mirror of https://github.com/CGAL/cgal
merge after discussion during 26th CGAL Dev Meeting
This commit is contained in:
Eric Berberich
parent
351e57621a
commit
ff2af3c58a
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@ -43,6 +43,25 @@ that takes as input a range of coefficients of type \ccc{Coefficient}, and an it
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\ccNestedType{Compare_1}{A model of the concept
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\ccc{AlgebraicKernel_d_1::Compare_1}.}
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TODO: The following might be redundant due to the signatures of \ccc{Solve_1}.
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Should be checked. They are introduced here to to symmetry
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\ccc{AlgebraicKernel_d_2}.
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\ccNestedType{IsCoprime_1}{A model of the concept
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\ccc{AlgebraicKernel_d_1::IsCoprime_1}.}
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\ccGlue
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\ccNestedType{IsSquareFree_1}{A model of the concept
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\ccc{AlgebraicKernel_d_1::IsSquareFree_1}.}
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\ccNestedType{MakeSquareFree_1}{A model of the concept
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\ccc{AlgebraicKernel_d_1::MakeSquareFree_1}.}
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\ccGlue
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\ccNestedType{MakeCoprime_1}{A model of the concept
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\ccc{AlgebraicKernel_d_1::MakeCoprime_1}.}
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\ccGlue
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\ccNestedType{SquareFreeFactorization_1}{A model of the concept
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\ccc{AlgebraicKernel_d_1::SquareFreeFactorization_1}.}
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\ccHasModels
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\ccSeeAlso
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@ -7,8 +7,6 @@ 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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$d$ in $\R^2$.
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\texttt{Question: what is the righ interface with AK for \ccc{compare_y_at_x_right}?}
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\ccTypes
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A model of \ccc{AlgebraicKernel_d_2} is supposed to provide
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@ -58,9 +56,38 @@ that constructs a polynomial from a range of coefficients of type
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\ccNestedType{Compare_xy_2}{A model of the concept
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\ccc{AlgebraicKernel_d_2::CompareXY_2}.}
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\ccNestedType{IsCoprime_2}{A model of the concept
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\ccc{AlgebraicKernel_d_2::IsCoprime_2}.}
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\ccGlue
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\ccNestedType{IsSquareFree_2}{A model of the concept
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\ccc{AlgebraicKernel_d_2::IsSquareFree_2}.}
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\ccNestedType{MakeSquareFree_2}{A model of the concept
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\ccc{AlgebraicKernel_d_2::MakeSquareFree_2}.}
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\ccGlue
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\ccNestedType{MakeCoprime_2}{A model of the concept
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\ccc{AlgebraicKernel_d_2::MakeCoprime_2}.}
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\ccGlue
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\ccNestedType{SquareFreeFactorization_2}{A model of the concept
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\ccc{AlgebraicKernel_d_2::SquareFreeFactorization_2}.}
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\begin{ccAdvanced}
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A bivariate polynomial can be seen an algebraic curve.
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Additional functors might be provided to implement the analysis of a single
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curve or a pair of curves using an $y$-per-$x$-view.
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In case a functor is not provided, it is set to \ccc{CGAL::Null_functor}.
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\ccNestedType{CurveAnalysis_2}{A model of concept
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\ccc{CurvePairAnalyis_2::CurveAnalysis_2}.}
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\ccGlue
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\ccNestedType{CurvePairAnalysis_2}{A model of concept
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\ccc{AlgebraicKernel_d_2::CurvePairAnalysis_2}.}
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\end{ccAdvanced}
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\ccHasModels
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\ccSeeAlso
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\end{ccRefConcept}
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@ -0,0 +1,68 @@
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\begin{ccRefConcept}{AlgebraicKernel_d_1::AlgebraicReal_1}
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\footnote{
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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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}
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\ccDefinition
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Concept to represent the root of a univariate polynomial
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that are models of \ccc{AlgebraicKernel_d_1::Polynomial_1}.
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\begin{ccAdvanced}
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TODO
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\ccTypes
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\ccTypedef{typedef typename AlgebraicReal_1::Boundary Boundary;}{A number type
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that is able to represent values between two AlgebraicReal\_1}
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\ccCreationVariable{ar}
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\ccAccessFunctions
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AlgebraicReal\_1 is required to have some functionality. The final question
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where to put such methods is undecided. Possibilities are
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member functions of AK, or traits like Algebraic\_structure\_traits that
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either exists in the AK, or outside of it.
