Concepts

Concepts
concept	AlgebraicStructureTraits_::Div
	AdaptableBinaryFunction computes the integral quotient of division with remainder. More...
concept	AlgebraicStructureTraits_::Divides
	AdaptableBinaryFunction, returns true if the first argument divides the second argument. More...
concept	AlgebraicStructureTraits_::DivMod
	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. More...
concept	AlgebraicStructureTraits_::Gcd
	AdaptableBinaryFunction providing the gcd. More...
concept	AlgebraicStructureTraits_::IntegralDivision
	AdaptableBinaryFunction providing an integral division. More...
concept	AlgebraicStructureTraits_::Inverse
	AdaptableUnaryFunction providing the inverse element with respect to multiplication of a Field. More...
concept	AlgebraicStructureTraits_::IsOne
	AdaptableUnaryFunction, returns true in case the argument is the one of the ring. More...
concept	AlgebraicStructureTraits_::IsSquare
	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. More...
concept	AlgebraicStructureTraits_::IsZero
	AdaptableUnaryFunction, returns true in case the argument is the zero element of the ring. More...
concept	AlgebraicStructureTraits_::KthRoot
	AdaptableBinaryFunction providing the k-th root. More...
concept	AlgebraicStructureTraits_::Mod
	AdaptableBinaryFunction computes the remainder of division with remainder. More...
concept	AlgebraicStructureTraits_::RootOf
	AdaptableFunctor computes a real root of a square-free univariate polynomial. More...
concept	AlgebraicStructureTraits_::Simplify
	This AdaptableUnaryFunction may simplify a given object. More...
concept	AlgebraicStructureTraits_::Sqrt
	AdaptableUnaryFunction providing the square root. More...
concept	AlgebraicStructureTraits_::Square
	AdaptableUnaryFunction, computing the square of the argument. More...
concept	AlgebraicStructureTraits_::UnitPart
	This AdaptableUnaryFunction computes the unit part of a given ring element. More...
concept	AlgebraicStructureTraits
	A model of AlgebraicStructureTraits reflects the algebraic structure of an associated type Type. More...
concept	EuclideanRing
	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. More...
concept	Field
	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 /=. More...
concept	FieldNumberType
	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. More...
concept	FieldWithKthRoot
	A model of FieldWithKthRoot is a FieldWithSqrt that has operations to take k-th roots. More...
concept	FieldWithRootOf
	A model of FieldWithRootOf is a FieldWithKthRoot with the possibility to construct it as the root of a univariate polynomial. More...
concept	FieldWithSqrt
	A model of FieldWithSqrt is a Field that has operations to take square roots. More...
concept	FractionTraits
	A model of FractionTraits is associated with a type Type. More...
concept	FractionTraits_::Decompose
	Functor decomposing a Fraction into its numerator and denominator. More...
concept	FractionTraits_::Compose
	AdaptableBinaryFunction, returns the fraction of its arguments. More...
concept	FractionTraits_::CommonFactor
	AdaptableBinaryFunction, finds great common factor of denominators. More...
concept	IntegralDomain
	IntegralDomain refines IntegralDomainWithoutDivision by providing an integral division. More...
concept	IntegralDomainWithoutDivision
	This is the most basic concept for algebraic structures considered within CGAL. More...
concept	RealEmbeddable
	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. More...
concept	RealEmbeddableTraits_::Abs
	AdaptableUnaryFunction computes the absolute value of a number. More...
concept	RealEmbeddableTraits_::Compare
	AdaptableBinaryFunction compares two real embeddable numbers. More...
concept	RealEmbeddableTraits_::IsNegative
	AdaptableUnaryFunction, returns true in case the argument is negative. More...
concept	RealEmbeddableTraits_::IsPositive
	AdaptableUnaryFunction, returns true in case the argument is positive. More...
concept	RealEmbeddableTraits_::Sgn
	This AdaptableUnaryFunction computes the sign of a real embeddable number. More...
concept	RealEmbeddableTraits_::ToDouble
	AdaptableUnaryFunction computes a double approximation of a real embeddable number. More...
concept	RealEmbeddableTraits_::ToInterval
	AdaptableUnaryFunction computes for a given real embeddable number \( x\) a double interval containing \( x\). This interval is represented by std::pair<double,double>. More...
concept	RealEmbeddableTraits
	A model of RealEmbeddableTraits is associated to a number type Type and reflects the properties of this type with respect to the concept RealEmbeddable. More...
concept	RingNumberType
	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. More...
concept	UniqueFactorizationDomain
	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.) More...