Exact algorithms for dealing with geometric objects are complicated, hard to implement in practice, and slow. Over the last 20 years a theory of geometric approximation algorithms has emerged. These algorithms tend to be simple, fast, and more robust than their exact counterparts. This book is the first to cover geometric approximation algorithms in detail. In addition, more traditional computational geometry techniques that are widely used in developing such algorithms, like sampling, linear programming, etc., are also surveyed. Other topics covered include approximate nearest-neighbor search, shape approximation, coresets, dimension reduction, and embeddings. The topics covered are relatively independent and are supplemented by exercises. Close to 200 color figures are included in the text to illustrate proofs and ideas.

/pdf/geometric-approximation-algorithms-2gufu3x4ym.pdf

Geometric Approximation Algorithms

We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, prune-and-search techniques for linear programming and related problems, and LP-type problems and their efficient solution. We then describe a wide range of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other query-type problems.

/pdf/efficient-algorithms-for-geometric-optimization-3y8u2zo19z.pdf

Efficient algorithms for geometric optimization

For twenty years, it has been clear that many datasets are excessively complex for applications such as real-time display, and that techniques for controlling the level of detail of models are crucial. More recently, there has been considerable interest in techniques for the automatic simplification of highly detailed polygonal models into faithful approximations using fewer polygons. Several effective techniques for the automatic simplification of polygonal models have been developed in recent years. This report begins with a survey of the most notable available algorithms. Iterative edge contraction algorithms are of particular interest because they induce a certain hierarchical structure on the surface. An overview of this hierarchical structure is presented,including a formulation relating it to minimum spanning tree construction algorithms. Finally, we will consider the most significant directions in which existing simplification methods can be improved, and a summary of other potential applications for the hierarchies resulting from simplification.

/pdf/multiresolution-modeling-survey-and-future-opportunities-2useesc33z.pdf

Multiresolution Modeling: Survey and Future Opportunities

Many applications in computer graphics and related fields can benefit from automatic simplification of complex polygonal surface models. Applications are often confronted with either very densely over-sampled surfaces or models too complex for the limited available hardware capacity. An effective algorithm for rapidly producing high-quality approximations of the original model is a valuable tool for managing data complexity. 
In this dissertation, I present my simplification algorithm, based on iterative vertex pair contraction. This technique provides an effective compromise between the fastest algorithms, which often produce poor quality results, and the highest-quality algorithms, which are generally very slow. For example, a 1000 face approximation of a 100,000 face model can be produced in about 10 seconds on a PentiumPro 200. The algorithm can simplify both the geometry and topology of manifold as well as non-manifold surfaces. In addition to producing single approximations, my algorithm can also be used to generate multiresolution representations such as progressive meshes and vertex hierarchies for view-dependent refinement. 
The foundation of my simplification algorithm, is the quadric error metric which I have developed. It provides a useful and economical characterization of local surface shape, and I have proven a direct mathematical connection between the quadric metric and surface curvature. A generalized form of this metric can accommodate surfaces with material properties, such as RGB color or texture coordinates. 
I have also developed a closely related technique for constructing a hierarchy of well-defined surface regions composed of disjoint sets of faces. This algorithm involves applying a dual form of my simplification algorithm to the dual graph of the input surface. The resulting structure is a hierarchy of face clusters which is an effective multiresolution representation for applications such as radiosity.

/pdf/quadric-based-polygonal-surface-simplification-4kownonry4.pdf

Quadric-based polygonal surface simplification

Many algorithms for reducing the number of triangles in a surface model have been proposed, but to date there has been little theoretical analysis of the approximations they produce. Previously we described an algorithm that simplifies polygonal models using a quadric error metric. This method is fast and produces high quality approximations in practice. Here we provide some theory to explain why the algorithm works as well as it does. Using methods from differential geometry and approximation theory, we show that in the limit as triangle area goes to zero on a differentiable surface, the quadric error is directly related to surface curvature. Also, in this limit, a triangulation that minimizes the quadric error metric achieves the optimal triangle aspect ratio in that it minimizes theL2 geometric error. This work represents a new theoretical approach for the analysis of simplification algorithms. © 1999 Elsevier Science B.V. All rights reserved.

/pdf/optimal-triangulation-and-quadric-based-surface-12ixg4p1jr.pdf

Optimal triangulation and quadric-based surface simplification

Given a set S of n points in {Re}{sup 3}, sampled from an unknown bivariate function f (x, y) (i.e., for each point p {element_of} S, z{sub p} = f (x{sub p}, y{sub p})), a piecewise-linear function g(x, y) is called an {epsilon}-approximation of f (x, y) if for every p {element_of} S, {vert_bar}f (x, y) - g (x, y){vert_bar} {le} {epsilon}. The problem of computing an {epsilon}-approximation with the minimum number of vertices is NP-Hard. We present a randomized algorithm that computes an {epsilon}-approximation of size O(c{sup 2} log{sup 2} c) in O(n{sup 2+{delta}} + c{sup 3} log{sup 2}c log n/c) expected time, where c is the size of the {epsilon}-approximation with the minimum number of vertices and {delta} is any arbitrarily small positive number. Under some reasonable assumptions, the size of the output is close to O(c log c) and the expected running time is O(n{sup 2+{delta}}). We have implemented a variant of this algorithm and include some empirical results.

An efficient algorithm for terrain simplification

Pankaj K. Agarwaly Pavan K. Desikanz Abstract An important problem in layered manufacturing is the choice of a good build direction, which in uences the quality and the cost of manufacturing the object. We present e cient algorithms for computing a build direction that optimizes the total area of faces that are in contact with supports. For a given convex polytope with n faces and for a parameter " > 0, we present an O((n="3) polylogn) algorithm that computes a build direction so that the total area of faces in contact with supports is at most (1+")A , where A is the minimum area of contact with supports for all build directions. For non-convex polyhedra, we provide evidence that the lower bound for any algorithm computing the optimal direction might be (n4). We also present a faster algorithm for some special cases. Our technique for convex polyhedra involves computing approximate levels in arrangement of lines with weights. For given parameters !; " > 0, we present an algorithm that computes a (1 + ")-approximate weighted !-level in an arrangement of n weighted lines in time O((n="3) polylogn).

Pavan K. Desikan

Papers

An efficient algorithm for terrain simplification

Approximation algorithms for layered manufacturing

An Efficient Algorithm for Terraine Simplification.