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\ccMethod{Boundary between(AlgebraicReal_1 ar2);}{
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returns a rational between \ccc{ar} and \ccc{ar2}
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\ccPrecond{ar != ar2}
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}
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\ccMethod{Boundary lower_bound();}{
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Gives the lower bound of the number.
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}
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\ccMethod{Boundary upper_bound();}{
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Gives the upper bound of the number.
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}
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\ccMethod{void refine();}{
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Refines the representation.
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}
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\ccMethod{void refine(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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\end{ccAdvanced}
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\ccOperations
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The comparison operator \ccc{==} must be provided.
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\ccFunction{bool operator ==(
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const AlgebraicKernel_d_1::Algebraic_real_1 & ar1,
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const AlgebraicKernel_d_1::Algebraic_real_1 & ar2);}{}
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\ccHasModels
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%\ccc{double, NiX::Algebraic_real}
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\end{ccRefConcept}
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@ -0,0 +1,190 @@
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\begin{ccRefConcept}{CurveAnalysis_2::CurveVerticalLine_1}
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\ccDefinition
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The \ccc{CurveVerticalLine_1} concept is meant to provide information
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about the intersections of a curve with a vertical line at a given
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finite $x$-coordinate. Note that a curve can have a vertical line component
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at this coordinate and non-vertical components may intersect
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the vertical line respectively.
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With the help of the concept's methods one is able
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to compute the local topology of the curve at the given vertical line.
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Note that vertical lines at $x = \pm\infty$ are not allowed, since different
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events (curve ends going to infinity with different non-horizontal asympotes)
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would have equal $y$-coordinate ($\pm\infty$), which confuses more than it
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helps. Note in addition that curve ends approaching the vertical asymptote
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introduce an event (depending on whether approaching $+\infty$ or $-\infty$ -
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but the event with coordinates $(x,-\infty)$, resp. $(x,+\infty)$, occur
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only once, if they occur, and they imply not to be associated with
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a instance of \ccc{Algebraic_real_2}.
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\ccTypes
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\ccNestedType{Algebraic_real_1}{A model of the concept
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\ccc{AlgebraicKernel_d_1::AlgebraicReal_1}.}
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\ccNestedType{Algebraic_real_2}{A model of the concept
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\ccc{AlgebraicKernel_d_2::Algebraic_real_2}.}
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\ccCreationVariable{cvl}
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\ccAccessFunctions
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\ccMethod{Algebraic_real_1 x();}{
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returns the $x$-coordinate of the vertical line (always a finite value).
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}
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\ccMethod{bool covers_line();}{
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returns \ccc{true} in case the given curve contains the vertical line as
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a component
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}
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\ccMethod{unsigned int number_of_events();}{
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returns number of distinct and finite intersections of a curve with a
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(intended) vertical line ignoring a real vertical line component
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of the curve at the given $x$-coordinate.
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}
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\ccMethod{Algebraic_real_2 get_algebraic_real_2(int j);}{
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returns an object of type \ccc{Algebraic_real_2} for the $j$-th event
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\ccPrecond{$0 \leq j < \mbox{num\_events()}$}.
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}
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\ccMethod{std::pair< unsigned int, unsigned int >
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get_number_of_incident_branches(int j);}{
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returns the number of branches of the curve connected to $j$-th event
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immediately to the left,
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to the right, respectively, as a pair of \ccc{unsigned int} ignoring
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vertical curve components at the given $x$-coordinate.
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\ccPrecond{$0 \leq j < \mbox{num\_events()}$}
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}
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\ccMethod{std::pair< unsigned int, unsigned int >
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get_number_of_branches_approaching_minus_infinity();}{
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returns the number of vertical asymptotes at $x$
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of the curve approaching $y=-\infty$
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from left and right. A vertical line being component of the curve is ignored.
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}
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\ccMethod{std::pair< unsigned int, unsigned int >
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get_number_of_branches_approaching_plus_infinity();}{
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returns the number of vertical asymptotes at $x$
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of the curve approaching $y=+\infty$
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from left and right. A vertical line being component of the curve is ignored.
