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 CGAL 5.0 - Algebraic Foundations
Class and Concept List
Here is the list of all concepts and classes of this package. Classes are inside the namespace CGAL. Concepts are in the global namespace.
[detail level 12]
 ▼NAlgebraicStructureTraits_ CDiv AdaptableBinaryFunction computes the integral quotient of division with remainder CDivides AdaptableBinaryFunction, returns true if the first argument divides the second argument CDivMod AdaptableFunctor computes both integral quotient and remainder of division with remainder. The quotient $$q$$ and remainder $$r$$ are computed such that $$x = q*y + r$$ and $$|r| < |y|$$ with respect to the proper integer norm of the represented ring. For integers this norm is the absolute value. For univariate polynomials this norm is the degree. In particular, $$r$$ is chosen to be $$0$$ if possible. Moreover, we require $$q$$ to be minimized with respect to the proper integer norm CGcd AdaptableBinaryFunction providing the gcd CIntegralDivision AdaptableBinaryFunction providing an integral division CInverse AdaptableUnaryFunction providing the inverse element with respect to multiplication of a Field CIsOne AdaptableUnaryFunction, returns true in case the argument is the one of the ring CIsSquare AdaptableBinaryFunction that computes whether the first argument is a square. If the first argument is a square the second argument, which is taken by reference, contains the square root. Otherwise, the content of the second argument is undefined CIsZero AdaptableUnaryFunction, returns true in case the argument is the zero element of the ring CKthRoot AdaptableBinaryFunction providing the k-th root CMod AdaptableBinaryFunction computes the remainder of division with remainder CRootOf AdaptableFunctor computes a real root of a square-free univariate polynomial CSimplify This AdaptableUnaryFunction may simplify a given object CSqrt AdaptableUnaryFunction providing the square root CSquare AdaptableUnaryFunction, computing the square of the argument CUnitPart This AdaptableUnaryFunction computes the unit part of a given ring element ▼NCGAL CAlgebraic_structure_traits An instance of Algebraic_structure_traits is a model of AlgebraicStructureTraits, where T is the associated type CCoercion_traits An instance of Coercion_traits reflects the type coercion of the types A and B, it is symmetric in the two template arguments CEuclidean_ring_tag Tag indicating that a type is a model of the EuclideanRing concept CField_tag Tag indicating that a type is a model of the Field concept CField_with_kth_root_tag Tag indicating that a type is a model of the FieldWithKthRoot concept CField_with_root_of_tag Tag indicating that a type is a model of the FieldWithRootOf concept CField_with_sqrt_tag Tag indicating that a type is a model of the FieldWithSqrt concept CFraction_traits An instance of Fraction_traits is a model of FractionTraits, where T is the associated type CIntegral_domain_tag Tag indicating that a type is a model of the IntegralDomain concept CIntegral_domain_without_division_tag Tag indicating that a type is a model of the IntegralDomainWithoutDivision concept CReal_embeddable_traits An instance of Real_embeddable_traits is a model of RealEmbeddableTraits, where T is the associated type CUnique_factorization_domain_tag Tag indicating that a type is a model of the UniqueFactorizationDomain concept ▼NFractionTraits_ CCommonFactor AdaptableBinaryFunction, finds great common factor of denominators CCompose AdaptableBinaryFunction, returns the fraction of its arguments CDecompose Functor decomposing a Fraction into its numerator and denominator ▼NRealEmbeddableTraits_ CAbs AdaptableUnaryFunction computes the absolute value of a number CCompare AdaptableBinaryFunction compares two real embeddable numbers CIsNegative AdaptableUnaryFunction, returns true in case the argument is negative CIsPositive AdaptableUnaryFunction, returns true in case the argument is positive CIsZero AdaptableUnaryFunction, returns true in case the argument is 0 CSgn This AdaptableUnaryFunction computes the sign of a real embeddable number CToDouble AdaptableUnaryFunction computes a double approximation of a real embeddable number CToInterval AdaptableUnaryFunction computes for a given real embeddable number $$x$$ a double interval containing $$x$$. This interval is represented by std::pair CAlgebraicStructureTraits A model of AlgebraicStructureTraits reflects the algebraic structure of an associated type Type CEuclideanRing A model of EuclideanRing represents an euclidean ring (or Euclidean domain). It is an UniqueFactorizationDomain that affords a suitable notion of minimality of remainders such that given $$x$$ and $$y \neq 0$$ we obtain an (almost) unique solution to $$x = qy + r$$ by demanding that a solution $$(q,r)$$ is chosen to minimize $$r$$. In particular, $$r$$ is chosen to be $$0$$ if possible CExplicitInteroperable Two types A and B are a model of the ExplicitInteroperable concept, if it is possible to derive a superior type for A and B, such that both types are embeddable into this type. This type is CGAL::Coercion_traits::Type CField A model of Field is an IntegralDomain in which every non-zero element has a multiplicative inverse. Thus, one can divide by any non-zero element. Hence division is defined for any divisor != 0. For a Field, we require this division operation to be available through operators / and /= CFieldNumberType The concept FieldNumberType combines the requirements of the concepts Field and RealEmbeddable. A model of FieldNumberType can be used as a template parameter for Cartesian kernels CFieldWithKthRoot A model of FieldWithKthRoot is a FieldWithSqrt that has operations to take k-th roots CFieldWithRootOf A model of FieldWithRootOf is a FieldWithKthRoot with the possibility to construct it as the root of a univariate polynomial CFieldWithSqrt A model of FieldWithSqrt is a Field that has operations to take square roots CFraction A type is considered as a Fraction, if there is a reasonable way to decompose it into a numerator and denominator. In this case the relevant functionality for decomposing and re-composing as well as the numerator and denominator type are provided by CGAL::Fraction_traits CFractionTraits A model of FractionTraits is associated with a type Type CFromDoubleConstructible A model of the concept FromDoubleConstructible is required to be constructible from the type double CFromIntConstructible A model of the concept FromIntConstructible is required to be constructible from int CImplicitInteroperable Two types A and B are a model of the concept ImplicitInteroperable, if there is a superior type, such that binary arithmetic operations involving A and B result in this type. This type is CGAL::Coercion_traits::Type. In case types are RealEmbeddable this also implies that mixed compare operators are available CIntegralDomain IntegralDomain refines IntegralDomainWithoutDivision by providing an integral division CIntegralDomainWithoutDivision This is the most basic concept for algebraic structures considered within CGAL CRealEmbeddable A model of this concepts represents numbers that are embeddable on the real axis. The type obeys the algebraic structure and compares two values according to the total order of the real numbers CRealEmbeddableTraits A model of RealEmbeddableTraits is associated to a number type Type and reflects the properties of this type with respect to the concept RealEmbeddable CRingNumberType The concept RingNumberType combines the requirements of the concepts IntegralDomainWithoutDivision and RealEmbeddable. A model of RingNumberType can be used as a template parameter for Homogeneous kernels CUniqueFactorizationDomain A model of UniqueFactorizationDomain is an IntegralDomain with the additional property that the ring it represents is a unique factorization domain (a.k.a. UFD or factorial ring), meaning that every non-zero non-unit element has a factorization into irreducible elements that is unique up to order and up to multiplication by invertible elements (units). (An irreducible element is a non-unit ring element that cannot be factored further into two non-unit elements. In a UFD, the irreducible elements are precisely the prime elements.)