songsenand
/
cgal
mirror of https://github.com/CGAL/cgal
1
0
Fork 0
Code Issues Packages Projects Releases Wiki Activity

Documentation

This commit is contained in:
Michael Kerber 2008-07-11 16:49:53 +00:00
parent d3d6a5c1eb
commit ca28e10eea
1 changed files with 283 additions and 146 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

429
Algebraic_kernel_d/include/CGAL/Algebraic_curve_kernel_2.h
View File

@ -9,13 +9,14 @@
//
// Author(s) : Eric Berberich <eric@mpi-inf.mpg.de>
// Pavel Emeliyanenko <asm@mpi-sb.mpg.de>
// Michael Kerber <mkerber@mpi-inf.mpg.de>
//
// ============================================================================
/*! \file Algebraic_curve_kernel_2.h
* \brief defines class \c Algebraic_curve_kernel_2
*
* Curve and curve pair analysis for algebraic plane curves
*
* A model for CGAL's AlgebraicKernelWithAnalysis_d_2 concept
*/
#ifndef CGAL_ALGEBRAIC_CURVE_KERNEL_2_H
@ -39,10 +40,43 @@
CGAL_BEGIN_NAMESPACE
/*!
* \b Algebraic_curve_kernel_2 is a model of CGAL's concept \c
* AlgebraicKernelWithAnalysis_d_2 which itself refines \c AlgebraicKernel_d_2.
* As such, it contains functionality
* for solving and manipulating (systems of) bivariate polynomials,
* of arbitrary degree,
* as required by the \c AlgebraicKernel_d_2 concept.
* Additionally, it contains functionality for the topological-geometric
* analysis of a single algebraic curve
* (given as the vanishing set of the polynomial),
* and of a pair of curves (given as a pair of polynomials), as required by the
* \c AlgebraicKernelWithAnalysis_d_2 concept. These two analyses are
* available via the types \c Curve_analysis_2 and Curve_pair_analysis_2.
*
* The given class is also a model of the \c CurveKernel_2 concept that is
* in turn required by the \c CurvedKernelViaAnalysis_2 concept
* (see the documentation of the corresponding package). Therefore,
* some types and methods of the class have both an "algebraic" name
* (demanded by \c CurveKernelWithAnalysis_d_2) and an "non-algebraic name
* (demanded by \c CurveKernel_2).
*
* \b Algebraic_curve_kernel_2 is a template class, and needs a model
* of the \c AlgebraicKernel_d_1 concept as parameter.
*
* Internally, the curve- and curve-pair analysis
* are the computational fundament of the kernel. That means, whenever
* a polynomial is considered within the kernel, the curve analysis
* of the corresponding algebraic curve is performed.
* The same holds for the curve pair analysis,
* when a kernel function deals with two polynomials,
* implicitly or explicitly (e.g. \c Solve_2, \c Sign_at_2).