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}
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\ccMethod{bool is_event();}{
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returns \ccc{true} if curve has vertical line component or
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curve $f$ has intersection with $f_y$ at $x$
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}
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\end{ccRefConcept}
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\begin{ccRefConcept}{AlgebraicKernel_d_2::CurveAnalysis_2}
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\footnote{There is no need
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for curves to be algebraic, hence the generic name. Only the number
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of events must be finite.}
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\ccDefinition
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The \ccc{AlgebraicKernel_d_2::CurveAnalysis_2}
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concept is meant to provide tools to analyse a single
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curve. An analysis is meant to describe the curve's interesting points and how
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they are connected. The analysis searches for {\it events}. Events only
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occur at a finite number of $x$-coordinates. Each such coordinate defines
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a \ccc{CurveVerticalLine_1} of an event. These coordinates also define open
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{\it intervals} on the $x$-axis. Different
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\ccc{CurveVerticalLine_1} at values within one
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such interval only differ in the values of the \ccc{Algebraic_real_2} entries.
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Topological information are equal for all $x$-coordinate inside such an
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open interval.
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\ccRefines{
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\ccc{DefaultConstructible, CopyConstructible, Assignable}
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}
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\ccTypes
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\ccNestedType{Algebraic_real_1}{A model of the concept
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\ccc{AlgebraicKernel_d_1::AlgebraicReal_1}.}
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\ccNestedType{Algebraic_real_2}{A model of the concept
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\ccc{AlgebraicKernel_d_2::Algebraic_real_2}.}
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\ccNestedType{Polynomial_2}{A model of the concept
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\ccc{AlgebraicKernel_d_2::Polynomial_2}.}
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\ccNestedType{Curve_vertical_line_1}{A model of the concept
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\ccc{CurveAnalysis_2::CurveVerticalLine_1}.}
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\ccCreation
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\ccCreationVariable{ca}
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\ccConstructor{CurveAnalysis_2(Polynomial_2 p);}
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{constructs an analysis for the curve defined by p. The polynomial must
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be squarefree.}
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\begin{ccAdvanced}
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\ccConstructor{template < class InputIterator >
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CurveAnalysis_2(Polynomial_2 p, InputIterator begin, InputIterator end);}{
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constructs an analysis for the curve defined by p. The polynomial must
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be squarefree. The iterator range [begin,end) contains factors of
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$\mbox{resultant}(p,p_y,y)$
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i.e., their roots define $x$-coordinates of events,
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which allows to simplify the real root isolation within this layer.
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The \ccc{value_type} of InputIterator is \ccc{Polynomial_1}.
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This constructor has been introduced to enable an upper layer (CK) to
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propagate additional knowledge on the problem.
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}
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\end{ccAdvanced}
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\ccAccessFunctions
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\ccMethod{Polynomial_2 get_polynomial_2();}{
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returns the defining polynomial of the anyalsis
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}
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\ccMethod{unsigned int number_of_vertical_lines_with_event();}{
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returns number of vertical lines that encode an event
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}
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\ccMethod{Curve_vertical_line_1 vertical_line_at_event(int i);}{
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returns an instance of \ccc{CurveVerticalLine_1} at the $i$-th event
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}
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\ccMethod{Curve_vertical_line_1 vertical_line_of_interval(int i);}{
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returns an instance of \ccc{CurveVerticalLine_1} of the $i$-th interval
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between $x$-events.
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}
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\ccMethod{Curve_vertical_line_1 vertical_line_for_x(Algebraic_real_1 x,
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CGAL::Sign perturb = CGAL::ZERO);}{
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returns vertical\_line\_at\_event(i), if $x$ hits $i$-th event, otherwise
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returns vertical\_line\_of\_interval(i), where $i$ is the id of the interval
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$x$ lies in. If $pertub$ is CGAL::NEGATIVE (CGAL::POSITIVE) and $x$ states
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an event, then vertical\_line\_of\_interval(i)
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(vertical\_line\_of\_interval(i+1)) is returned.
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\ccPrecond{$x$ is finite}
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}
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\ccMethod{Curve_vertical_line_1 vertical_line_at_exact_x(Algebraic_real_1 x);}{
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returns an instance of \ccc{CurveVerticalLine_1} at the given $x$.
