Algebraic_kernel_for_circles (instead of Algebraic_kernel)

Algebraic_kernel_for_circles/doc_tex/Algebraic_kernel_for_circles_ref/AlgFunctorsConstruct.tex

@@ -0,0 +1,153 @@
+\begin{ccRefConcept}{AlgebraicKernelForCircles_2_2::ConstructPolynomial_1_2}
+
+\ccDefinition
+
+\ccCreationVariable{cpol}
+
+A model \ccVar\ of this type must provide:
+
+\ccMethod{AlgebraicKernelForCircles_2_2::Polynomial_1_2
+	operator()(const AlgebraicKernelForCircles_2_2::RT a,
+		const AlgebraicKernelForCircles_2_2::RT b,
+		const AlgebraicKernelForCircles_2_2::RT c);}
+{Constructs polynomial \ccc{ax+by+c}.}
+
+\end{ccRefConcept}
+
+\begin{ccRefConcept}{AlgebraicKernelForCircles_2_2::ConstructPolynomialForCircles_2_2}
+
+\ccCreationVariable{cpol}
+
+A model \ccVar\ of this type must provide:
+
+\ccMethod{AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2
+	operator()(const AlgebraicKernelForCircles_2_2::FT a,
+		const AlgebraicKernelForCircles_2_2::FT b,
+		const AlgebraicKernelForCircles_2_2::FT rsq);}
+{Constructs polynomial \ccc{(x-a)^2 + (y-b)^2 - rsq}.}
+
+\end{ccRefConcept}
+
+\begin{ccRefConcept}{AlgebraicKernelForCircles_2_2::Solve}
+
+\ccDefinition
+
+\ccCreationVariable{sol}
+
+A model \ccVar\ of this type must provide:
+
+\ccMethod{template < class OutputIterator >
+    OutputIterator
+    operator()(const AlgebraicKernelForCircles_2_2::Polynomial_1_2 &p1,
+	     const AlgebraicKernelForCircles_2_2::Polynomial_1_2 &p2,
+	     OutputIterator res);}
+{Copies in the output iterator the common roots of \ccc{p1} and \ccc{p2}, with their multiplicity, as objects of type \ccc{std::pair< AlgebraicKernelForCircles_2_2::RootForCircles_2_2, int>}.} \footnote{???}
+
+\ccMethod{template < class OutputIterator >
+    OutputIterator
+    operator()(const AlgebraicKernelForCircles_2_2::Polynomial1_2 &p1,
+	     const AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2 &p2,
+	     OutputIterator res);}
+{Same as previous.} 
+
+\ccMethod{template < class OutputIterator >
+    OutputIterator
+    operator()(const AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2 &p1,
+	     const AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2 &p2,
+	     OutputIterator res);}
+{Same as previous.} 
+
+\ccHasModels
+
+\ccc{Algebraic_kernel_for_circles_2_2::Solve;}
+
+\ccSeeAlso
+
+\ccRefIdfierPage{CGAL::solve}
+
+\end{ccRefConcept}
+\begin{ccRefConcept}{AlgebraicKernel_4_2::Solve}
+
+\ccDefinition
+
+\ccCreationVariable{sol}
+
