques used in computational geometry are all covered: polygon triangualtions, convex hulls, Voronoi diagrams, arrangements, geometric searching, and motion planning. The self-contained treatment presumes only an elementary knowledge of mathematics, but it reaches topics on the frontier of current research. Thus professional programmers will find it a useful tutorial.

Computational Geometry in C.

We experiment with exact integer arithmetic to implement primitives for geometric algorithms. Naive use of exact arithmetic—either modular or multiprecision integer—increases execution time dramatically over the use of floating-point arithmetic. By combining tuned multiprecision integer arithmetic and a floating-point filter based on interval analysis, we can obtain the effect of exact integer arithmetic at a cost close to that of floating-point arithmetic. We describe an experimental expression compiler that conveniently packages our techniques.

Efficient exact arithmetic for computational geometry

We describe a method for inferring constraints that are desirable for a given (rough) drawing and then modifying the drawing to satisfy the constraints wherever possible. The method has been implemented as part of an online graphics editor running under the UNIX™ operating system and it has undergone modifications in response to user input. Although the framework we discuss is general, the current implementation is polygon-oriented. The relations examined are: approximate equality of the slope or length of sides, collinearity of sides, and vertical and horizontal alignment of points.

An automatic beautifier for drawings and illustrations

Geometric algorithms are usually described assuming that arithmetic operations are performed exactly on real numbers. A program implemented using a naive substitution of floating-point arithmetic for real arithmetic can fail, since geometric primitives depend upon sign-evaluation and may not be reliable if evaluated approximately. Geometric primitives are reliable if evaluated exactly with integer arithmetic, but this degrades performance since software extended-precision arithmetic is required.

/pdf/static-analysis-yields-efficient-exact-integer-arithmetic-177wuhi0gf.pdf

Static analysis yields efficient exact integer arithmetic for computational geometry

A programming language tha t includes special constructs for drawing pictures is discussed. The language has been designed so that programs to draw pictures can reflect the structure of those pictures. Picture elements are called boxes. They are defmed by declaring algebraic relationships tha t should exist among their significant points and requesting actions to be performed at those points. The algebraic relationships are processed by an equation solver to determine the absolute location of significant points. A new mechanism of variable reference and scope makes the expression of relationships tha t should exist among picture elements more direct than it is in conventional languages. Some common picture operations--drawing lines using a pattern, filling in areas with textures, and obscuring parts of a picture by other par ts--are all expressed readily once the mechanism for defining and using boxes is in place. The language has been implemented, and used by several authors to prepare figures for their papers and books. It was also used to prepare this paper.

Christopher J. Van Wyk

Papers

Computational Geometry in C.

Efficient exact arithmetic for computational geometry

An automatic beautifier for drawings and illustrations

Static analysis yields efficient exact integer arithmetic for computational geometry

A High-Level Language for Specifying Pictures