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\ccPrecond{$x$ is finite}
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}
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Note that the access methods to vertical lines are not redundant! The ones
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using an id-value are efficient for speed-ups (caching, avoids to search for
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some $x$, or the unique representative vertical line for an interval).
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The others enable a user to compute vertical lines for given $x$,
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i.e., also to access $y$-coordinates at given $x$.
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%\begin{ccAdvanced}
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%
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%\ccMethod{int find(Algebraic_real_2 s);}{
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% returns the index of the event at the vertical line defined by
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% $s$'s $x$-coordinate, or -1 if $s$ is does not lie on the curve.
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%}
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%
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%\end{ccAdvanced}
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\end{ccRefConcept}
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@ -0,0 +1,222 @@
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\begin{ccRefConcept}{CurvePairAnalysis_2::CurvePairVerticalLine_1}
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\ccDefinition
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The \ccc{CurvePairVerticalLine_1} concept is meant to provide information
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about the intersections of a pair of curves with a (intended) vertical line
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(ignoring vertical lines of the curves themselves). Each intersection of a
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curve with the vertical line defined by some given $x$ induces an event.
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An event can be asked for its coordinates (\ccc{Algebraic_real_2})
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and the involved curve(s).
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Note that the involvment also holds for curve ends approaching the
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vertical asymptote. Again \ccc{CurvePairVerticalLine_1} at $x = \pm\infty$ are
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not allowed.
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\ccTypes
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\ccNestedType{Algebraic_real_1}{Model of the concept
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\ccc{AlgebraicKernel_d_1::AlgebraicReal_1}.}
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\ccNestedType{Algebraic_real_2}{Model of the concept
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\ccc{AlgebraicKernel_d_2::AlgebraicReal_2}.}
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\ccCreationVariable{cpvl}
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\ccAccessFunctions
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\ccMethod{Algebraic_real_1 x();}{
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returns the $x$-coordinate of the vertical line (always a finite value).
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}
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\ccMethod{unsigned int number_of_events();}{
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returns number of distinct and finite intersections of a pair of
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curves with a
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(intended) vertical line ignoring a real vertical line component
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of the curves at the given $x$-coordinate.
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}
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\ccMethod{unsigned int get_event_of_curve(int k, bool c);}{
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returns the $y$-position of the $k$-th event of the $c$-th
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(0 or 1) curve in the sequence of events. Note that each event
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is formed by the first, the second, or both curves.
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\ccPrecond{$0 \leq k < \mbox{number of arcs defined for the curve at x()}$}
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}
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\ccMethod{unsigned int get_multiplicity_of_intersection(int j);}{
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returns the multiplicity of intersection defined at event with position $j$.
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May return $0$ in case multiplicity is unknown.
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\ccPrecond{There is an intersection of both curves at $j$-th event.}
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\ccPrecond{$0 \leq j < \mbox{num\_events()}$}
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}
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\ccMethod{std::pair< int, int > get_curves_at_event(int j);}{
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returns a pair of int indicating whether event $j$ is formed by
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which arc numbers of the first and the second curve, or $-1$, if the
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corresponding curve is not involved.
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\ccPrecond{$0 \leq j < \mbox{num\_events()}$}
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}
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Note that this interface mainly rewrites $\{f,g,x\}^n$ in a different way -
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using ints. Actually the CPVL has to compute this sequence, but for
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the interface it is nicer to have it already here to avoid conversion objects
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(introducing additional constructor calls, cache-misses, lookup and so on) in
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the next layer on top (CK). Obviously, the ints can be computed in a generic
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way from a sequence, so that it makes sense to offer a default implementation
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providing such conversion. There is no need to document this fact on the
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conceptual view.
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\ccMethod{bool is_event();}{
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returns \ccc{true} if a curve has an event or in case there is an
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intersection of both curves.
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}
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\ccMethod{bool is_intersection();}{
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returns \ccc{true} if there is an intersection of both curves.