+A model \ccVar\ of this type must provide:
+
+\ccMethod{template < class OutputIterator >
+    OutputIterator
+    operator()(const AlgebraicKernel_4_2::Polynomial_2_2 &p1,
+	     const AlgebraicKernel_4_2::Polynomial_2_2 &p2,
+	     OutputIterator res);}
+{Copies in the output iterator the common roots of \ccc{p1} and \ccc{p2}, with their multiplicity, as objects of type \ccc{std::pair< AlgebraicKernel_4_2::RootOf_4, int>}.} 
+
+\ccHasModels
+
+\ccc{Algebraic_kernel_for_circles_2_2::Solve;}
+
+\ccSeeAlso
+
+\ccRefIdfierPage{CGAL::solve}
+
+\end{ccRefConcept}
+
+\begin{ccRefConcept}{AlgebraicKernelForCircles_2_2::XCriticalPoints}
+
+\ccDefinition
+
+\ccCreationVariable{xcrit}
+
+A model \ccVar\ of this type must provide:
+
+\ccMethod{template < class OutputIterator >
+    OutputIterator
+    operator()(const AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2 &p,
+	     OutputIterator res);}
+{Copies in the output iterator the \ccc{x}-critical points of polynomial 
+\ccc{p}, as objects of type \ccc{AlgebraicKernelForCircles_2_2::RootForCircles_2_2}.} 
+
+\ccMethod{template < class OutputIterator >
+    AlgebraicKernelForCircles_2_2::RootForCircles_2_2
+    operator()(const AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2 &p,
+	     bool i);}
+{Computes the \ccc{i}th \ccc{x}-critical point of polynomial \ccc{p}.} 
+
+\ccHasModels
+
+\ccc{Algebraic_kernel_for_circles_2_2::X_critical_points;}
+
+\ccSeeAlso
+
+\ccRefIdfierPage{CGAL::x_critical_points}
+
+\end{ccRefConcept}
+\begin{ccRefConcept}{AlgebraicKernelForCircles_2_2::YCriticalPoints}
+
+\ccDefinition
+
+\ccCreationVariable{ycrit}
+
+A model \ccVar\ of this type must provide:
+
+\ccMethod{template < class OutputIterator >
+    OutputIterator
+    operator()(const AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2 &p,
+	     OutputIterator res);}
+{Copies in the output iterator the \ccc{y}-critical points of polynomial 
+\ccc{p}, as objects of type \ccc{AlgebraicKernelForCircles_2_2::RootForCircles_2_2}.} 
+
+\ccMethod{template < class OutputIterator >
+    AlgebraicKernelForCircles_2_2::RootForCircles_2_2
+    operator()(const AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2 &p,
+	     bool i);}
+{Computes the \ccc{i}th \ccc{y}-critical point of polynomial \ccc{p}.} 
+
+\ccHasModels
+
+\ccc{Algebraic_kernel_for_circles_2_2::Y_critical_points;}
+
+\ccSeeAlso
+
+\ccRefIdfierPage{CGAL::y_critical_points}
+
+\end{ccRefConcept}