*/
#if CGAL_ACK_USE_EXACUS
template < class AlgebraicCurvePair_2, class AlgebraicKernel_1 >
template < class AlgebraicCurvePair_2, class AlgebraicKernel_d_1 >
#else
template < class AlgebraicKernel_1 >
template < class AlgebraicKernel_d_1 >
#endif
class Algebraic_curve_kernel_2 {
@ -62,9 +96,7 @@ public:
//!@{
//! type of 1D algebraic kernel
typedef AlgebraicKernel_1 Algebraic_kernel_1;
//! type of X_coordinate
typedef AlgebraicKernel_d_1 Algebraic_kernel_1;
#if CGAL_ACK_USE_EXACUS
// type of an internal curve pair
@ -74,29 +106,31 @@ public:
typedef typename AlgebraicCurvePair_2::Algebraic_curve_2 Internal_curve_2;
#endif
//! type of x-coordinate
#if CGAL_ACK_USE_EXACUS
typedef typename Internal_curve_2::X_coordinate X_coordinate_1;
#else
typedef typename Algebraic_kernel_1::Algebraic_real_1 X_coordinate_1;
#endif
//! type of y-coordinate
typedef X_coordinate_1 Y_coordinate_1;
//! type of coefficient
//! type of polynomial coefficient
typedef typename Algebraic_kernel_1::Coefficient Coefficient;
//! myself
// myself
#if CGAL_ACK_USE_EXACUS
typedef Algebraic_curve_kernel_2<AlgebraicCurvePair_2, AlgebraicKernel_1>
typedef Algebraic_curve_kernel_2<AlgebraicCurvePair_2, AlgebraicKernel_d_1>
Self;
#else
typedef Algebraic_curve_kernel_2<AlgebraicKernel_1> Self;
typedef Algebraic_curve_kernel_2<AlgebraicKernel_d_1> Self;
#endif
// Boundary type
typedef typename Algebraic_kernel_1::Boundary Boundary;
//! new CGAL univariate polynomial type
//! CGAL univariate polynomial type
typedef ::CGAL::Polynomial<Coefficient> Polynomial_1;
//! new CGAL bivariate polynomial type
@ -106,27 +140,45 @@ public:
typedef ::CGAL::Polynomial_traits_d< Polynomial_2 >
Polynomial_traits_2;
typedef typename Polynomial_traits_2::
Innermost_coefficient Innermost_coefficient;
//! type of a curve point
/*!
* \brief type of a curve point, a model for the
* \c AlgebraicKernel_d_2::AlgebraicReal_2 concept
*/
typedef CGALi::Xy_coordinate_2<Self> Xy_coordinate_2;
//! type of 1-curve analysis
/*!
* type of the curve analysis, a model for the
* \c AlgebraicKernelWithAnalysis_d_2::CurveAnalysis_2 concept
*/
#if CGAL_ACK_USE_EXACUS
typedef CGALi::Curve_analysis_2<Self> Curve_analysis_2;
#else
typedef Curve_analysis_2<Self> Curve_analysis_2;
#endif
//! type of 2-curve analysis
/*!
* type of the curve pair analysis, a model for the
* \c AlgebraicKernelWithAnalysis_d_2::CurvePairAnalysis_2 concept
*/
#if CGAL_ACK_USE_EXACUS
typedef CGALi::Curve_pair_analysis_2<Self> Curve_pair_analysis_2;
#else
typedef Curve_pair_analysis_2<Self> Curve_pair_analysis_2;
#endif
//! berfriending representations to make protected typedefs available
//! traits class used for approximations of x-coordinate
typedef CGALi::Algebraic_real_traits<X_coordinate_1> X_real_traits_1;
//! traits class used for approximations of y-coordinates
#if CGAL_ACK_USE_EXACUS
typedef CGALi::Algebraic_real_traits_for_y
<Xy_coordinate_2,Internal_curve_pair_2> Y_real_traits_1;
#else
typedef CGALi::Algebraic_real_traits_for_y
<Xy_coordinate_2,CGAL::Null_functor> Y_real_traits_1;
#endif
// berfriending representations to make protected typedefs available
friend class CGALi::Curve_analysis_2_rep<Self>;
friend class CGALi::Curve_pair_analysis_2_rep<Self>;