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}
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\end{ccRefConcept}
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\begin{ccRefConcept}{AlgebraicKernel_d_2::CurvePairAnalysis_2}
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\ccDefinition
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|
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The \ccc{CurvePairAnalysis_2} concept is meant to provide tools to analyse
|
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a pair of curves. An analysis is meant to describe the curve pair's
|
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interesting points and how they are connected.
|
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The analysis searches for {\it events}. Events only
|
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occur at a finite number of $x$-coordinate. Each such coordinate is
|
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covered by a \ccc{CurvePairVerticalLine_1},
|
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originated by the events of a single curve
|
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and also the intersections of two curves.
|
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These coordinates also define open {\it intervals}
|
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on the $x$-axis. \ccc{CurvePairVerticalLine_1}
|
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at values in between one such interval
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differ only in the values of the \ccc{Algebraic_real_2} entries. Topological
|
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information are equal.
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|
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\ccRefines{
|
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\ccc{DefaultConstructible, CopyConstructible, Assignable}
|
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}
|
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\ccTypes
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\ccNestedType{Algebraic_real_1}{Model of the concept
|
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\ccc{AlgebraicKernel_d_1::AlgebraicReal_1}.}
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\ccNestedType{Algebraic_real_2}{Model of the concept
|
||||
\ccc{AlgebraicKernel_d_2::AlgebraicReal_2}.}
|
||||
|
||||
\ccNestedType{Curve_pair_vertical_line_1}{Model of the concept
|
||||
\ccc{CurvePairAnalysis_2::CurvePairVerticalLine_1}.}
|
||||
|
||||
\ccNestedType{Curve_analysis_2}{Model of the concept
|
||||
\ccc{CurvePairAnalysis::CurveAnalysis_2}.}
|
||||
|
||||
\ccCreation
|
||||
\ccCreationVariable{cp}
|
||||
|
||||
\ccConstructor{CurvePairAnalysis_2(Curve_analysis_2 ca1, Curve_analysis_2 ca2);}
|
||||
{constructs an analysis for the curve-pair defined by analyses given by
|
||||
ca1 and ca2. The polynomials defining the analyses must be squarefree and
|
||||
coprime.
|
||||
}
|
||||
|
||||
\begin{ccAdvanced}
|
||||
|
||||
\ccConstructor{template < class InputIterator >
|
||||
CurvePairAnalysis_2(Curve_analysis_2 ca1, Curve_analysis_2 ca2,
|
||||
InputIterator begin, InputIterator end);}{
|
||||
constructs an analysis for the pair of curves defined by $ca1$ and $ca2$.
|
||||
The polynomials definining the analyses must be squarefree and coprime.
|
||||
The iterator range [begin,end) contains factors of $\mbox{resultant}(p,q,y)$,
|
||||
i.e., define $x$-coordinates,
|
||||
which allows to simplify the real root isolation within this layer.
|
||||
The \ccc{value_type} of InputIterator is \ccc{Polynomial_1}.
|
||||
This constructor has been introduced to enable an upper layer (CK) to use
|
||||
additional knowledge on the problem.
|
||||
}
|
||||
|
||||
\end{ccAdvanced}
|
||||
|
||||
\ccAccessFunctions
|
||||
|
||||
\ccMethod{Curve_analysis_2 get_curve_analysis(bool c);}{
|
||||
returns curve analyis for $c$-''th'' curve (0 or 1)
|
||||
}
|
||||
|
||||
\ccMethod{unsigned int number_of_vertical_lines_with_event();}{
|
||||
returns number of vertical lines that encode an event
|
||||
}
|
||||
|
||||
|
||||
\ccMethod{int event_of_curve_analysis(unsigned int i, bool c);}{
|
||||
Given the $i$-th event of the curve pair
|
||||
this method returns the id of the event of the corresponding curve
|
||||
analysis $c$ (0 or 1), or $-1$, if the curve has no event at this coordinate.
|
||||
}
|
||||
|
||||
\ccMethod{Curve_pair_vertical_line vertical_line_at_event(int i);}{
|
||||
returns an instance of \ccc{CurvePairVerticalLine_1} at the $i$-th event
|
||||
\ccPrecond{$0 \leq i < \mbox{num\_vertical\_lines\_with\_event()}$}
|
||||
}
|
||||
|
||||
\ccMethod{Curve_pair_vertical_line vertical_line_of_interval(int i);}{
|
||||
returns an instance of \ccc{CurvePairVerticalLine_1} of the $i$-th interval
|
||||
between $x$-events.
