mirror of https://github.com/CGAL/cgal
better names, refined concepts for AK
This commit is contained in:
Eric Berberich
parent
3879478407
commit
0e5d766ca3
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@ -1,19 +1,26 @@
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\begin{ccRefConcept}{AlgebraicCurveKernel_2}
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\begin{ccRefConcept}{AlgebraicKernelBase_2}
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\ccDefinition
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The \ccc{AlgebraicCurveKernel_2} concept is meant to provide the curved
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kernel with all the algebraic functionalities on bivariate polynomials
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required for the manipulation of arcs of algebraic curves of general degree
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$\R^2$ using an $y$-per-$x$-view on the curves.
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The \ccc{AlgebraicKernelBase_2} concept is meant to provide the curved
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kernel with very fundamental algebraic functionalities on uni- and bivariate
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polynomials.
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\ccTypes
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A model of \ccc{AlgebraicCurveKernel_2} is supposed to provide
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A model of \ccc{AlgebraicKernelBase_2} is supposed to provide
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\ccNestedType{Coefficient}{A model of \ccc{IntegralDomain}. }
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\ccNestedType{Polynomial_1}{A model of \ccc{Polynomial_1}, for
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univariate polynomials when the \ccc{Coefficient} type is an
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\ccc{IntegralDomain}. It must also be possible to perform \ccc{Canonicalize,
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GcdUpToConstantFactor,
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IntegralDivUpToConstantFactor,
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MakeSquareFreeUpToConstantFactor, and
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SquareFreeFactorizationUpToConstantFactor.
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Also needed some PolynomialConstruction concept.}
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}
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\ccNestedType{Polynomial_2}{A model of \ccc{Polynomial_2}, for
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bivariate polynomials on an \ccc{IntegralDomain} coefficient type.
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it must also be possible to perform \ccc{Canonicalize,
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@ -21,24 +28,17 @@ A model of \ccc{AlgebraicCurveKernel_2} is supposed to provide
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IntegralDivUpToConstantFactor,
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MakeSquareFreeUpToConstantFactor, and
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SquareFreeFactorizationUpToConstantFactor} on this type - maybe using a
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traits class. Also needed some PolynomialConstruction concept
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traits class. Also needed some PolynomialConstruction concept.
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}
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% TODO: Boundary: RealEmbeddable that is 'dense R'
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\ccNestedType{Solution_1}{A model of
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\ccc{AlgebraicKernelBase_2::Solution_1}, for real roots of univariate
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polynomials of type \ccc{Polynomial_1}.}
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\ccNestedType{Solution_2}{A model of
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\ccc{AlgebraicCurveKernel_2::Solution_2}, for solutions of systems
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\ccc{AlgebraicKernelBase_2::Solution_2}, for solutions of systems
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of two bivariate polynomials of type \ccc{Polynomial_2}.}
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\ccNestedType{CurvePairAnalysis_2}{A model of
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\ccc{AlgebraicCurveKernel_2::CurvePairAnalysis_2},
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for analysing a pair of curves
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defined as two analyses of type \ccc{CurveAnalysis_2}.}
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\ccNestedType{CurveAnalysis_2}{A model of
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\ccc{CurvePairAnalyis_2::CurveAnalysis_2}, for analysing single
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curves defined as bivariate polynomials of type \ccc{Polynomial_2}.}
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\ccNestedType{Compare_x_2}{A model of the concept
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\ccc{AlgebraicKernel_d_2::CompareX_2}.}
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\ccGlue
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@ -49,4 +49,78 @@ curves defined as bivariate polynomials of type \ccc{Polynomial_2}.}
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\ccc{AlgebraicKernel_d_2::CompareXY_2}.}
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\end{ccRefConcept}
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\begin{ccRefConcept}{AlgebraicKernelBasic_2}
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\footnote{Adapted the concepts \ccc{AlgebraicKernel_d_1} and
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\ccc{AlgebraicKernel_d_2} and re-assigned types/functors
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to Base and Basic part.}
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\ccDefinition
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The \ccc{AlgebraicKernelBasic_2} concept is meant to provide the curved
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kernel with all the algebraic functionalities on uni- and bivariate polynomials
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required for the manipulation of arcs of algebraic curves of general degree
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$d$ in $\R^2$.
