Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 14 doc

318 E. Fogel, M. Teillaud In a geometric algorithm each computational step is either a construction step or a conditional step based on the result of a predicate. The former produces a new geometric object such as the intersection point of two segments. The latter typically computes the sign of an expression used by the program control. Different computational paths lead to results with different combinatorial characteristics. Although numerical errors can sometimes be tolerated and interpreted as small perturbations in the input, they may lead to invalid combinatorial structures or inconsistent state during a program ex- ecution. Thus, it suffices to ensure that all predicates are evaluated correctly to eliminate inconsistencies and guarantee combinatorially correct results. Exact Geometric Computation (EGC), as summarized by Yap [346], sim- ply amounts to ensuring that we never err in predicate evaluations. EGC represents a significant relaxation from the naive concept of numerical ex- actness. We only need to compute to sufficient precision to make the correct predicate evaluation. This has led to the development of several techniques such as precision-driven computation, lazy evaluation, adaptive computation, and floating point filters, some of which are implemented in Cgal, such as numerical filtering. Here, computation is carried out using a number type that supports only inexact arithmetic (e.g., double floating point), while ap- plying a filter that indicates whether the result is exact. If the filter fails, the computation is re-done using exact arithmetic. Switching between number types and exact computation techniques, and choosing the appropriate components that best suit the application needs, is conveniently enabled through the generic programming paradigm, as it typically requires only a minor code change reflected in the instantiating of just a few data types. 8.4 Cgal Contents Cgal is written in C++ according to the generic programming paradigm described above. It has a common programming style, which is very similar to that of the STL. Its Application Programming Interface (API) is homo- geneous, and allows for a convenient and consistent interfacing with other software packages and applications. . The library consists of about 500,000 lines of code divided among approx- imately 150 classes. Cgal also comes with numerous examples and demos. The manual has about 3,000 pages. There are roughly 50 chapters that are grouped in several parts for a rough description. The first part is the kernels [155], which consist of constant size non- modifiable geometric primitive objects and operations on these objects. The objects are represented both as stand-alone classes that are instantiated by a kernel class, and as members of the kernel classes. The latter option allows for more flexibility and adaptability of the kernel. 8 Appendix - Generic Programming and The Cgal Library 319 In addition, Cgal offers a collection of basic geometric data structures and algorithms such as convex hull, polygons and polyhedra and operations on them (Boolean operations, polygon offsetting), 2D arrangements, 2D and 3D triangulations, Voronoi diagrams, surface meshing and surface subdivi- sion, search structures, geometric optimization, interpolation, and kinetic data structures. These data structures and algorithms are parameterized by traits classes, that define the interface between them and the primitives they use. In many cases, the kernel can be used as a traits class, or the kernel classes provided in Cgal can be used as components of traits classes for these data structures and algorithms. The third part of the library consists of non-geometric support facilities, such as circulators, random generators, I/O support for debugging and for interfacing Cgal with various visualization tools. This part also provides the user with number type support. Cgal kernel classes are parameterized by number types. Instantiating a kernel with a particular number type is a trade-off between efficiency and ac- curacy. The choice depends on the algorithm implementation and the expected input data to be handled. Number types must fulfil certain requirements, so that they can be successfully used by the kernel code. The list of requirements establishes a concept of a number type. A few number-type concepts have been introduced by Cgal, e.g., RingNumberType and FieldNumberType. Nat- urally, number types have evident semantic constraints. That is, they should be meaningful in the sense that they approximate some subfield of the real numbers. Cgal provides several models of its number-type concepts, some of them implement techniques to expedite exact computation mentioned in the previous paragraph. Cgal also provides a glue layer that adapts number- type classes implemented by external libraries as models of its number-type concepts. The above describes the accessibility model of Cgal at the time this book was written. Constant and persistent improvement to the source code and the didactic manuals, review of packages by the Editorial board and exhaustive testing, through the years led to a state of excellent quality internationally recognized as an unrivalled tool in its field. At the time these lines are written, Cgal already has a foothold in many domains related to computational geometry and could be found in many academic and research institutes as well as commercial entities. Release 3.1 was downloaded more than 14.500 times, and the public discussion list counts more than 950 subscribed users. Acknowledgements The development of Cgal was supported by two European Projects Cgal and Galia during three years in total (1996–1999). Several sites have kept on working on Cgal after the European support stopped. 320 E. Fogel, M. Teillaud The new European project ACS (Algorithms for Complex Shapes with cer- tified topology and numerics) 4 provides again partial support for new research and developments in Cgal. 4 http://acs.cs.rug.nl/. References The page numbers where each reference is cited are listed in brackets at the end of each item. 1. Boost, C++ libraries. http://www.boost.org. [154, 316] 2. 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