|
||||
\ccPrecond{$0 \leq i leq \mbox{num\_vertical\_lines\_with\_event()}$}
|
||||
}
|
||||
|
||||
\ccMethod{Curve_pair_vertical_line vertical_line_for_x(Algebraic_real_1 x,
|
||||
CGAL::Sign perturb = CGAL::ZERO);}{
|
||||
returns vertical\_line\_at\_event(i), if $x$ hits $i$-th event, otherwise
|
||||
returns vertical\_line\_of\_interval(i), where $i$ is the id of the interval
|
||||
$x$ lies in.
|
||||
If $pertub$ is CGAL::NEGATIVE (CGAL::POSITIVE) and $x$ states
|
||||
an event, then vertical\_line\_of\_interval(i)
|
||||
(vertical\_line\_of\_interval(i+1)) is returned.
|
||||
\ccPrecond{$x$ is finite}
|
||||
}
|
||||
|
||||
\ccMethod{Curve_pair_vertical_line vertical_line_at_exact_x(Algebraic_real_1 x);}{
|
||||
returns an instance of \ccc{CurvePairVerticalLine_1} at the given $x$
|
||||
\ccPrecond{$x$ is finite}
|
||||
}
|
||||
|
||||
Note that the access methods to vertical lines are not redundant! The ones
|
||||
using an id-value are efficient for speed-ups (caching, avoids to search some
|
||||
$x$, or the unique representative vertical line for an interval). The others
|
||||
enable a user to compute vertical lines for given $x$.
|
||||
|
||||
\begin{ccAdvanced}
|
||||
|
||||
\ccMethod{Algebraic_real_2_const_iterator solve_begin();}{
|
||||
returns iterator running over all finite intersection of the two curves.
|
||||
}
|
||||
|
||||
\ccMethod{Algebraic_real_2_const_iterator solve_end();}{
|
||||
returns past-end-value for all finite intersections of the two curves.
|
||||
}
|
||||
|
||||
%\ccMethod{int find(Algebraic_real_2 s);}{
|
||||
% returns the index of the event at the vertical line defined by
|
||||
% $s$'s $x$-coordinate, or -1 if $s$ is does not lie on any curve.
|
||||
%}
|
||||
|
||||
\end{ccAdvanced}
|
||||
|
||||
|
||||
%Additionally note that we talk about 4 sequences of $x$-critical lines here.
|
||||
%
|
||||
%\begin{itemize}
|
||||
%\item The sequence of CurveVerticalLine\_1 of $p$ where each CVL has been
|
||||
%converted and merged with $q$ to a CVPL.
|
||||
%\item The sequence of CurveVerticalLine\_1 of $q$ where each CVL has been
|
||||
%converted and merged with $p$ to a CPVL.
|
||||
%\item The sequence of CurvePairVerticalLine\_1 of the intersections of
|
||||
%$p$ and $q$
|
||||
%\item The merged sequence of CurvePairVerticalLine_1 of the former three
|
||||
%sequences. The method event\_id\_of\_x helps to find indeces in the first
|
||||
%three sequences starting from an index in this sequence!
