#include <Concepts/AlgebraicKernel_d_2.h>

Definition

A model of the AlgebraicKernel_d_2 concept gathers necessary tools for solving and handling bivariate polynomial systems of general degree \( d\).

Refines:

AlgebraicKernel_d_1

CopyConstructible

Assignable

See also: AlgebraicKernel_d_1

Classes
class	ApproximateAbsoluteX_2
	A model of `AlgebraicKernel_d_2::ApproximateAbsoluteX_2` is an `AdaptableBinaryFunction` that computes an approximation of the \( x\)-coordinate of an `AlgebraicKernel_d_2::Algebraic_real_2` value with respect to a given absolute precision. More...

class	ApproximateAbsoluteY_2
	A model of `AlgebraicKernel_d_2::ApproximateAbsoluteY_2` is an `AdaptableBinaryFunction` that computes an approximation of the \( y\)-coordinate of an `AlgebraicKernel_d_2::Algebraic_real_2` value with respect to a given absolute precision. More...

class	ApproximateRelativeX_2
	A model of `AlgebraicKernel_d_2::ApproximateRelativeX_2` is an `AdaptableBinaryFunction` that computes an approximation of the \( x\)-coordinate of an `AlgebraicKernel_d_2::Algebraic_real_2` value with respect to a given relative precision. More...

class	ApproximateRelativeY_2
	A model of `AlgebraicKernel_d_2::ApproximateRelativeY_2` is an `AdaptableBinaryFunction` that computes an approximation of the \( y\)-coordinate of an `AlgebraicKernel_d_2::Algebraic_real_2` value with respect to a given relative precision. More...

class	BoundBetweenX_2
	Computes a number of type `AlgebraicKernel_d_1::Bound` in-between the first coordinates of two `AlgebraicKernel_d_2::AlgebraicReal_2`. More...

class	BoundBetweenY_2
	Computes a number of type `AlgebraicKernel_d_1::Bound` in-between the second coordinates of two `AlgebraicKernel_d_2::AlgebraicReal_2`. More...

class	CompareX_2
	Compares the first coordinates of `AlgebraicKernel_d_2::Algebraic_real_2`s. More...

class	CompareXY_2
	Compares `AlgebraicKernel_d_2::Algebraic_real_2`s lexicographically. More...

class	CompareY_2
	Compares the second coordinated of `AlgebraicKernel_d_2::Algebraic_real_2`s. More...

class	ComputePolynomialX_2
	Computes a univariate square free polynomial \( p\), such that the first coordinate of a given `AlgebraicKernel_d_2::Algebraic_real_2` is a real root of \( p\). More...

class	ComputePolynomialY_2
	Computes a univariate square free polynomial \( p\), such that the second coordinate of a given `AlgebraicKernel_d_2::Algebraic_real_2` is a real root of \( p\). More...

class	ComputeX_2
	Computes the first coordinate of an `AlgebraicKernel_d_2::AlgebraicReal_2`. More...

class	ComputeY_2
	Computes the second coordinate of an `AlgebraicKernel_d_2::AlgebraicReal_2`. More...

class	ConstructAlgebraicReal_2
	Constructs an `AlgebraicKernel_d_2::Algebraic_real_2`. More...

class	IsCoprime_2
	Computes whether a given pair of bivariate polynomials is coprime. More...

class	Isolate_2
	Computes an isolating box for a given `AlgebraicKernel_d_2::Algebraic_real_2`. More...

class	IsolateX_2
	Computes an isolating interval for the first coordinate of an `AlgebraicKernel_d_2::Algebraic_real_2` with respect to the real roots of a univariate polynomial. More...

class	IsolateY_2
	Computes an isolating interval for the second coordinate of an `AlgebraicKernel_d_2::Algebraic_real_2` with respect to the real roots of a univariate polynomial. More...

class	IsSquareFree_2
	Computes whether the given bivariate polynomial is square free. More...

class	IsZeroAt_2
	Computes whether an `AlgebraicKernel_d_2::Polynomial_2` is zero at a given `AlgebraicKernel_d_2::Algebraic_real_2`. More...

class	MakeCoprime_2
	Computes for a given pair of bivariate polynomials \( p_1\), \( p_2\) their common part \( g\) and coprime parts \( q_1\), \( q_2\) respectively. More...

class	MakeSquareFree_2
	Returns a square free part of a bivariate polynomial. More...

class	NumberOfSolutions_2
	Computes the number of real solutions of the given bivariate polynomial system. More...

class	SignAt_2
	Computes the sign of a bivariate polynomial `AlgebraicKernel_d_2::Polynomial_2` at a value of type `AlgebraicKernel_d_2::Algebraic_real_2`. More...

class	Solve_2
	Computes the real zero-dimensional solutions of a bivariate polynomial system. The multiplicity stored in the output iterator is the multiplicity in the system. More...

class	SquareFreeFactorize_2
	Computes a square free factorization of an `AlgebraicKernel_d_2::Polynomial_2`. More...

