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@ -18,8 +18,8 @@ predicates. The former replace constructors of the kernel classes and
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constructive procedures in the kernel. There are also function objects
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replacing operators, especially for equality testing.
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\hasModel `Cartesian_d<FieldNumberType>`
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\hasModel `Homogeneous_d<RingNumberType>`
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\hasModel `CGAL::Cartesian_d<FieldNumberType>`
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\hasModel `CGAL::Homogeneous_d<RingNumberType>`
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*/
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class Kernel_d {
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public:
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@ -1,7 +1,7 @@
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namespace CGAL {
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/*!
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\mainpage dD Geometry Kernel
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\mainpage dD Geometry %Kernel
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\anchor Chapter_dD_Geometry_Kernel
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\autotoc
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\authors Michael Seel
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@ -69,7 +69,7 @@ This extends the syntax of random access iterators to input iterators.
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If we index the tuple as above then we require that
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\f$ ++^{(d+1)}\mathit{first} = \mathit{last}\f$.
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# Kernel Representations #
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# %Kernel Representations #
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Our object of study is the \f$ d\f$-dimensional affine Euclidean space,
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where \f$ d\f$ is a parameter of our geometry. Objects in that space are
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@ -94,7 +94,7 @@ For a point with Cartesian coordinates
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\f$ (c_0,c_1,\ldots,c_{d-1})\f$ a
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possible homogeneous representation is
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\f$ (c_0,c_1,\ldots,c_{d-1},1)\f$.
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Homogeneous coordinates in fact allow to represent
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%Homogeneous coordinates in fact allow to represent
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objects in a more general space, the projective space \f$ \mathbb{P}^d\f$.
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In \cgal, we do not compute in projective geometry. Rather, we use
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homogeneous coordinates to avoid division operations,
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@ -150,7 +150,7 @@ must be available as well. It must work for exact (i.e., no
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remainder) integer divisions only. Furthermore, both number types
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should fulfill \cgal's requirements on a number type.
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## Cartesian Kernel ##
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## %Cartesian %Kernel ##
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With `Cartesian_d<FieldNumberType,LinearAlgebra>` you can choose
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Cartesian representation of coordinates. The type
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@ -170,7 +170,7 @@ and `Cartesian<FieldNumberType>::RT` are mapped to number type
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type `LinearAlgebra`. `Cartesian<FieldNumberType>` uses
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reference counting internally to save copying costs.
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## Homogeneous Kernel ##
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## %Homogeneous %Kernel ##
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As we mentioned before, homogeneous coordinates permit to avoid
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division operations in numerical computations, since the additional
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@ -222,7 +222,7 @@ parameterized with a number type, such as
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`Cartesian_d<double>` or `Homogeneous_d<leda_integer>`.
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</OL>
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## Kernel as a Traits Class ##
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## %Kernel as a Traits Class ##
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Algorithms and data structures in the basic library of \cgal are
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parameterized by a traits class that subsumes the objects on which the
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@ -235,7 +235,7 @@ algorithm. In some other cases, the algorithms or data structures
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needs more than is provided by a kernel. In these cases, a kernel can
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not be used as a traits class.
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## Choosing a Kernel ##
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## Choosing a %Kernel ##
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If you start with integral Cartesian coordinates,
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many geometric computations will involve integral numerical values
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@ -274,7 +274,7 @@ geometric objects of the kernel that you would like to use with the
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representation class, i.e., `CGAL/Cartesian_d.h` or
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`CGAL/Homogeneous_d.h`
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# Kernel Geometry #
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# %Kernel Geometry #
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## Points and Vectors ##
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@ -317,7 +317,7 @@ Note that these constructions do not involve any performance overhead
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for the conversion with the currently available representation
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classes.
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## Kernel Objects ##
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## %Kernel Objects ##
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Besides points (`Point_d<R>`), vectors (`Vector_d<R>`), and
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directions (`Direction_d<R>`), \cgal provides lines, rays,
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@ -384,7 +384,7 @@ objects (also called functors and provided by a kernel class).
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\cgal provides predicates for the orientation of
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point sets (`orientation`), for comparing points according to some
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given order, especially for comparing Cartesian coordinates
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given order, especially for comparing %Cartesian coordinates
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(e.g. `lexicographically_xy_smaller`), in-sphere tests, and
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predicates to compare distances.
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