From 730f1b5682a0b7b83fcde799fd0b62768e8f643e Mon Sep 17 00:00:00 2001 From: Andreas Fabri Date: Tue, 25 Sep 2012 13:32:09 +0000 Subject: [PATCH] unlink --- Kernel_d/doc/Kernel_d/CGAL/Cartesian_d.h | 2 +- Kernel_d/doc/Kernel_d/CGAL/Homogeneous_d.h | 4 +-- .../doc/Kernel_d/CGAL/Kernel_d/Direction_d.h | 4 +-- .../doc/Kernel_d/CGAL/Kernel_d/Hyperplane_d.h | 2 +- Kernel_d/doc/Kernel_d/CGAL/Kernel_d/Point_d.h | 14 ++++---- .../doc/Kernel_d/CGAL/Kernel_d/Vector_d.h | 36 +++++++++---------- Kernel_d/doc/Kernel_d/CGAL/predicates_d.h | 10 +++--- .../Kernel--Compare_lexicographically_d.h | 4 +-- .../Concepts/Kernel--Component_accessor_d.h | 2 +- ...ernel--ConstructCartesianConstIterator_d.h | 4 +-- .../Concepts/Kernel--Less_coordinate_d.h | 4 +-- .../Kernel--Less_lexicographically_d.h | 2 +- ...ernel--Less_or_equal_lexicographically_d.h | 2 +- .../Kernel_d/Concepts/Kernel--Orientation_d.h | 2 +- Kernel_d/doc/Kernel_d/Concepts/Kernel_d.h | 2 +- Kernel_d/doc/Kernel_d/Kernel_d.txt | 34 +++++++++--------- 16 files changed, 64 insertions(+), 64 deletions(-) diff --git a/Kernel_d/doc/Kernel_d/CGAL/Cartesian_d.h b/Kernel_d/doc/Kernel_d/CGAL/Cartesian_d.h index d054bae5e71..353574586c8 100644 --- a/Kernel_d/doc/Kernel_d/CGAL/Cartesian_d.h +++ b/Kernel_d/doc/Kernel_d/CGAL/Cartesian_d.h @@ -4,7 +4,7 @@ namespace CGAL { /*! \ingroup PkgKernelDKernels -A model for `Kernel_d` that uses Cartesian coordinates to represent the +A model for `Kernel_d` that uses %Cartesian coordinates to represent the geometric objects. In order for `Cartesian_d` to model Euclidean geometry in \f$ E^d\f$ , for some mathematical field \f$ E\f$ (e.g., the rationals \f$\mathbb{Q}\f$ or the reals \f$\mathbb{R}\f$), the template parameter `FieldNumberType` diff --git a/Kernel_d/doc/Kernel_d/CGAL/Homogeneous_d.h b/Kernel_d/doc/Kernel_d/CGAL/Homogeneous_d.h index da6712d36cf..fd0f530e636 100644 --- a/Kernel_d/doc/Kernel_d/CGAL/Homogeneous_d.h +++ b/Kernel_d/doc/Kernel_d/CGAL/Homogeneous_d.h @@ -16,11 +16,11 @@ the kernel is only an approximation of Euclidean geometry. \models ::Kernel_d -\sa `CGAL::Cartesian_d` +\sa `CGAL::Cartesian_d` */ template< typename RingNumberType > -class Homogeneous { +class Homogeneous_d { public: /// @} diff --git a/Kernel_d/doc/Kernel_d/CGAL/Kernel_d/Direction_d.h b/Kernel_d/doc/Kernel_d/CGAL/Kernel_d/Direction_d.h index 54233eb4829..5aef8bbda03 100644 --- a/Kernel_d/doc/Kernel_d/CGAL/Kernel_d/Direction_d.h +++ b/Kernel_d/doc/Kernel_d/CGAL/Kernel_d/Direction_d.h @@ -7,10 +7,10 @@ A `Direction_d` is a vector in the \f$ d\f$-dimensional vector space where we forget about its length. We represent directions in \f$ d\f$-dimensional space as a tuple \f$ (h_0,\ldots,h_d)\f$ of variables of type `RT` which we call the homogeneous coordinates of the -direction. The coordinate \f$ h_d\f$ must be positive. The Cartesian +direction. The coordinate \f$ h_d\f$ must be positive. The %Cartesian coordinates of a direction are \f$ c_i = h_i/h_d\f$ for \f$ 0 \le i < d\f$, which are of type `FT`. Two directions are equal if their -Cartesian coordinates are positive multiples of each other. Directions +%Cartesian coordinates are positive multiples of each other. Directions are in one-to-one correspondence to points on the unit sphere. ### Downward compatibility ### diff --git a/Kernel_d/doc/Kernel_d/CGAL/Kernel_d/Hyperplane_d.h b/Kernel_d/doc/Kernel_d/CGAL/Kernel_d/Hyperplane_d.h index 9dfb910dc58..649bf0db65a 100644 --- a/Kernel_d/doc/Kernel_d/CGAL/Kernel_d/Hyperplane_d.h +++ b/Kernel_d/doc/Kernel_d/CGAL/Kernel_d/Hyperplane_d.h @@ -8,7 +8,7 @@ in \f$ d\f$ - dimensional space. A hyperplane \f$ h\f$ is represented by coefficients \f$ (c_0,c_1,\ldots,c_d)\f$ of type `RT`. At least one of \f$ c_0\f$ to \f$ c_{ d - 1 }\f$ must be non-zero. The plane equation is \f$ \sum_{ 0 \le i < d } c_i x_i + c_d = 0\f$, where \f$ x_0\f$ to \f$ x_{d-1}\f$ are -Cartesian point coordinates. For a particular \f$ x\f$ the sign of \f$ \sum_{ +%Cartesian point coordinates. For a particular \f$ x\f$ the sign of \f$ \sum_{ 0 \le i < d } c_i x_i + c_d\f$ determines the position of a point \f$ x\f$ with respect to the hyperplane (on the hyperplane, on the negative side, or on the positive side). diff --git a/Kernel_d/doc/Kernel_d/CGAL/Kernel_d/Point_d.h b/Kernel_d/doc/Kernel_d/CGAL/Kernel_d/Point_d.h index 30c337be049..db093a78e31 100644 --- a/Kernel_d/doc/Kernel_d/CGAL/Kernel_d/Point_d.h +++ b/Kernel_d/doc/Kernel_d/CGAL/Kernel_d/Point_d.h @@ -10,7 +10,7 @@ space in dimension \f$ d\f$. A point \f$ p = (p_0,\ldots,p_{ d - 1 })\f$ in h_i/h_d\f$, which is of type `FT`. The homogenizing coordinate \f$ h_d\f$ is positive. -We call \f$ p_i\f$, \f$ 0 \leq i < d\f$ the \f$ i\f$-th Cartesian coordinate and +We call \f$ p_i\f$, \f$ 0 \leq i < d\f$ the \f$ i\f$-th %Cartesian coordinate and \f$ h_i\f$, \f$ 0 \le i \le d\f$, the \f$ i\f$-th homogeneous coordinate. We call \f$ d\f$ the dimension of the point. @@ -43,7 +43,7 @@ typedef Hidden_type LA; /*! a read-only iterator for the -Cartesian coordinates. +%Cartesian coordinates. */ typedef Hidden_type Cartesian_const_iterator; @@ -73,7 +73,7 @@ Point_d(int d, Origin); /*! introduces a variable -`p` of type `Point_d` in dimension `d`. If `size [first,last) == d` this creates a point with Cartesian coordinates +`p` of type `Point_d` in dimension `d`. If `size [first,last) == d` this creates a point with %Cartesian coordinates `set [first,last)`. If `size [first,last) == d+1` the range specifies the homogeneous coordinates \f$ H = set [first,last) = (\pm h_0, \pm h_1, \ldots, \pm h_d)\f$ @@ -127,7 +127,7 @@ returns the dimension of `p`. int dimension() ; /*! -returns the \f$ i\f$-th Cartesian +returns the \f$ i\f$-th %Cartesian coordinate of `p`. \pre \f$ 0 \leq i < d\f$. @@ -135,7 +135,7 @@ coordinate of `p`. FT cartesian(int i) ; /*! -returns the \f$ i\f$-th Cartesian +returns the \f$ i\f$-th %Cartesian coordinate of `p`. \pre \f$ 0 \leq i < d\f$. @@ -152,14 +152,14 @@ RT homogeneous(int i) ; /*! returns an -iterator pointing to the zeroth Cartesian coordinate \f$ p_0\f$ of +iterator pointing to the zeroth %Cartesian coordinate \f$ p_0\f$ of `p`. */ Cartesian_const_iterator cartesian_begin() ; /*! returns an -iterator pointing beyond the last Cartesian coordinate of `p`. +iterator pointing beyond the last %Cartesian coordinate of `p`. */ Cartesian_const_iterator cartesian_end() ; diff --git a/Kernel_d/doc/Kernel_d/CGAL/Kernel_d/Vector_d.h b/Kernel_d/doc/Kernel_d/CGAL/Kernel_d/Vector_d.h index 30f9457316c..da42f0a371f 100644 --- a/Kernel_d/doc/Kernel_d/CGAL/Kernel_d/Vector_d.h +++ b/Kernel_d/doc/Kernel_d/CGAL/Kernel_d/Vector_d.h @@ -7,7 +7,7 @@ An instance of data type `Vector_d` is a vector of Euclidean space in dimension \f$ d\f$. A vector \f$ r = (r_0,\ldots,r_{ d - 1})\f$ can be represented in homogeneous coordinates \f$ (h_0,\ldots,h_d)\f$ of number type `RT`, such that \f$ r_i = h_i/h_d\f$ which is of type `FT`. We -call the \f$ r_i\f$'s the Cartesian coordinates of the vector. The +call the \f$ r_i\f$'s the %Cartesian coordinates of the vector. The homogenizing coordinate \f$ h_d\f$ is positive. This data type is meant for use in computational geometry. It realizes @@ -45,7 +45,7 @@ typedef Hidden_type LA; /*! a read-only iterator for the -Cartesian coordinates. +%Cartesian coordinates. */ typedef Hidden_type Cartesian_const_iterator; @@ -81,7 +81,7 @@ Vector_d(int d, Null_vector); /*! introduces a variable `v` of type `Vector_d` in dimension `d`. If -`size [first,last) == d` this creates a vector with Cartesian +`size [first,last) == d` this creates a vector with %Cartesian coordinates `set [first,last)`. If `size [first,last) == p+1` the range specifies the homogeneous coordinates \f$ H = set [first,last) = (\pm h_0, \pm h_1, \ldots, \pm h_d)\f$ where the sign chosen is the sign of \f$ h_d\f$. @@ -140,7 +140,7 @@ returns the dimension of `v`. int dimension() ; /*! -returns the \f$ i\f$-th Cartesian +returns the \f$ i\f$-th %Cartesian coordinate of `v`. \pre \f$ 0 \leq i < d\f$. @@ -148,7 +148,7 @@ coordinate of `v`. FT cartesian(int i) ; /*! -returns the \f$ i\f$-th Cartesian +returns the \f$ i\f$-th %Cartesian coordinate of `v`. \pre \f$ 0 \leq i < d\f$. @@ -171,13 +171,13 @@ FT squared_length() ; /*! returns an -iterator pointing to the zeroth Cartesian coordinate of `v`. +iterator pointing to the zeroth %Cartesian coordinate of `v`. */ Cartesian_const_iterator cartesian_begin() ; /*! returns an -iterator pointing beyond the last Cartesian coordinate of `v`. +iterator pointing beyond the last %Cartesian coordinate of `v`. */ Cartesian_const_iterator cartesian_end() ; @@ -215,50 +215,50 @@ Vector_d transform(const Aff_transformation_d& t) /*! multiplies all -Cartesian coordinates by `n`. +%Cartesian coordinates by `n`. */ Vector_d& operator*=(const RT& n) ; /*! multiplies all -Cartesian coordinates by `r`. +%Cartesian coordinates by `r`. */ Vector_d& operator*=(const FT& r) ; /*! returns the vector -with Cartesian coordinates \f$ v_i/n, 0 \leq i < d\f$. +with %Cartesian coordinates \f$ v_i/n, 0 \leq i < d\f$. */ Vector_d operator/(const RT& n) ; /*! returns the vector -with Cartesian coordinates \f$ v_i/r, 0 \leq i < d\f$. +with %Cartesian coordinates \f$ v_i/r, 0 \leq i < d\f$. */ Vector_d operator/(const FT& r) ; /*! divides all -Cartesian coordinates by `n`. +%Cartesian coordinates by `n`. */ Vector_d& operator/=(const RT& n) ; /*! divides all -Cartesian coordinates by `r`. +%Cartesian coordinates by `r`. */ Vector_d& operator/=(const FT& r) ; /*! inner product, i.e., \f$ \sum_{ 0 \le i < d } v_i w_i\f$, where \f$ v_i\f$ and \f$ w_i\f$ are the -Cartesian coordinates of \f$ v\f$ and \f$ w\f$ respectively. +%Cartesian coordinates of \f$ v\f$ and \f$ w\f$ respectively. */ FT operator* (const Vector_d& w) ; /*! returns the -vector with Cartesian coordinates \f$ v_i+w_i, 