\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 4.13 - Algebraic Kernel
CGAL::Algebraic_kernel_d_2< Coeff > Class Template Reference

#include <CGAL/Algebraic_kernel_d_2.h>


This class gathers necessary tools for solving and handling bivariate polynomial systems of general degree \( d\).

This class is based on an algorithm computing a geometric-topological analysis of a single curve [5] and of a pair of curves [4]. The main idea behind both analyses is to compute the critical x-coordinates of curves and curve pairs by projection (resultants), and compute additional information about the critical fibers using subresultants and Sturm-Habicht sequences [7]. With that information, the fiber at critical x-coordinates is computed by a variant of the Bitstream Descartes method. See also [8] for a comprehensive description of these techniques.

A point \( p\) of type Algebraic_real_2 is represented by its \( x\)-coordinate \( x_0\) (as described in the Algebraic_kernel_d_1 paragraph above), an algebraic curve where \( p\) lies on, and an integer \( i\), denoting that \( p\) is the \( i\)th point in the fiber at \( x_0\), counted from the bottom (ignoring a possible vertical line at \( x_0\)). This determines the point uniquely, but the \( y\)-coordinate is not stored internally in terms of an Algebraic_real_1 object. Querying such a representation by calling Compute_y_2 is a time-consuming step, and should be avoided for efficiency reasons if possible. Note that this representation is not exposed in the interface.

The template argument Coeff determines the coefficient type of the kernel, which is also the innermost coefficient type of the supported polynomials.

Currently, the following coefficient types are supported:


The template argument type can also be set to Sqrt_extension<NT,ROOT>, where NT is one of the types listed above. ROOT should be one of the integer types. See also the documentation of Sqrt_extension<NT,ROOT>.

Is Model Of:
See also


typedef unspecified_type Coefficient
 Same type as the template argument Coeff.
typedef unspecified_type Polynomial_2
 A model of AlgebraicKernel_d_2::Polynomial_2
typedef unspecified_type Algebraic_real_2
 A model of AlgebraicKernel_d_2::AlgebraicReal_2
typedef unspecified_type Bound
 The choice of Coeff also determines the provided bound type. More...
typedef unspecified_type Multiplicity_type
 The multiplicity type is int.

Member Typedef Documentation

◆ Bound

template<typename Coeff >
typedef unspecified_type CGAL::Algebraic_kernel_d_2< Coeff >::Bound

The choice of Coeff also determines the provided bound type.

In case of Coeff is