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On the design of CGAL a computational geometry algorithms library
 SOFTW. – PRACT. EXP
, 1999
"... CGAL is a Computational Geometry Algorithms Library written in C++, which is being developed by research groups in Europe and Israel. The goal is to make the large body of geometric algorithms developed in the field of computational geometry available for industrial application. We discuss the major ..."
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Cited by 97 (16 self)
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CGAL is a Computational Geometry Algorithms Library written in C++, which is being developed by research groups in Europe and Israel. The goal is to make the large body of geometric algorithms developed in the field of computational geometry available for industrial application. We discuss the major design goals for CGAL, which are correctness, flexibility, easeofuse, efficiency, and robustness, and present our approach to reach these goals. Generic programming using templates in C++ plays a central role in the architecture of CGAL. We give a short introduction to generic programming in C++, compare it to the objectoriented programming paradigm, and present examples where both paradigms are used effectively in CGAL. Moreover, we give an overview of the current structure of the CGALlibrary and consider software engineering aspects in the CGALproject.
Geometric Algebra: A Computational Framework for Geometrical Applications
, 2002
"... Geometric algebra is a consistent computational framework in which to define geometric primitives and their relationships. This algebraic approach contains all geometric operators and permits specification of constructions in a totally coordinatefree manner. Since it contains primitives of any dime ..."
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Cited by 29 (4 self)
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Geometric algebra is a consistent computational framework in which to define geometric primitives and their relationships. This algebraic approach contains all geometric operators and permits specification of constructions in a totally coordinatefree manner. Since it contains primitives of any dimensionality (rather than just vectors) it has no special cases: all intersections of primitives are computed with one general incidence operator. This paper gives an introduction to the elements of geometric algebra to aid assessment of its potential for geometric programming. It contains no really new results, but collects known elements of relevance to computer graphics. Keywords: Geometric algebra, geometric programming. 1
Reversing Subdivision Rules: Local Linear Conditions and Observations on Inner Products
 Journal of Computational and Applied Mathematics
, 1999
"... In a previous work [32] we investigated how to reverse subdivision rules using global least squares fitting. This led to multiresolution structures that could be viewed as semiorthogonal wavelet systems whose inner product was that for finitedimensional Cartesian vector space. We produced simple an ..."
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Cited by 27 (17 self)
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In a previous work [32] we investigated how to reverse subdivision rules using global least squares fitting. This led to multiresolution structures that could be viewed as semiorthogonal wavelet systems whose inner product was that for finitedimensional Cartesian vector space. We produced simple and sparse reconstruction filters, but had to appeal to matrix factorization to obtain an efficient, exact decomposition. We also made some observations on how the inner product that defines the semiorthogonality inuences the sparsity of the reconstruction filters. In this work we carry the investigation further by studying biorthogonal systems based upon subdivision rules and local least squares fitting problems that reverse the subdivision. We are able to produce multiresolution structures for some common univariate subdivision rules that have both sparse reconstruction and decomposition filters. Three will be presented here  for quadratic and cubic Bspline subdivision and for the 4point interpola...
Multiresolution Curve and Surface Representation: Reversing Subdivision Rules by LeastSquares Data Fitting
, 1998
"... This work explores how three techniques for defining and representing curves and surfaces can be related efficiently. The techniques are subdivision, leastsquares data fitting, and wavelets. We show how leastsquares data fitting can be used to "reverse" a subdivision rule, how this revers ..."
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Cited by 25 (13 self)
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This work explores how three techniques for defining and representing curves and surfaces can be related efficiently. The techniques are subdivision, leastsquares data fitting, and wavelets. We show how leastsquares data fitting can be used to "reverse" a subdivision rule, how this reversal is related to wavelets, how this relationship can provide a multilevel representation, and how the decomposition/reconstruction process can be carried out in linear time and space through the use of a matrix factorization. Some insights that this work brings forth are that the inner product used in a multiresolution analysis influences the support of a wavelet, that wavelets can be constructed by straightforward matrix observations, and that matrix partitioning and factorization can provide alternatives to inverses or duals for building efficient decomposition and reconstruction processes. We illustrate our findings using an example curve, greyscale image, and tensorproduct surface. Keywords: Su...