0 \leq i < d\f$. +vector with %Cartesian coordinates \f$ v_i+w_i, 0 \leq i < d\f$. */ Vector_d operator+(const Vector_d& w) ; @@ -270,7 +270,7 @@ Vector_d& operator+=(const Vector_d& w) ; /*! returns the -vector with Cartesian coordinates \f$ v_i-w_i, 0 \leq i < d\f$. +vector with %Cartesian coordinates \f$ v_i-w_i, 0 \leq i < d\f$. */ Vector_d operator-(const Vector_d& w) ; @@ -293,12 +293,12 @@ vector. bool is_zero() ; /*! -returns the vector with Cartesian coordinates \f$ n v_i\f$. +returns the vector with %Cartesian coordinates \f$ n v_i\f$. */ Vector_d operator*(const RT& n, const Vector_d& v); /*! -returns the vector with Cartesian coordinates \f$ r v_i, 0 \leq i < +returns the vector with %Cartesian coordinates \f$ r v_i, 0 \leq i < d\f$. */ Vector_d operator*(const FT& r, const Vector_d& v); diff --git a/Kernel_d/doc/Kernel_d/CGAL/predicates_d.h b/Kernel_d/doc/Kernel_d/CGAL/predicates_d.h index f1c0031205f..99bee1b9e33 100644 --- a/Kernel_d/doc/Kernel_d/CGAL/predicates_d.h +++ b/Kernel_d/doc/Kernel_d/CGAL/predicates_d.h @@ -31,9 +31,9 @@ affine_rank(ForwardIterator first, ForwardIterator last); /*! -Compares the Cartesian +Compares the %Cartesian coordinates of points `p` and `q` lexicographically -in ascending order of its Cartesian components `p[i]` and +in ascending order of its %Cartesian components `p[i]` and `q[i]` for \f$ i = 0,\ldots,d-1\f$. \pre `p.dimension() == q.dimension()`. @@ -89,7 +89,7 @@ const Point_d& p); returns `true` iff `p` is -lexicographically smaller than `q` with respect to Cartesian +lexicographically smaller than `q` with respect to %Cartesian lexicographic order of points. \pre `p.dimension() == q.dimension()`. @@ -102,7 +102,7 @@ Point_d& q); returns `true` iff \f$ p\f$ is -lexicographically smaller than \f$ q\f$ with respect to Cartesian +lexicographically smaller than \f$ q\f$ with respect to %Cartesian lexicographic order of points or equal to \f$ q\f$. \pre `p.dimension() == q.dimension()`. @@ -146,7 +146,7 @@ determines the orientation of the points of the tuple `A = tuple [first,last)` w 1 & 1 & 1 & 1 \\ A[0] & A[1] & \dots& A[d] \end{array} \right| \f] -where `A[i]` denotes the Cartesian coordinate vector of +where `A[i]` denotes the %Cartesian coordinate vector of the \f$ i\f$-th point in \f$ A\f$. \pre `size [first,last) == d+1` and `A[i].dimension() == d` \f$ \forall0 \leq i \leq d\f$. diff --git a/Kernel_d/doc/Kernel_d/Concepts/Kernel--Compare_lexicographically_d.h b/Kernel_d/doc/Kernel_d/Concepts/Kernel--Compare_lexicographically_d.h index a35bb7fe905..f06997c2a67 100644 --- a/Kernel_d/doc/Kernel_d/Concepts/Kernel--Compare_lexicographically_d.h +++ b/Kernel_d/doc/Kernel_d/Concepts/Kernel--Compare_lexicographically_d.h @@ -14,9 +14,9 @@ public: /// @{ /*! -Compares the Cartesian coordinates of +Compares the %Cartesian coordinates of points `p` and `q` lexicographically in ascending -order of its Cartesian components `p[i]` and `q[i]` for \f$ i = +order of its %Cartesian components `p[i]` and `q[i]` for \f$ i = 0,\ldots,d-1\f$. \pre The objects are of the same dimension. diff --git a/Kernel_d/doc/Kernel_d/Concepts/Kernel--Component_accessor_d.h b/Kernel_d/doc/Kernel_d/Concepts/Kernel--Component_accessor_d.h index 8912679c44a..2cc4da68d9f 100644 --- a/Kernel_d/doc/Kernel_d/Concepts/Kernel--Component_accessor_d.h +++ b/Kernel_d/doc/Kernel_d/Concepts/Kernel--Component_accessor_d.h @@ -28,7 +28,7 @@ Kernel_d::RT homogeneous(const Kernel_d::Point_d& p, int i); /*! -returns the ith Cartesian coordinate of \f$ p\f$. +returns the ith %Cartesian coordinate of \f$ p\f$. \pre `0 <= i < dimension(p)`. */ diff --git a/Kernel_d/doc/Kernel_d/Concepts/Kernel--ConstructCartesianConstIterator_d.h b/Kernel_d/doc/Kernel_d/Concepts/Kernel--ConstructCartesianConstIterator_d.h index d9ea3a0da9d..5457c1ad6a4 100644 --- a/Kernel_d/doc/Kernel_d/Concepts/Kernel--ConstructCartesianConstIterator_d.h +++ b/Kernel_d/doc/Kernel_d/Concepts/Kernel--ConstructCartesianConstIterator_d.h @@ -13,13 +13,13 @@ A model for this must provide: class Kernel_d::ConstructCartesianConstIterator_d { public: /*! -returns an iterator on the 0'th Cartesian coordinate of `p`. +returns an iterator on the 0'th %Cartesian coordinate of `p`. */ Kernel_d::Cartesian_const_iterator_d operator()(const Kernel_d::Point_d &p); /*! -returns the past the end iterator of the Cartesian coordinates of `p`. +returns the past the end iterator of the %Cartesian coordinates of `p`. */ Kernel_d::Cartesian_const_iterator_d operator()(const Kernel_d::Point_d &p, int); diff --git a/Kernel_d/doc/Kernel_d/Concepts/Kernel--Less_coordinate_d.h b/Kernel_d/doc/Kernel_d/Concepts/Kernel--Less_coordinate_d.h index ab26065a9af..4e6c66b6094 100644 --- a/Kernel_d/doc/Kernel_d/Concepts/Kernel--Less_coordinate_d.h +++ b/Kernel_d/doc/Kernel_d/Concepts/Kernel--Less_coordinate_d.h @@ -14,9 +14,9 @@ public: /// @{ /*! -returns `true` iff the \f$ i\f$th Cartesian coordinate +returns `true` iff the \f$ i\f$th %Cartesian coordinate of `p` is -smaller than the \f$ i\f$th Cartesian coordinate of `q`. +smaller than the \f$ i\f$th %Cartesian coordinate of `q`. \pre `p` and `q` have the same dimension. */ diff --git a/Kernel_d/doc/Kernel_d/Concepts/Kernel--Less_lexicographically_d.h b/Kernel_d/doc/Kernel_d/Concepts/Kernel--Less_lexicographically_d.h index 794dc433d7b..63e48d1b25e 100644 --- a/Kernel_d/doc/Kernel_d/Concepts/Kernel--Less_lexicographically_d.h +++ b/Kernel_d/doc/Kernel_d/Concepts/Kernel--Less_lexicographically_d.h @@ -15,7 +15,7 @@ public: /*! returns `true` iff `p` is -lexicographically smaller than `q` with respect to Cartesian +lexicographically smaller than `q` with respect to %Cartesian lexicographic order of points. \pre `p` and `q` have the same dimension. diff --git a/Kernel_d/doc/Kernel_d/Concepts/Kernel--Less_or_equal_lexicographically_d.h b/Kernel_d/doc/Kernel_d/Concepts/Kernel--Less_or_equal_lexicographically_d.h index 9605e646af6..3f03c3819cc 100644 --- a/Kernel_d/doc/Kernel_d/Concepts/Kernel--Less_or_equal_lexicographically_d.h +++ b/Kernel_d/doc/Kernel_d/Concepts/Kernel--Less_or_equal_lexicographically_d.h @@ -15,7 +15,7 @@ public: /*! returns `true` iff \f$ p\f$ is -lexicographically smaller than \f$ q\f$ with respect to Cartesian +lexicographically smaller than \f$ q\f$ with respect to %Cartesian lexicographic order of points or equal to \f$ q\f$. \pre `p` and `q` have the same dimension. diff --git a/Kernel_d/doc/Kernel_d/Concepts/Kernel--Orientation_d.h b/Kernel_d/doc/Kernel_d/Concepts/Kernel--Orientation_d.h index ea2cb92f437..863adc43463 100644 --- a/Kernel_d/doc/Kernel_d/Concepts/Kernel--Orientation_d.h +++ b/Kernel_d/doc/Kernel_d/Concepts/Kernel--Orientation_d.h @@ -21,7 +21,7 @@ determines the orientation of the points of the tuple 1 & 1 & 1 & 1 \\ A[0] & A[1] & \dots& A[d] \end{array} \right| \f] -where `A[i]` denotes the Cartesian coordinate vector of +where `A[i]` denotes the %Cartesian coordinate vector of the \f$ i\f$-th point in \f$ A\f$. \pre `size [first,last) == d+1` and `A[i].dimension() == d` \f$ \forall0 \leq i \leq d\f$. diff --git a/Kernel_d/doc/Kernel_d/Concepts/Kernel_d.h b/Kernel_d/doc/Kernel_d/Concepts/Kernel_d.h index 76086be5b9d..424e913c377 100644 --- a/Kernel_d/doc/Kernel_d/Concepts/Kernel_d.h +++ b/Kernel_d/doc/Kernel_d/Concepts/Kernel_d.h @@ -44,7 +44,7 @@ typedef Hidden_type RT; /*! a type that allows to iterate over -the Cartesian coordinates +the %Cartesian coordinates */ typedef Hidden_type Cartesian_const_iterator_d; diff --git a/Kernel_d/doc/Kernel_d/Kernel_d.txt b/Kernel_d/doc/Kernel_d/Kernel_d.txt index 709e9a83810..95a03a157fb 100644 --- a/Kernel_d/doc/Kernel_d/Kernel_d.txt +++ b/Kernel_d/doc/Kernel_d/Kernel_d.txt @@ -74,23 +74,23 @@ If we index the tuple as above then we require that Our object of study is the \f$ d\f$-dimensional affine Euclidean space, where \f$ d\f$ is a parameter of our geometry. Objects in that space are sets of points. A common way to represent the points is the use of -Cartesian coordinates, which assumes a reference +%Cartesian coordinates, which assumes a reference frame (an origin and \f$ d\f$ orthogonal axes). In that framework, a point is represented by a \f$ d\f$-tuple \f$ (c_0,c_1,\ldots,c_{d-1})\f$, and so are vectors in the underlying linear space. Each point is -represented uniquely by such Cartesian +represented uniquely by such %Cartesian coordinates. Another way to represent points is by homogeneous coordinates. In that framework, a point is represented by a \f$ (d+1)\f$-tuple \f$ (h_0,h_1,\ldots,h_d)\f$. Via the formulae \f$ c_i = h_i/h_d\f$, -the corresponding point with Cartesian coordinates +the corresponding point with %Cartesian coordinates \f$ (c_0,c_1,\ldots,c_{d-1})\f$ can be computed. Note that homogeneous coordinates are not unique. For \f$ \lambda\ne 0\f$, the tuples \f$ (h_0,h_1,\ldots,h_d)\f$ and \f$ (\lambda\cdot h_0,\lambda\cdot h_1,\ldots,\lambda\cdot h_d)\f$ represent the same point. -For a point with Cartesian coordinates +For a point with %Cartesian coordinates \f$ (c_0,c_1,\ldots,c_{d-1})\f$ a possible homogeneous representation is \f$ (c_0,c_1,\ldots,c_{d-1},1)\f$. @@ -114,11 +114,11 @@ all kernel objects `Kernel_object`, the types `CGAL::Kernel_object` and `R::Kernel_object` are identical. \cgal offers two families of concrete models for the concept -representation class, one based on the Cartesian +representation class, one based on the %Cartesian representation of points and one based on the homogeneous representation of points. The interface of the kernel objects is designed such that it works well with both -Cartesian and homogeneous representation, for +%Cartesian and homogeneous representation, for example, points have a constructor with a range of coordinates plus a common denominator (the \f$ d+1\f$ homogeneous coordinates of the point). The common interfaces