AlgebraicKernel_d_2 Concept Reference

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

Concepts
concept	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...
concept	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...
concept	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...
concept	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...
concept	BoundBetweenX_2
	Computes a number of type AlgebraicKernel_d_1::Bound in-between the first coordinates of two AlgebraicKernel_d_2::AlgebraicReal_2. More...
concept	BoundBetweenY_2
	Computes a number of type AlgebraicKernel_d_1::Bound in-between the second coordinates of two AlgebraicKernel_d_2::AlgebraicReal_2. More...
concept	CompareX_2
	Compares the first coordinates of AlgebraicKernel_d_2::Algebraic_real_2s. More...
concept	CompareXY_2
	Compares AlgebraicKernel_d_2::Algebraic_real_2s lexicographically. More...
concept	CompareY_2
	Compares the second coordinated of AlgebraicKernel_d_2::Algebraic_real_2s. More...
concept	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...
concept	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...
concept	ComputeX_2
	Computes the first coordinate of an AlgebraicKernel_d_2::AlgebraicReal_2. More...
concept	ComputeY_2
	Computes the second coordinate of an AlgebraicKernel_d_2::AlgebraicReal_2. More...
concept	ConstructAlgebraicReal_2
	Constructs an AlgebraicKernel_d_2::Algebraic_real_2. More...
concept	IsCoprime_2
	Computes whether a given pair of bivariate polynomials is coprime. More...
concept	Isolate_2
	Computes an isolating box for a given AlgebraicKernel_d_2::Algebraic_real_2. More...
concept	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...
concept	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...
concept	IsSquareFree_2
	Computes whether the given bivariate polynomial is square free. More...
concept	IsZeroAt_2
	Computes whether an AlgebraicKernel_d_2::Polynomial_2 is zero at a given AlgebraicKernel_d_2::Algebraic_real_2. More...
concept	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...
concept	MakeSquareFree_2
	Returns a square free part of a bivariate polynomial. More...
concept	NumberOfSolutions_2
	Computes the number of real solutions of the given bivariate polynomial system. More...
concept	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...
concept	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...
concept	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.
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.
typedef unspecified_type	Compute_polynomial_x_2
	A model of AlgebraicKernel_d_2::ComputePolynomialX_2.
typedef unspecified_type	Compute_polynomial_y_2
	A model of AlgebraicKernel_d_2::ComputePolynomialY_2.
typedef unspecified_type	Isolate_2
	A model of AlgebraicKernel_d_2::Isolate_2.
typedef unspecified_type	Isolate_x_2
	A model of AlgebraicKernel_d_2::IsolateX_2.
typedef unspecified_type	Isolate_y_2
	A model of AlgebraicKernel_d_2::IsolateY_2.
typedef unspecified_type	Is_square_free_2
	A model of AlgebraicKernel_d_2::IsSquareFree_2.
typedef unspecified_type	Make_square_free_2
	A model of AlgebraicKernel_d_2::MakeSquareFree_2.
typedef unspecified_type	Square_free_factorize_2
	A model of AlgebraicKernel_d_2::SquareFreeFactorize_2.
typedef unspecified_type	Is_coprime_2
	A model of AlgebraicKernel_d_2::IsCoprime_2.
typedef unspecified_type	Make_coprime_2
	A model of AlgebraicKernel_d_2::MakeCoprime_2.
typedef unspecified_type	Solve_2
	A model of AlgebraicKernel_d_2::Solve_2.
typedef unspecified_type	Number_of_solutions_2
	A model of AlgebraicKernel_d_2::NumberOfSolutions_2.
typedef unspecified_type	Sign_at_2
	A model of AlgebraicKernel_d_2::SignAt_2.
typedef unspecified_type	Compare_x_2
	A model of AlgebraicKernel_d_2::CompareX_2.
typedef unspecified_type	Compare_y_2
	A model of AlgebraicKernel_d_2::CompareY_2.
typedef unspecified_type	Compare_xy_2
	A model of AlgebraicKernel_d_2::CompareXY_2.
typedef unspecified_type	Bound_between_x_2
	A model of AlgebraicKernel_d_2::BoundBetweenX_2.
typedef unspecified_type	Bound_between_y_2
	A model of AlgebraicKernel_d_2::BoundBetweenY_2.
typedef unspecified_type	Approximate_absolute_x_2
	A model of AlgebraicKernel_d_2::ApproximateAbsoluteX_2.
typedef unspecified_type	Approximate_absolute_y_2
	A model of AlgebraicKernel_d_2::ApproximateAbsoluteY_2.
typedef unspecified_type	Approximate_relative_x_2
	A model of AlgebraicKernel_d_2::ApproximateRelativeX_2.
typedef unspecified_type	Approximate_relative_y_2
	A model of AlgebraicKernel_d_2::ApproximateRelativeY_2.

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

Member Typedef Documentation

Algebraic_real_2

typedef unspecified_type AlgebraicKernel_d_2::Algebraic_real_2

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

A model of DefaultConstructible, CopyConstructible and Assignable.