@ -151,7 +203,7 @@ protected:
//! \name private functors
//!@{
//! polynomial canonicalizer
//! polynomial canonicalizer, needed for the cache
template <class Poly>
struct Poly_canonicalizer : public Unary_function< Poly, Poly >
{
@ -177,7 +229,7 @@ protected:
};
// to remove a confusion with Curve_pair_2
// NOT a curve pair in our notation, simply a std::pair of Curve_analysis_2
typedef std::pair<Curve_analysis_2, Curve_analysis_2> Pair_of_curves_2;
//! orders pair items by ids
@ -215,9 +267,8 @@ protected:
typedef CGALi::LRU_hashed_map<Polynomial_2,
Curve_analysis_2, CGALi::Poly_hasher,
std::equal_to<Polynomial_2>,
Poly_canonicalizer<Polynomial_2> > Curve_cache;
Poly_canonicalizer<Polynomial_2> > Curve_cache_2;
//CGALi::Curve_pair_hasher_2<Curve_analysis_2>,
//! type of curve pair analysis cache
typedef CGALi::LRU_hashed_map<Pair_of_curves_2,
Curve_pair_analysis_2, CGALi::Pair_hasher, Pair_id_equal_to,
@ -250,6 +301,7 @@ protected:
std::less<Polynomial_2>,
std::less<Polynomial_2> > Polynomial_2_compare;
//! Cache for gcd computations
typedef CGAL::Cache<Pair_of_polynomial_2,
Polynomial_2,
Gcd<Polynomial_2>,
@ -257,24 +309,22 @@ protected:
Polynomial_2_compare> Gcd_cache_2;
//!@}
protected:
//!\name protected methods and data types
public:
//!\name cache access functions
//!@{
//! access to the static gcd_cache
static Gcd_cache_2& gcd_cache_2() {
static Gcd_cache_2 cache;
return cache;
}
//!@}
public:
//!\name cache access functions
//!@{
//! access to the static curve cache
static Curve_cache& curve_cache()
static Curve_cache_2& curve_cache_2()
{
static Curve_cache _m_curve_cache;
return _m_curve_cache;
static Curve_cache_2 _m_curve_cache_2;
return _m_curve_cache_2;
}
//! access to the static curve pair cache
@ -290,7 +340,7 @@ public:
//! \brief default constructor
Algebraic_curve_kernel_2() //: _m_curve_cache()
Algebraic_curve_kernel_2() //: _m_curve_cache_2()
{ }
/*! \brief
@ -302,14 +352,14 @@ public:
Curve_analysis_2 operator()
(const Polynomial_2& f) const {
return Self::curve_cache()(f);
return Self::curve_cache_2()(f);
}
};
CGAL_Algebraic_Kernel_cons(Construct_curve_2, construct_curve_2_object);
/*! \brief
* constructs \c Curve_pair_analysis_2 from pair of 1-curve analysis,
* constructs \c Curve_pair_analysis_2 from pair of one curve analyses,
* caching is used when appropriate
*/
struct Construct_curve_pair_2 :
@ -326,20 +376,8 @@ public:
};
CGAL_Algebraic_Kernel_cons(Construct_curve_pair_2,
construct_curve_pair_2_object);
//! traits class for \c X_coordinate
typedef CGALi::Algebraic_real_traits<X_coordinate_1> X_real_traits_1;
//! traits class for \c Xy_coorinate_2
#if CGAL_ACK_USE_EXACUS
typedef CGALi::Algebraic_real_traits_for_y
<Xy_coordinate_2,Internal_curve_pair_2> Y_real_traits_1;
#else
typedef CGALi::Algebraic_real_traits_for_y
<Xy_coordinate_2,CGAL::Null_functor> Y_real_traits_1;
#endif
//! returns the first coordinate of \c Xy_coordinate_2
//! returns the x-coordinate of an \c Xy_coordinate_2 object
struct Get_x_2 :
public Unary_function<Xy_coordinate_2, X_coordinate_1> {
@ -349,7 +387,18 @@ public:
};
CGAL_Algebraic_Kernel_cons(Get_x_2, get_x_2_object);
//! returns the second coordinate of \c Xy_coordinate_2
/*!