parameterized with a representation class allow @@ -153,21 +153,21 @@ should fulfill \cgal's requirements on a number type. ## %Cartesian %Kernel ## With `Cartesian_d` you can choose -Cartesian representation of coordinates. The type +%Cartesian representation of coordinates. The type `LinearAlgebra` must me a linear algebra module working on numbers of type `FieldNumberType`. The second parameter defaults to module delivered with the kernel so for short a user can just write `Cartesian_d` when not providing her own linear algebra. -When you choose Cartesian representation you have +When you choose %Cartesian representation you have to declare at least the type of the coordinates. A number type used with the `Cartesian_d` representation class should be a field -type as described above. Both `Cartesian::FT` -and `Cartesian::RT` are mapped to number type +type as described above. Both `Cartesian_d::FT` +and `Cartesian_d::RT` are mapped to number type `FieldNumberType`. `Cartesian_d::LA` is mapped to the -type `LinearAlgebra`. `Cartesian` uses +type `LinearAlgebra`. `Cartesian_d` uses reference counting internally to save copying costs. ## %Homogeneous %Kernel ## @@ -178,7 +178,7 @@ coordinate can serve as a common denominator. Avoiding divisions can be useful for exact geometric computation. With `Homogeneous_d` you can choose homogeneous representation of coordinates with the kernel objects. -As for Cartesian representation you have to declare +As for %Cartesian representation you have to declare at the same time the type used to store the homogeneous coordinates. Since the homogeneous representation allows one to avoid the divisions, the number type associated with a homogeneous @@ -193,7 +193,7 @@ is no issue. However, some operations provided by this kernel involve division operations, for example computing squared distances or returning a -Cartesian coordinate. To keep the requirements on +%Cartesian coordinate. To keep the requirements on the number type parameter of `Homogeneous` low, the number type `Quotient` is used instead. This number type turns a ring type into a field type. It maintains numbers as @@ -237,7 +237,7 @@ not be used as a traits class. ## Choosing a %Kernel ## -If you start with integral Cartesian coordinates, +If you start with integral %Cartesian coordinates, many geometric computations will involve integral numerical values only. Especially, this is true for geometric computations that evaluate only predicates, which are tantamount to determinant @@ -253,12 +253,12 @@ type, incorrect results might arise due to overflow. If new points are to be constructed, for example the intersection point of two lines, computation of -Cartesian coordinates usually involves divisions, -so you need to use a field type with Cartesian +%Cartesian coordinates usually involves divisions, +so you need to use a field type with %Cartesian representation or have to switch to homogeneous representation. `double` is a possible, but imprecise field type. You can also put any ring type into `Quotient` to get a field type and put it -into `Cartesian`, but you better put the ring type into +into `Cartesian_d`, but you better put the ring type into `Homogeneous`. `leda_rational` and `leda_real` are valid field types, too.