\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 4.11.3 - Algebraic Kernel
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AlgebraicKernel_d_2 Concept Reference
Bivariate Algebraic Kernel

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

conceptApproximateAbsoluteX_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...
 
conceptApproximateAbsoluteY_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...
 
conceptApproximateRelativeX_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...
 
conceptApproximateRelativeY_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...
 
conceptBoundBetweenX_2
 Computes a number of type AlgebraicKernel_d_1::Bound in-between the first coordinates of two AlgebraicKernel_d_2::AlgebraicReal_2. More...
 
conceptBoundBetweenY_2
 Computes a number of type AlgebraicKernel_d_1::Bound in-between the second coordinates of two AlgebraicKernel_d_2::AlgebraicReal_2. More...
 
conceptCompareX_2
 Compares the first coordinates of AlgebraicKernel_d_2::Algebraic_real_2s. More...
 
conceptCompareXY_2
 Compares AlgebraicKernel_d_2::Algebraic_real_2s lexicographically. More...
 
conceptCompareY_2
 Compares the second coordinated of AlgebraicKernel_d_2::Algebraic_real_2s. More...
 
conceptComputePolynomialX_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...
 
conceptComputePolynomialY_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...
 
conceptComputeX_2
 Computes the first coordinate of an AlgebraicKernel_d_2::AlgebraicReal_2. More...
 
conceptComputeY_2
 Computes the second coordinate of an AlgebraicKernel_d_2::AlgebraicReal_2. More...
 
conceptConstructAlgebraicReal_2
 Constructs an AlgebraicKernel_d_2::Algebraic_real_2. More...
 
conceptIsCoprime_2
 Computes whether a given pair of bivariate polynomials is coprime. More...
 
conceptIsolate_2
 Computes an isolating box for a given AlgebraicKernel_d_2::Algebraic_real_2. More...
 
conceptIsolateX_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...
 
conceptIsolateY_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...
 
conceptIsSquareFree_2
 Computes whether the given bivariate polynomial is square free. More...
 
conceptIsZeroAt_2
 Computes whether an AlgebraicKernel_d_2::Polynomial_2 is zero at a given AlgebraicKernel_d_2::Algebraic_real_2. More...
 
conceptMakeCoprime_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...
 
conceptMakeSquareFree_2
 Returns a square free part of a bivariate polynomial. More...
 
conceptNumberOfSolutions_2
 Computes the number of real solutions of the given bivariate polynomial system. More...
 
conceptSignAt_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...
 
conceptSolve_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...
 
conceptSquareFreeFactorize_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

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.