* \brief returns the y-coordinate of \c Xy_coordinate_2 object
*
* \attention{This method returns the y-coordinate in isolating interval
* representation. Calculating such a representation is usually a time-
* consuming taks, since it is against the "y-per-x"-view that we take
* in our kernel. Therefore, it is recommended, if possible,
* to use the functors
* \c Lower_boundary_y_2 and \c Upper_boundary_y_2 instead that
* return approximation of the y-coordinate. The approximation can be
* made arbitrarily good by iteratively calling \c Refine_y_2.}
*/
struct Get_y_2 :
public Unary_function<Xy_coordinate_2, X_coordinate_1> {
@ -359,55 +408,67 @@ public:
};
CGAL_Algebraic_Kernel_cons(Get_y_2, get_y_2_object);
//! Refines the x-coordinate of an Xy_coordinate_2 object
struct Refine_x_2 :
public Unary_function<Xy_coordinate_2, void> {
//! \brief returns at least half's the current interval of the first
//! coordinate of \c r
//!
//! note that an interval may also degenerate to a single point
/*!
* \brief Refines the approximation of the x-coordinate by at least
* a factor 2 (i.e., the isolating intervals has at most half its
* original size).
*
* note that an interval may also collaps to a single point
*/
void operator()(const Xy_coordinate_2& r) const {
r.refine_x();
}
//! \brief refines the current interval of the first coordinate of \c r
//! w.r.t. given relative precision
//!
//! that is:
//! <tt>|lower - upper|/|r.x()| <= 2^(-rel_prec)</tt>
/*!
* \brief refines the x-coordinate's interval of \c r
* w.r.t. given relative precision
*
* that is:
* <tt>|lower - upper|/|r.x()| <= 2^(-rel_prec)</tt>
*/
void operator()(Xy_coordinate_2& r, int rel_prec) const {
r.refine_x(rel_prec);
}
};
CGAL_Algebraic_Kernel_pred(Refine_x_2, refine_x_2_object);
//! Refines the y-coordinate of an Xy_coordinate_2 object
struct Refine_y_2 :
public Unary_function<Xy_coordinate_2, void> {
//! \brief returns at least half's the current interval of the second
//! coordinate of \c r
//!
//! note that an interval may also degenerate to a single point
/*!
* \brief Refines the approximation of the y-coordinate by at least
* a factor 2 (i.e., the isolating intervals has at most half its
* original size).
*
* note that an interval may also collaps to a single point
*/
void operator()(const Xy_coordinate_2& r) const {
typename Y_real_traits_1::Refine()(r);
}
//! \brief refines the current interval of the second coordinate of
//! \c r w.r.t. given relative precision
//!
//! that is:
//! <tt>|lower - upper|/|r.y()| <= 2^(-rel_prec)</tt>
/*!
* \brief refines the x-coordinate's interval of \c r
* w.r.t. given relative precision
*
* that is:
* <tt>|lower - upper|/|r.x()| <= 2^(-rel_prec)</tt>
*/
void operator()(Xy_coordinate_2& r, int rel_prec) const {
typename Y_real_traits_1::Refine()(r, rel_prec);
}
};
CGAL_Algebraic_Kernel_pred(Refine_y_2, refine_y_2_object);
//! computes the current lower boundary of the first coordinate of \c r
//! a lower boundary of the x-coordinate of \c r
struct Lower_boundary_x_2 {
typedef Xy_coordinate_2 agrument_type;
typedef typename Algebraic_kernel_1::Boundary result_type;
typedef Xy_coordinate_2 argument_type;
typedef Boundary result_type;
result_type operator()(const Xy_coordinate_2& r) {
return typename X_real_traits_1::Lower_boundary()(r.x());
@ -415,11 +476,11 @@ public:
};
CGAL_Algebraic_Kernel_cons(Lower_boundary_x_2, lower_boundary_x_2_object);
//! computes the current upper boundary of the first coordinate of \c r
//! an upper boundary of the x-coordinate of \c r
struct Upper_boundary_x_2 {
typedef Xy_coordinate_2 agrument_type;
typedef typename Algebraic_kernel_1::Boundary result_type;
