\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - 3D Spherical Geometry Kernel

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AlgebraicKernelForSpheres Concept Reference
Algebraic Concepts

## Definition

The `AlgebraicKernelForSpheres` concept is meant to provide the curved kernel with all the algebraic functionalities required for the manipulation of spheres, circles, and circular arcs in 3D.

Has Models:
  CGAL::Algebraic_kernel_for_spheres_2_3

See Also
  `SphericalKernel`
  `CGAL::Spherical_kernel_3<Kernel,AlgebraicKernelForSpheres>`

## Concepts
concept | CompareX
concept | CompareXY
concept | CompareXYZ
concept | CompareY
concept | CompareZ
concept | ConstructPolynomial_1_3
concept | ConstructPolynomialForSpheres_2_3
concept | ConstructPolynomialsForLines_3
concept | Polynomial_1_3
| Concept to represent trivariate polynomials of degree 1 whose coefficients are of a type that is a model of the concept `RingNumberType`. More...
concept | PolynomialForSpheres_2_3
| Concept to represent trivariate polynomials of degree up to 2 capable of storing equations of spheres, whose center's coordinates, as well as the square of the radius, are of a type that is a model of the concept `FieldNumberType`. More...
concept | PolynomialsForCircles_3
| Concept to represent a system of polynomials on `FieldNumberType` capable of storing equations of circles. More...
concept | PolynomialsForLines_3
| Concept to represent a system of polynomials on `FieldNumberType` capable of storing equations of lines. More...
concept | RootForSpheres_2_3
| Concept to represent the roots of a system of three equations of degree 2 in three variables `x`, `y` and `z` that are models of concept `AlgebraicKernelForSpheres::PolynomialForSpheres_2_3`. More...
concept | SignAt
concept | Solve
concept | XCriticalPoints
concept | YCriticalPoints
concept | ZCriticalPoints

## Types
A model of `AlgebraicKernelForSpheres` is supposed to provide
typedef unspecified_type | RT
| A model of `RingNumberType`.
typedef unspecified_type | FT
| A model of `FieldNumberType``<RT>`.
typedef unspecified_type | Polynomial_1_3
| A model of `AlgebraicKernelForSpheres::Polynomial_1_3`, for trivariate polynomials of degree up to 1.
typedef unspecified_type | Polynomial_for_spheres_2_3
| A model of `AlgebraicKernelForSpheres::PolynomialForSpheres_2_3`, for trivariate polynomials of degree up to 2 that can store equations of spheres.
typedef unspecified_type | Polynomials_for_lines_3
| A model of `AlgebraicKernelForSpheres::PolynomialsForLines_3`, for systems of polynomials that can store equations of lines in 3D.
typedef unspecified_type | Root_of_2
| A model of `RootOf_2`, for algebraic numbers of degree up to 2.
typedef unspecified_type | Root_for_spheres_2_3
| A model of `AlgebraicKernelForSpheres::RootForSpheres_2_3`, for solutions of systems of three models of `AlgebraicKernelForSpheres::PolynomialForSpheres_2_3`.
typedef unspecified_type | Construct_polynomial_1_3
| A model of `AlgebraicKernelForSpheres::ConstructPolynomial_1_3`.
typedef unspecified_type | Construct_polynomial_for_spheres_2_3
| A model of `AlgebraicKernelForSpheres::ConstructPolynomialForSpheres_2_3`.
typedef unspecified_type | Construct_polynomials_for_lines_3
| A model of `AlgebraicKernelForSpheres::ConstructPolynomialsForLines_3`.
typedef unspecified_type | Compare_x
| A model of the concept `AlgebraicKernelForSpheres::CompareX`.
typedef unspecified_type | Compare_y
| A model of the concept `AlgebraicKernelForSpheres::CompareY`.
typedef unspecified_type | Compare_z
| A model of the concept `AlgebraicKernelForSpheres::CompareZ`.
typedef unspecified_type | Compare_xy
| A model of the concept `AlgebraicKernelForSpheres::CompareXY`.
typedef unspecified_type | Compare_xyz
| A model of the concept `AlgebraicKernelForSpheres::CompareXYZ`.
typedef unspecified_type | Sign_at
| A model of the concept `AlgebraicKernelForSpheres::SignAt`.
typedef unspecified_type | X_critical_points
| A model of the concept `AlgebraicKernelForSpheres::XCriticalPoints`.
typedef unspecified_type | Y_critical_points
| A model of the concept `AlgebraicKernelForSpheres::YCriticalPoints`.
typedef unspecified_type | Z_critical_points
| A model of the concept `AlgebraicKernelForSpheres::ZCriticalPoints`.
typedef unspecified_type | Solve
| A model of the concept `AlgebraicKernelForSpheres::Solve`.