Types
typedef unspecified_type	Polynomial_2
	A bivariate polynomial that is a model of `Polynomial_d`, where `CGAL::Polynomial_traits_d<Polynomial_2>::Innermost_coefficient_type` is `AlgebraicKernel_d_1::Coefficient`. More...

typedef unspecified_type	Algebraic_real_2
	A type that is used to represent real solutions of bivariate zero dimensional polynomial systems. More...

Functors
typedef unspecified_type	Construct_algebraic_real_2
	A model of `AlgebraicKernel_d_2::ConstructAlgebraicReal_2`. More...

typedef unspecified_type	Compute_polynomial_x_2
	A model of `AlgebraicKernel_d_2::ComputePolynomialX_2`. More...

typedef unspecified_type	Compute_polynomial_y_2
	A model of `AlgebraicKernel_d_2::ComputePolynomialY_2`. More...

typedef unspecified_type	Isolate_2
	A model of `AlgebraicKernel_d_2::Isolate_2`. More...

typedef unspecified_type	Isolate_x_2
	A model of `AlgebraicKernel_d_2::IsolateX_2`. More...

typedef unspecified_type	Isolate_y_2
	A model of `AlgebraicKernel_d_2::IsolateY_2`. More...

typedef unspecified_type	Is_square_free_2
	A model of `AlgebraicKernel_d_2::IsSquareFree_2`. More...

typedef unspecified_type	Make_square_free_2
	A model of `AlgebraicKernel_d_2::MakeSquareFree_2`. More...

typedef unspecified_type	Square_free_factorize_2
	A model of `AlgebraicKernel_d_2::SquareFreeFactorize_2`. More...

typedef unspecified_type	Is_coprime_2
	A model of `AlgebraicKernel_d_2::IsCoprime_2`. More...

typedef unspecified_type	Make_coprime_2
	A model of `AlgebraicKernel_d_2::MakeCoprime_2`. More...

typedef unspecified_type	Solve_2
	A model of `AlgebraicKernel_d_2::Solve_2`. More...

typedef unspecified_type	Number_of_solutions_2
	A model of `AlgebraicKernel_d_2::NumberOfSolutions_2`. More...

typedef unspecified_type	Sign_at_2
	A model of `AlgebraicKernel_d_2::SignAt_2`. More...

typedef unspecified_type	Compare_x_2
	A model of `AlgebraicKernel_d_2::CompareX_2`. More...

typedef unspecified_type	Compare_y_2
	A model of `AlgebraicKernel_d_2::CompareY_2`. More...

typedef unspecified_type	Compare_xy_2
	A model of `AlgebraicKernel_d_2::CompareXY_2`. More...

typedef unspecified_type	Bound_between_x_2
	A model of `AlgebraicKernel_d_2::BoundBetweenX_2`. More...

typedef unspecified_type	Bound_between_y_2
	A model of `AlgebraicKernel_d_2::BoundBetweenY_2`. More...

typedef unspecified_type	Approximate_absolute_x_2
	A model of `AlgebraicKernel_d_2::ApproximateAbsoluteX_2`. More...

typedef unspecified_type	Approximate_absolute_y_2
	A model of `AlgebraicKernel_d_2::ApproximateAbsoluteY_2`. More...

typedef unspecified_type	Approximate_relative_x_2
	A model of `AlgebraicKernel_d_2::ApproximateRelativeX_2`. More...

typedef unspecified_type	Approximate_relative_y_2
	A model of `AlgebraicKernel_d_2::ApproximateRelativeY_2`. More...

Operations
For each of the function objects above, there must exist a member function that requires no arguments and returns an instance of that function object. The name of the member function is the uncapitalized name of the type returned with the suffix `_object` appended. For example, for the function object `AlgebraicKernel_d_2::Bound_betweenX_2` the following member function must exist:
AlgebraicKernel_d_2::Bound_between_x_2	bound_between_x_2_object () const