Modeling 3D Euclidean geometry
 IEEE Computer Graphics and Applications
, 2003
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Spline Overlay Surfaces
 University of Waterloo
, 1991
"... We consider the construction of spline features on spline surfaces. 1 The approach taken is a generalization of the hierarchical surface introduced in [Forsey88]. Features are regarded as splinedefined vector displacement fields that are overlain on existing surfaces. No assumption is made that t ..."
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Cited by 8 (1 self)
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We consider the construction of spline features on spline surfaces. 1 The approach taken is a generalization of the hierarchical surface introduced in [Forsey88]. Features are regarded as splinedefined vector displacement fields that are overlain on existing surfaces. No assumption is made that the overlays are derived from the base surface. They may be applied with any orientation in a nonhierarchical fashion. In particular, we present a "cheap" version of the concept in which the displacement field is mapped to the base surface approximately, through the mapping of its control vectors alone. The result is a feature that occupies the appropriate position in space with respect to the base surface. It may be manipulated and rendered as an independent spline, thus avoiding the costs of a true displacement mapping. This approach is useful for prototyping and previewing during design. When a finished product is desired, of course, true displacement mapping is employed. 1 Introduction ...
A Coordinate Free Geometry ADT
, 1997
"... An algebra for geometric reasoning is developed that is amenable to software implementation. The features of the algebra are chosen to support geometric programming of the variety found in computer graphics and computer aided geometric design applications. The implementation of the algebra in C++ is ..."
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Cited by 5 (1 self)
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An algebra for geometric reasoning is developed that is amenable to software implementation. The features of the algebra are chosen to support geometric programming of the variety found in computer graphics and computer aided geometric design applications. The implementation of the algebra in C++ is described, and several examples illustrating the use of this software are given. 1 Introduction Traditionally, computer graphics packages are implemented using homogeneous coordinates and a matrix package. Although this practice is widespread and successful, it does have its shortcomings. The basic problem with the traditional coordinatebased approach is due to differences between matrix computations and geometric reasoning. Although graphics programs require reasoning in affine and projective geometries, use of a matrix package places the programmer in a vector space setting; the geometric interpretation of these calculations is left to the imagination and discipline of the programmer (w...
The Mathematics of Graphical Transformations: Vector Geometric and CoordinateBased Approaches,” DesignLab
, 1997
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Geometry Freedom in Geometric Computation  Towards HigherOrder Genericity through Purely Combinatorial Geometric Algorithms
"... Any geometric algorithm can be dissected into combinatorial parts and geometric parts. The two parts intertwine in both the description and the implementation of the algorithm. We consider the benefits of relaxing this tight coupling. Coordinate freedom is widely considered the most important princi ..."
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Any geometric algorithm can be dissected into combinatorial parts and geometric parts. The two parts intertwine in both the description and the implementation of the algorithm. We consider the benefits of relaxing this tight coupling. Coordinate freedom is widely considered the most important principle in designing and implementing geometric systems. By ensuring that client code not manipulate individual coordinates and by developing two foundations for homogeneous and Cartesian coordinates, switching from one to the other can be easily performed after the system has been completed. We take another step and show that geometry freedom is possible. By removing the geometric classes from the implementation of a geometric algorithm, the algorithm becomes purely combinatorial. An arbitrary Euclidean or spherical geometry is then used as a parameter to the combinatorial algorithm to produce a geometric system in that geometry. Geometric freedom is helpful, for instance, when a geographic input is no longer constrained to a small area of Earth and one wishes to use spherical instead of Euclidean geometry. We apply geometry freedom to three classical problems. For the first two problems—convex hulls and Delaunay triangulations—the algorithms become generic with respect to the geometry. For the third—binary space partitioning—the algorithm becomes generic with respect to both the geometry and the dimension.