typedef Boundary result_type;
result_type operator()(const Xy_coordinate_2& r) {
return typename X_real_traits_1::Upper_boundary()(r.x());
@ -427,11 +488,11 @@ public:
};
CGAL_Algebraic_Kernel_cons(Upper_boundary_x_2, upper_boundary_x_2_object);
//! computes the current lower boundary of the second coordinate of \c r
//! a lower boundary of the x-coordinate of \c r
struct Lower_boundary_y_2 {
typedef Xy_coordinate_2 agrument_type;
typedef typename Algebraic_kernel_1::Boundary result_type;
typedef Boundary result_type;
result_type operator()(const Xy_coordinate_2& r) {
return typename Y_real_traits_1::Lower_boundary()(r);
@ -439,11 +500,11 @@ public:
};
CGAL_Algebraic_Kernel_cons(Lower_boundary_y_2, lower_boundary_y_2_object);
//! computes the current lower boundary of the second coordinate of \c r
//! an upper boundary of the y-coordinate of \c r
struct Upper_boundary_y_2 {
typedef Xy_coordinate_2 agrument_type;
typedef typename Algebraic_kernel_1::Boundary result_type;
typedef Boundary result_type;
result_type operator()(const Xy_coordinate_2& r) {
return typename Y_real_traits_1::Upper_boundary()(r);
@ -451,13 +512,17 @@ public:
};
CGAL_Algebraic_Kernel_cons(Upper_boundary_y_2, upper_boundary_y_2_object);
//! returns the number of boundary type in-between x-coordinates of two
//! Xy_coordinate_2 objects
/*!
* \brief returns a value of type \c Boundary that lies between
* the x-coordinates of the \c Xy_coordinate_2s.
*
* \pre{The x-coordinates must not be equal}
*/
struct Boundary_between_x_2 {
typedef Xy_coordinate_2 first_agrument_type;
typedef Xy_coordinate_2 second_agrument_type;
typedef typename Algebraic_kernel_1::Boundary result_type;
typedef Boundary result_type;
result_type operator()(const Xy_coordinate_2& r1,
const Xy_coordinate_2& r2) const {
@ -468,13 +533,17 @@ public:
CGAL_Algebraic_Kernel_cons(Boundary_between_x_2,
boundary_between_x_2_object);
//! returns the number of boundary type in-between y-coordinates of two
//! Xy_coordinate_2 objects
/*!
* \brief returns a value of type \c Boundary that lies between
* the y-coordinates of the \c Xy_coordinate_2s.
*
* \pre{The y-coordinates must not be equal}
*/
struct Boundary_between_y_2 {
typedef Xy_coordinate_2 first_agrument_type;
typedef Xy_coordinate_2 second_agrument_type;
typedef typename Algebraic_kernel_1::Boundary result_type;
typedef Boundary result_type;
result_type operator()(const Xy_coordinate_2& r1,
const Xy_coordinate_2& r2) const {
@ -494,13 +563,21 @@ public:
return x1.compare(x2);
}
Comparison_result operator()(const Xy_coordinate_2& xy1,
const Xy_coordinate_2& xy2) const {
const Xy_coordinate_2& xy2) const {
return (*this)(xy1.x(), xy2.x());
}
};
CGAL_Algebraic_Kernel_pred(Compare_x_2, compare_x_2_object);
//! \brief comparison of y-coordinates of two points
/*!
* \brief comparison of y-coordinates of two points
*
* \attention{If both points have different x-coordinates, this method
* has to translate both y-coordinates
* into isolating interval representations which is a time-consuming
* operation (compare the documentation of the \c Get_y_2 functor)
* If possible, it is recommended to avoid this functor for efficiency.}
*/
struct Compare_y_2 :
public Binary_function< Xy_coordinate_2, Xy_coordinate_2,
Comparison_result > {
@ -529,9 +606,12 @@ public:
return Compare_y_2((Self *)this);
}
//! lexicographical comparison of two objects of type \c Xy_coordinate_2
//!
//! \c equal_x specifies that only y-coordinates need to be compared
/*!
* \brief lexicographical comparison of two \c Xy_coordinate_2 objects
*
* \param equal_x if set, the points are assumed
* to have equal x-coordinates, thus only the y-coordinates are compared.
