In addition, Cgal offers a collection of basic geometric data structuresand algorithms such as convex hull, polygons and polyhedra and operationson them Boolean operations, polygon offsett
Trang 1In a geometric algorithm each computational step is either a tion step or a conditional step based on the result of a predicate The formerproduces a new geometric object such as the intersection point of two seg-ments The latter typically computes the sign of an expression used by theprogram control Different computational paths lead to results with differentcombinatorial characteristics Although numerical errors can sometimes betolerated and interpreted as small perturbations in the input, they may lead
construc-to invalid combinaconstruc-torial structures or inconsistent state during a program ecution Thus, it suffices to ensure that all predicates are evaluated correctly
ex-to eliminate inconsistencies and guarantee combinaex-torially correct results.Exact Geometric Computation (EGC), as summarized by Yap [346], sim-ply amounts to ensuring that we never err in predicate evaluations EGCrepresents a significant relaxation from the naive concept of numerical ex-actness We only need to compute to sufficient precision to make the correctpredicate evaluation This has led to the development of several techniquessuch as precision-driven computation, lazy evaluation, adaptive computation,and floating point filters, some of which are implemented in Cgal, such asnumerical filtering Here, computation is carried out using a number typethat 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, thecomputation is re-done using exact arithmetic
Switching between number types and exact computation techniques, andchoosing the appropriate components that best suit the application needs,
is conveniently enabled through the generic programming paradigm, as ittypically requires only a minor code change reflected in the instantiating ofjust 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 geneous, and allows for a convenient and consistent interfacing with othersoftware packages and applications
homo-The library consists of about 500,000 lines of code divided among imately 150 classes Cgal also comes with numerous examples and demos.The manual has about 3,000 pages There are roughly 50 chapters that aregrouped in several parts for a rough description
approx-The first part is the kernels [155], which consist of constant size modifiable geometric primitive objects and operations on these objects Theobjects are represented both as stand-alone classes that are instantiated by
non-a kernel clnon-ass, non-and non-as members of the kernel clnon-asses The lnon-atter option non-allowsfor more flexibility and adaptability of the kernel
Trang 2In addition, Cgal offers a collection of basic geometric data structuresand algorithms such as convex hull, polygons and polyhedra and operations
on them (Boolean operations, polygon offsetting), 2D arrangements, 2D and3D triangulations, Voronoi diagrams, surface meshing and surface subdivi-sion, search structures, geometric optimization, interpolation, and kinetic datastructures These data structures and algorithms are parameterized by traitsclasses, 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 classesprovided in Cgal can be used as components of traits classes for these datastructures 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 forinterfacing Cgal with various visualization tools This part also provides theuser with number type support
Cgal kernel classes are parameterized by number types Instantiating akernel with a particular number type is a trade-off between efficiency and ac-curacy The choice depends on the algorithm implementation and the expectedinput data to be handled Number types must fulfil certain requirements, sothat they can be successfully used by the kernel code The list of requirementsestablishes 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 realnumbers Cgal provides several models of its number-type concepts, some
of them implement techniques to expedite exact computation mentioned inthe previous paragraph Cgal also provides a glue layer that adapts number-type classes implemented by external libraries as models of its number-typeconcepts
The above describes the accessibility model of Cgal at the time this bookwas written Constant and persistent improvement to the source code and thedidactic manuals, review of packages by the Editorial board and exhaustivetesting, through the years led to a state of excellent quality internationallyrecognized as an unrivalled tool in its field At the time these lines are writ-ten, Cgal already has a foothold in many domains related to computationalgeometry 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 Cgaland Galia during three years in total (1996–1999) Several sites have kept onworking on Cgal after the European support stopped
Trang 3The new European project ACS (Algorithms for Complex Shapes with tified topology and numerics)4provides again partial support for new researchand developments in Cgal.
cer-4http://acs.cs.rug.nl/
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