|
||||
%\end{itemize}
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -0,0 +1,14 @@
|
|||
\begin{ccRefConcept}{AlgebraicKernel_d_1::IsCoprime_1}
|
||||
|
||||
\ccCreationVariable{fo}
|
||||
|
||||
A model \ccVar\ of this type must provide:
|
||||
|
||||
\ccMethod{
|
||||
bool
|
||||
operator()(const AlgebraicKernel_d_1::Polynomial_1 & p);}
|
||||
{}
|
||||
|
||||
\ccSeeAlso
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -0,0 +1,14 @@
|
|||
\begin{ccRefConcept}{AlgebraicKernel_d_2::IsCoprime_2}
|
||||
|
||||
\ccCreationVariable{fo}
|
||||
|
||||
A model \ccVar\ of this type must provide:
|
||||
|
||||
\ccMethod{
|
||||
bool
|
||||
operator()(const AlgebraicKernel_d_2::Polynomial_2 & p);}
|
||||
{}
|
||||
|
||||
\ccSeeAlso
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -0,0 +1,14 @@
|
|||
\begin{ccRefConcept}{AlgebraicKernel_d_1::IsSquareFree_1}
|
||||
|
||||
\ccCreationVariable{fo}
|
||||
|
||||
A model \ccVar\ of this type must provide:
|
||||
|
||||
\ccMethod{
|
||||
bool
|
||||
operator()(const AlgebraicKernel_d_1::Polynomial_1 & p);}
|
||||
{}
|
||||
|
||||
\ccSeeAlso
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -0,0 +1,14 @@
|
|||
\begin{ccRefConcept}{AlgebraicKernel_d_2::IsSquareFree_2}
|
||||
|
||||
\ccCreationVariable{fo}
|
||||
|
||||
A model \ccVar\ of this type must provide:
|
||||
|
||||
\ccMethod{
|
||||
bool
|
||||
operator()(const AlgebraicKernel_d_2::Polynomial_2 & p);}
|
||||
{}
|
||||
|
||||
\ccSeeAlso
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -0,0 +1,18 @@
|
|||
\begin{ccRefConcept}{AlgebraicKernel_d_1::MakeCoprime_1}
|
||||
|
||||
\ccCreationVariable{fo}
|
||||
|
||||
A model \ccVar\ of this type must provide:
|
||||
|
||||
\ccMethod{template < class OutputIterator >
|
||||
void
|
||||
operator()(const AlgebraicKernel_d_1::Polynomial_1 & p1,
|
||||
const AlgebraicKernel_d_1::Polynomial_1 & p2,
|
||||
AlgebraicKernel_d_1::Polynomial_1 & common,
|
||||
OutputIterator oi1
|
||||
OutputIterator oi2);}
|
||||
{}
|
||||
|
||||
\ccSeeAlso
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -0,0 +1,18 @@
|
|||
\begin{ccRefConcept}{AlgebraicKernel_d_2::MakeCoprime_2}
|
||||
|
||||
\ccCreationVariable{fo}
|
||||
|
||||
A model \ccVar\ of this type must provide:
|
||||
|
||||
\ccMethod{template < class OutputIterator >
|
||||
void
|
||||
operator()(const AlgebraicKernel_d_2::Polynomial_2 & p1,
|
||||
const AlgebraicKernel_d_2::Polynomial_2 & p2,
|
||||
AlgebraicKernel_d_2::Polynomial_2 & common,
|
||||
OutputIterator oi1
|
||||
OutputIterator oi2);}
|
||||
{}
|
||||
|
||||
\ccSeeAlso
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -0,0 +1,15 @@
|
|||
\begin{ccRefConcept}{AlgebraicKernel_d_1::MakeSquareFree_1}
|
||||
|
||||
\ccCreationVariable{fo}
|
||||
|
||||
A model \ccVar\ of this type must provide:
|
||||
|
||||
\ccMethod{template < class OutputIterator >
|
||||
void
|
||||
operator()(const AlgebraicKernel_d_1::Polynomial_1 & p,
|
||||
OutputIterator oi);}
|
||||
{}
|
||||
|
||||
\ccSeeAlso
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -0,0 +1,15 @@
|
|||
\begin{ccRefConcept}{AlgebraicKernel_d_2::MakeSquareFree_2}
|
||||
|
||||
\ccCreationVariable{fo}
|
||||
|
||||
A model \ccVar\ of this type must provide:
|
||||
|
||||
\ccMethod{template < class OutputIterator >
|
||||
void
|
||||
operator()(const AlgebraicKernel_d_2::Polynomial_2 & p,
|
||||
OutputIterator oi);}
|
||||
{}
|
||||
|
||||
\ccSeeAlso
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -0,0 +1,16 @@
|
|||
\begin{ccRefConcept}{AlgebraicKernel_d_1::SquareFreeFactorization_1}
|
||||
|
||||
\ccCreationVariable{fo}
|
||||
|
||||
A model \ccVar\ of this type must provide:
|
||||
|
||||
\ccMethod{template < class OutputIterator1, class OutputIterator2 >
|
||||
void
|
||||
operator()(const AlgebraicKernel_d_1::Polynomial_1 & p1,
|
||||
OutputIterator oi1
|
||||
OutputIterator oi2);}
|
||||
{oi1 returns factors, oi2 returns multiplicities}
|
||||
|
||||
\ccSeeAlso
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -0,0 +1,16 @@
|
|||
\begin{ccRefConcept}{AlgebraicKernel_d_2::SquareFreeFactorization_2}
|
||||
|
||||
\ccCreationVariable{fo}
|
||||
|
||||
A model \ccVar\ of this type must provide:
|
||||
|
||||
\ccMethod{template < class OutputIterator1, class OutputIterator2 >
|
||||
void
|
||||
operator()(const AlgebraicKernel_d_2::Polynomial_2 & p1,
|
||||
OutputIterator oi1
|
||||
OutputIterator oi2);}
|
||||
{oi1 returns factors, oi2 returns multiplicities}
|
||||
|
||||
\ccSeeAlso
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
|
@ -0,0 +1,34 @@
|
|||
|
||||
- IsCoprime, IsSquareFree, MakeCoprime, MakeSquareFree,
|
||||
SquareFreeFactorization - documentation needed as discussed in Inria
|
||||
|
||||
- Solve_2 - similar to Solve_1
|
||||
|
||||
- Polynomial - there are things to coordinate with PolynomialTraits
|
||||
and/or PolynomialToolBox, what about Construct_polynomial?