*/
struct Compare_xy_2 :
public Binary_function<Xy_coordinate_2, Xy_coordinate_2,
Comparison_result > {
@ -577,30 +657,33 @@ public:
return Compare_xy_2((Self *)this);
}
//! \brief checks whether curve has only finitely many self-intersection
//! points, i.e., it has no self-overlapped continuous parts
//!
//! for algerbaic curves this means that supporting polynomial is
//! square-free
/*!
* \brief checks whether the curve induced by \c p
* has only finitely many self-intersection points
*
* In algebraic terms, it is checked whether
* the polynomial \c p is square free.
*/
struct Has_finite_number_of_self_intersections_2 :
public Unary_function< Polynomial_2, bool > {
bool operator()(const Polynomial_2& p) const {
//typename Polynomial_traits_2::Is_square_free is_square_free;
CGAL_error_msg("is_square_free is not yet supported\n");
return true; //is_square_free(p);
typename Polynomial_traits_2::Is_square_free is_square_free;
return is_square_free(p);
}
};
CGAL_Algebraic_Kernel_pred(Has_finite_number_of_self_intersections_2,
has_finite_number_of_self_intersections_2_object);
//! \brief checks whether a curve pair has finitely many intersections,
//! in other words, whether two curves have no continuous common part
//!
//! in case of algerbaic curves: checks whether supporting polynomials are
//! coprime
/*!
* \brief checks whether two curves induced bt \c f and \c g
* habe finitely many intersections.
*
* In algebraic terms, it is checked whether
* the two polynomials \c f and \c g are coprime.
*/
struct Has_finite_number_of_intersections_2 :
public Binary_function< Polynomial_2, Polynomial_2, bool > {
@ -617,28 +700,27 @@ public:
CGAL_Algebraic_Kernel_pred(Has_finite_number_of_intersections_2,
has_finite_number_of_intersections_2_object);
//! set of various curve and curve pair decomposition functions
//! Various curve and curve pair decomposition functions
struct Decompose_2 {
//! default constructor
Decompose_2(/*Self *pkernel_2*/)
{ }
//! \brief returns a curve without self-overlapping parts
//!
//! in case of algebraic curves computes square-free part of supporting
//! polynomial
//! returns the square free part of the curve induced by \c p
Polynomial_2 operator()(const Polynomial_2& p) {
typename Polynomial_traits_2::Make_square_free msf;
return msf(p);
}
//! \brief computes a square-free factorization of a curve \c c,
//! returns the number of pairwise coprime square-free factors
//!
//! returns square-free pairwise coprime factors in \c fit and
//! multiplicities in \c mit. Template argument type of \c fit is
//! \c Curve_analysis_2, and \c mit is \c int
/*!
* \brief computes a square-free factorization of a curve \c c,
* returns the number of pairwise coprime square-free factors
*
* returns square-free pairwise coprime factors in \c fit and
* multiplicities in \c mit. The value type of \c fit is
* \c Curve_analysis_2, the value type of \c mit is \c int
*/
template< class OutputIterator1, class OutputIterator2 >
int operator()(const Curve_analysis_2& ca,
OutputIterator1 fit, OutputIterator2 mit ) const {
@ -657,12 +739,17 @@ public:
}
/*!\brief
* computes for a given pair of curves \c ca1 and \c ca2 their
* common part \c oib and coprime parts \c oi1 and \c oi2
* respectively; returns \c true if the curves were decomposed
* Decomposes two curves \c ca1 and \c ca2 into common part
* and coprime parts
*
* returns true if \c ca1 and \c ca2 are coprime. Template argument
* type of \c oi{1,2,b} is \c Curve_analysis_2
* The common part of the curves \c ca1 and \c ca2 is written in
* \c oib, the coprime parts are written to \c oi1 and \c oi2,
* respectively.