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\ccNestedType{Solve_1}{A model of the concept
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\ccc{AlgebraicKernelBasic_1::Solve_1}.}
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\ccNestedType{Sign_at_1}{A model of the concept
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\ccc{AlgebraicKernelBasic_1::SignAt_1}.}
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\ccNestedType{Derivative_1}{A model of the concept
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\ccc{AlgebraicKernelBasic_1::Derivative}.}
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\ccNestedType{Compare_1}{A model of the concept
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\ccc{AlgebraicKernelBasic_1::Compare_1}.}
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\textbf{Question: what is the righ interface with AK
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for \ccc{compare_y_at_x_right}?}
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\ccNestedType{Solve_2}{A model of the concept
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\ccc{AlgebraicKernelBasic_2::Solve_2}.}
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\ccNestedType{Sign_at_2}{A model of the concept
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\ccc{AlgebraicKernelBasic_2::SignAt_2}.}
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\ccNestedType{Derivative_x_2}{A model of the concept
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\ccc{AlgebraicKernelBasic_2::DerivativeX_2}.}
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\ccNestedType{Derivative_y_2}{A model of the concept
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\ccc{AlgebraicKernelBasic_2::DerivativeY_2}.}
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\ccNestedType{X_critical_points_2}{A model of the concept
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\ccc{AlgebraicKernelBasic_2::XCriticalPoints_2}.}
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\ccGlue
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\ccNestedType{Y_critical_points_2}{A model of the concept
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\ccc{AlgebraicKernelBasic_2::YCriticalPoints_2}.}
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\end{ccRefConcept}
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\begin{ccRefConcept}{AlgebraicKernelCCPA_2}
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\ccDefinition
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The \ccc{AlgebraicKernelCCPA_2} concept refines the \ccc{AlgebraicKernelBase_2}
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concept with functionality on bivariate polynomials
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required for the manipulation of arcs of algebraic curves of general degree
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$\R^2$ using an $y$-per-$x$-view on the curves.
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\ccRefines
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\ccc{AlgebraicKernelBase_2}
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\ccTypes
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\ccNestedType{CurvePairAnalysis_2}{A model of
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\ccc{AlgebraicKernelCCPA_2::CurvePairAnalysis_2},
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for analysing a pair of curves
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defined as two analyses of type \ccc{CurveAnalysis_2}.}
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\ccNestedType{CurveAnalysis_2}{A model of
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\ccc{CurvePairAnalyis_2::CurveAnalysis_2}, for analysing single
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curves defined as bivariate polynomials of type \ccc{Polynomial_2}.}
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{\small TODO: Boundary: RealEmbeddable that is 'dense R'??}
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\end{ccRefConcept}
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@ -6,11 +6,22 @@ The concept \ccc{Solution_1} is meant to store a coordinate ($x$, or $y$) of
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a point on curve.
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\ccRefines
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\ccc{RealEmbeddable}
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\ccc{DefaultConstructible, Assignable, LessThanComparable, CompactifiedNumber}
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Remark: Michael Hemmer proposed to extend the concept \ccc{RealEmbeddable}
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with the notion of compactification (adding/dealing with $\pm\infty$). A
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valid model has to implement handling of infinity.
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The concept \ccc{CompactifiedNumber} has not been introduced so far. Michael
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Hemmer will introduce it as basic tool as soon as possible. For
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now we assume that a number of this type is either finite, and if not,
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we can ask, wether it represents $\pm\infty$.
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There are reasons why we introduce \ccc{Solution_1} over
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\ccc{AlgebraicReal_1}\footnote{Not yet introduced so far}:
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\begin{itemize}
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\item It may suffices to use a simpler number type ($Q[\sqrt{2}]$).
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\item Keep concept minimal, \ccc{AlgebraicReal_1} will quite certainly
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contain more than needed here.
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\item \ccc{AlgebraicReal_1} won't be a refinement of \ccc{CompaticiedNumber}.
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\ccc{double} is.
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\end{itemize}
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\begin{ccAdvanced}
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@ -38,7 +49,7 @@ following methods:
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\ccPrecond{sol1 != s}
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}
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\ccMethod{std::pair<Boundary,Boundary> s1.approximate(int prec);}{
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\ccMethod{std::pair<Boundary,Boundary> approximate(int prec);}{
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Refines the representation to the given precision (binary digits
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after point). Internally the precision can already be higher.
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}
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@ -48,4 +59,11 @@ Maybe, we can have it orthogonal to the normal concept.
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\end{ccAdvanced}
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Remark: Note that this concept only deals with the interface to upper
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layers. There might be additional requirements for number types to
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implement a model of this concept.
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\ccHasModels
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\ccc{double, NiX::Compactified< NiX::Algebraic_real >}
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\end{ccRefConcept}
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@ -5,6 +5,18 @@
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The concept \ccc{Solution_2} is meant to store both coordinates of a point
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on a curve.
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There are reasons why we introduce \ccc{Solution_2} over
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\ccc{AlgebraicReal_2}:
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\begin{itemize}
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\item \ccc{AlgebraicReal_2} seems to be pair of \ccc{AlgebraicReal_1}, while
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\ccc{Solution_2} has members to access \ccc{Solution_1} for $x$ and $y$.
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With the name we state better to be more generic.
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\item Keep concept minimal, in contrast to \ccc{AlgebraicReal_2} that
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seems to be quite powerful.
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\item \ccc{Solution_2} is ``compactified'' (with the help of
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\ccc{Solution_1}) in contrast to \ccc{AlgebraicReal_2}.
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\end{itemize}
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\ccCreationVariable{sol2}
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\ccAccessFunctions
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