|
||||
What about Derivative?
|
||||
|
||||
- Is Derivative and XCriticalPoints/YCriticalPoints redundant?
|
||||
|
||||
|
||||
- AlgebraicReal_1 - what about the functions upper_bound, lower_bound
|
||||
etc. See .tex file!
|
||||
|
||||
- AlgebraicReal_2 - Do we provide .x() and .y() method??
|
||||
|
||||
- Names of CurveAnalysis_2, CurveVerticalLine_1 and
|
||||
CurvePairAnalysis_2, CurvePairVerticalLine_1?
|
||||
- Improvements of Interface
|
||||
- Improvements on Documentation
|
||||
- Use "int" at all places, or stick to distinction of unsigned int + int?
|
||||
|
||||
|
||||
Notes:
|
||||
- CA_2 and CPA_2 require squarefree and coprime input!
|
||||
- Events at infinity are only important for a single curve. Since
|
||||
inf doesn't play a role in CPA_2 we discuss it as a special case in CA_2.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
|
@ -1,10 +1,16 @@
|
|||
\ccRefChapter{Algebraic Kernel}
|
||||
|
||||
\input{Algebraic_kernel_d_ref/AlgebraicKernel_d_1}
|
||||
\input{Algebraic_kernel_d_ref/AlgebraicReal_1}
|
||||
\input{Algebraic_kernel_d_ref/SolveC_1}
|
||||
\input{Algebraic_kernel_d_ref/SignAt_1}
|
||||
\input{Algebraic_kernel_d_ref/DerivativeC_1}
|
||||
\input{Algebraic_kernel_d_ref/CompareC_1}
|
||||
\input{Algebraic_kernel_d_ref/IsSquareFree_1}
|
||||
\input{Algebraic_kernel_d_ref/IsCoprime_1}
|
||||
\input{Algebraic_kernel_d_ref/MakeSquareFree_1}
|
||||
\input{Algebraic_kernel_d_ref/MakeCoprime_1}
|
||||
\input{Algebraic_kernel_d_ref/SquareFreeFactorization_1}
|
||||
|
||||
\input{Algebraic_kernel_d_ref/AlgebraicKernel_d_2}
|
||||
\input{Algebraic_kernel_d_ref/AlgebraicReal_2}
|
||||
|
|
@ -13,4 +19,10 @@
|
|||
\input{Algebraic_kernel_d_ref/SolveC_2}
|
||||
\input{Algebraic_kernel_d_ref/SignAt_2}
|
||||
\input{Algebraic_kernel_d_ref/DerivativeC_2}
|
||||
|
||||
\input{Algebraic_kernel_d_ref/IsSquareFree_2}
|
||||
\input{Algebraic_kernel_d_ref/IsCoprime_2}
|
||||
\input{Algebraic_kernel_d_ref/MakeSquareFree_2}
|
||||
\input{Algebraic_kernel_d_ref/MakeCoprime_2}
|
||||
\input{Algebraic_kernel_d_ref/SquareFreeFactorization_2}
|
||||
\input{Algebraic_kernel_d_ref/CurveAnalysis_2}
|
||||
\input{Algebraic_kernel_d_ref/CurvePairAnalysis_2}
|
||||
|
|
|
|||
Loading…
Reference in New Issue