*
* \return {true, if the two curves were not coprime (i.e., have a
* non-trivial common part}
*
* The value type of \c oi{1,2,b} is \c Curve_analysis_2
*/
template < class OutputIterator >
bool operator()(const Curve_analysis_2& ca1,
@ -709,7 +796,7 @@ public:
const Polynomial_2& g = ca2.polynomial_2();
if(f == g) {
std::cout << "WARNING, both curves are equal, but have different representations!" << std::endl;
// both curves are equal, but have different representations!
CGAL_assertion(false);
return false;
}
@ -759,17 +846,29 @@ public:
//! \name types and functors for \c CurvedKernelViaAnalysis_2
//!@{
//! Algebraic name
typedef X_coordinate_1 Algebraic_real_1;
//! Algebraic name
typedef Xy_coordinate_2 Algebraic_real_2;
//! Algebraic name
typedef Has_finite_number_of_self_intersections_2 Is_square_free_2;
//! Algebraic name
typedef Has_finite_number_of_intersections_2 Is_coprime_2;
//! Algebraic name
typedef Decompose_2 Make_square_free_2;
//! Algebraic name
typedef Decompose_2 Square_free_factorization;
//! Algebraic name
typedef Decompose_2 Make_coprime_2;
//! \brief computes the derivative w.r.t. the first (innermost) variable
//! computes the derivative w.r.t. x
struct Derivative_x_2 :
public Unary_function< Polynomial_2, Polynomial_2 > {
@ -781,7 +880,7 @@ public:
};
CGAL_Algebraic_Kernel_cons(Derivative_x_2, derivative_x_2_object);
//! \brief computes the derivative w.r.t. the first (outermost) variable
//! \brief computes the derivative w.r.t. y
struct Derivative_y_2 :
public Unary_function< Polynomial_2, Polynomial_2 > {
@ -793,16 +892,23 @@ public:
};
CGAL_Algebraic_Kernel_cons(Derivative_y_2, derivative_y_2_object);
/*!
* \brief computes the x-critical points of of a curve/a polynomial
*
* An x-critical point (x,y) of \c f (or its induced curve)
* satisfies f(x,y) = f_y(x,y) = 0,
* where f_y means the derivative w.r.t. y.
* In pariticular, each singular point is x-critical.
*/
struct X_critical_points_2 :
public Binary_function< Curve_analysis_2,
std::iterator<std::output_iterator_tag, Xy_coordinate_2>,
std::iterator<std::output_iterator_tag, Xy_coordinate_2> > {
//! \brief copies in the output iterator the x-critical points of
//! polynomial \c p as objects of type \c Xy_coordinate_2
//!
//! all points (x, y) with f(x,y) = fy(x,y) = 0 are x-critical points
//! (i.e, singularities are also counted)
/*!
* \brief writes the x-critical points of \c ca_2 into \c oi
*/
template <class OutputIterator>
OutputIterator operator()(const Curve_analysis_2& ca_2,
OutputIterator oi) const {
@ -844,7 +950,7 @@ public:
return oi;
}
//! \brief computes the ith x-critical point of polynomial \c p
//! \brief computes the \c i-th x-critical point of \c ca
Xy_coordinate_2 operator()(const Curve_analysis_2& ca, int i) const
{
std::vector<Xy_coordinate_2> x_points;
@ -856,13 +962,22 @@ public:
CGAL_Algebraic_Kernel_cons(X_critical_points_2,
x_critical_points_2_object);
/*!
* \brief computes the y-critical points of of a curve/a polynomial
*
* An y-critical point (x,y) of \c f (or its induced curve)
* satisfies f(x,y) = f_x(x,y) = 0,
* where f_x means the derivative w.r.t. x.
* In pariticular, each singular point is y-critical.
*/
struct Y_critical_points_2 :
public Binary_function< Curve_analysis_2,
std::iterator<std::output_iterator_tag, Xy_coordinate_2>,
std::iterator<std::output_iterator_tag, Xy_coordinate_2> > {
//! \brief copies in the output iterator the y-critical points of
//! polynomial \c p as objects of type \c Xy_coordinate_2
/*!
* \brief writes the y-critical points of \c ca_2 into \c oi
*/
template <class OutputIterator>
OutputIterator operator()(const Curve_analysis_2& ca_2,
OutputIterator oi) const
@ -916,8 +1031,8 @@ public:
}
return oi;
}
//! \brief computes the ith y-critical point of polynomial \c p
//! \brief computes the \c i-th x-critical point of \c ca
Xy_coordinate_2 operator()(const Curve_analysis_2& ca, int i) const
{
std::vector<Xy_coordinate_2> y_points;
@ -929,11 +1044,12 @@ public:
CGAL_Algebraic_Kernel_cons(Y_critical_points_2,
y_critical_points_2_object);
/*!\brief
* computes the sign of a bivariate polynomial \c p evaluated at the root
* \c r of a system of two bivariate polynomial equations
/*!
* \brief sign computation of a point and a curve
*
* returns a value convertible to \c CGAL::Sign
* computes the sign of a point \c p, evaluate at the polynomial
* that defines a curve \c c. If the result is 0, the point lies on the
* curve. Returns a value convertible to \c CGAL::Sign
*/
struct Sign_at_2 :
public Binary_function< Curve_analysis_2, Xy_coordinate_2, Sign > {
@ -944,8 +1060,14 @@ public:
typedef CGAL::Polynomial<Boundary> Poly_rat_1;
typedef CGAL::Polynomial<Poly_rat_1> Poly_rat_2;
Sign operator()(const Polynomial_2& f,
const Xy_coordinate_2& r) const {
return (*this)(Construct_curve_2()(f),r);
}
Sign operator()(const Curve_analysis_2& ca_2,
const Xy_coordinate_2& r) const {
const Xy_coordinate_2& r) const {
if(ca_2.is_identical(r.curve()) || _test_exact_zero(ca_2, r))
return CGAL::ZERO;
@ -1027,16 +1149,10 @@ public:
};
CGAL_Algebraic_Kernel_pred(Sign_at_2, sign_at_2_object);
/*!\brief
* copies in the output iterator \c roots the common roots of polynomials
* \c p1 and \c p2 and copies in the output iterator \c mults their
* respective multiplicity as intergers, in the same order
/*!
* \brief computes solutions of systems of two 2 equations and 2 variables
*
* template argument type of \c roots is \c Xy_coordinate_2 , returns the
* pair of respective past-the-end iterators
*
* \pre p1 and p2 are square-free and the set of solutions of the system
* is 0-dimensional
* \pre the polynomials must be square-free and coprime
*/
struct Solve_2 {
@ -1049,6 +1165,24 @@ public:
typedef std::pair<third_argument_type, fourth_argument_type>
result_type;
/*!
* \brief solves the system (f=0,g=0)
*
* All solutions of the system are written into \c roots
* (whose value type is \c Xy_coordinate_2). The multiplicities
* are written into \c mults (whose value type is \c int)
*/
template <class OutputIteratorRoots, class OutputIteratorMult>
std::pair<OutputIteratorRoots, OutputIteratorMult>
operator()
(const Polynomial_2& f, const Polynomial_2& g,
OutputIteratorRoots roots, OutputIteratorMult mults) const {
return
(*this)(Construct_curve_2()(f),Construct_curve_2()(g),
roots,mults);
}
//! Version with curve analyses
template <class OutputIteratorRoots, class OutputIteratorMult>
std::pair<OutputIteratorRoots, OutputIteratorMult>
operator()(const Curve_analysis_2& ca1, const Curve_analysis_2& ca2,
@ -1099,6 +1233,9 @@ public:
};
CGAL_Algebraic_Kernel_cons(Solve_2, solve_2_object);
/*!
* \brief Construct a curve with the roles of x and y interchanged.
*/
struct Swap_x_and_y_2 {
typedef Polynomial_2 argument_type;