Algebraic_kernel_for_circles/doc_tex/Algebraic_kernel_for_circles_ref/AlgFunctorsPredicates.tex

@@ -0,0 +1,100 @@
+\begin{ccRefConcept}{AlgebraicKernelForCircles_2_2::CompareX}
+
+\ccDefinition
+
+\ccCreationVariable{cmpx}
+
+A model \ccVar\ of this type must provide:
+
+\ccMethod{template < class OutputIterator >
+    CGAL::Comparison_result
+    operator()(const AlgebraicKernelForCircles_2_2::RootForCircles_2_2 & r1,
+	     const AlgebraicKernelForCircles_2_2::RootForCircles_2_2 & r2);}
+{Compares the \ccc{x} variables of two \ccc{RootForCircles_2_2}.}
+
+\ccHasModels
+
+\ccc{Algebraic_kernel_for_circles_2_2::Compare_x;}
+
+\ccSeeAlso
+
+\ccRefIdfierPage{CGAL::compare_x}
+
+\end{ccRefConcept}
+\begin{ccRefConcept}{AlgebraicKernelForCircles_2_2::CompareY}
+
+\ccDefinition
+
+\ccCreationVariable{cmpy}
+
+A model \ccVar\ of this type must provide:
+
+\ccMethod{template < class OutputIterator >
+    CGAL::Comparison_result
+    operator()(const AlgebraicKernelForCircles_2_2::RootForCircles_2_2 & r1,
+	     const AlgebraicKernelForCircles_2_2::RootForCircles_2_2 & r2);}
+{Compares the \ccc{y} variables of two \ccc{RootForCircles_2_2}.}
+
+\ccHasModels
+
+\ccc{Algebraic_kernel_for_circles_2_2::Compare_y;}
+
+\ccSeeAlso
+
+\ccRefIdfierPage{CGAL::compare_y}
+
+\end{ccRefConcept}
+\begin{ccRefConcept}{AlgebraicKernelForCircles_2_2::CompareXY}
+
+\ccDefinition
+
+\ccCreationVariable{cmpxy}
+
+A model \ccVar\ of this type must provide:
+
+\ccMethod{template < class OutputIterator >
+    CGAL::Comparison_result
+    operator()(const AlgebraicKernelForCircles_2_2::RootForCircles_2_2 & r1,
+	     const AlgebraicKernelForCircles_2_2::RootForCircles_2_2 & r2);}
+{Compares two \ccc{RootForCircles_2_2} lexicographically.}
+
+\ccHasModels
+
+\ccc{Algebraic_kernel_for_circles_2_2::Compare_xy;}
+
+\ccSeeAlso
+
+\ccRefIdfierPage{CGAL::compare_xy}
+
+\end{ccRefConcept}
+
+\begin{ccRefConcept}{AlgebraicKernelForCircles_2_2::SignAt}
+
+\ccDefinition
+
+\ccCreationVariable{sign}
+
+A model \ccVar\ of this type must provide:
+
+\ccMethod{template < class OutputIterator >
+    CGAL::Sign
+    operator()(const AlgebraicKernelForCircles_2_2::Polynomial_1_2 & p,
+	     const AlgebraicKernelForCircles_2_2::RootForCircles_2_2 & r);}
+{Computes the sign of polynomial \ccc{p} evaluated at a root \ccc{r}.}
+
+\ccMethod{template < class OutputIterator >
+    CGAL::Sign
+    operator()(const AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2 & p,
+	     const AlgebraicKernelForCircles_2_2::RootForCircles_2_2 & r);}
+{Same as previous.}
+
+\ccHasModels
+
+\ccc{Algebraic_kernel_for_circles_2_2::Sign_at;}
+
+\ccSeeAlso
+
+\ccRefIdfierPage{CGAL::sign_at}
+
+\end{ccRefConcept}
+

Algebraic_kernel_for_circles/doc_tex/Algebraic_kernel_for_circles_ref/AlgebraicKernelForCircles_2_2.tex

@@ -0,0 +1,76 @@
+\begin{ccRefConcept}{AlgebraicKernelForCircles_2_2} 
+
+\ccDefinition
+
+The \ccc{AlgebraicKernelForCircles_2_2} concept is meant to provide the
+curved kernel with all the algebraic functionalities required for the
+manipulation of circular arcs. 
+
+\ccTypes
+
+A model of \ccc{AlgebraicKernelForCircles_2_2} is supposed to provide
+
+\ccNestedType{RT}{A model of \ccc{RingNumberType}. }%In addition, the
+%class \ccc{Root_of_traits_2<RT>} must be defined and provide a nested
+%type \ccc{Type} which must be the same as \ccc{Root_of_2} and a
+%function \ccc{make_root_of_2(RT,RT,RT,bool)} whose return type is
+%\ccc{Type}.} 
+\ccGlue
+\ccNestedType{FT}{A model of \ccc{FieldNumberType}\ccc{<RT>}.}
+\footnote{concept template...?} 
+
+\ccNestedType{Polynomial_1_2}{A model of 
+\ccc{AlgebraicKernelForCircles_2_2::Polynomial_1_2}, for bivariate polynomials of degree up
+to~1.}
+\ccGlue
+\ccNestedType{Polynomial_for_circles_2_2}{A model of
+\ccc{AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2}, for bivariate polynomials 
+of degree up to~2 that can store equations of circles.} 
+
+\ccNestedType{Root_of_2}{A model of 
+\ccc{AlgebraicKernelForCircles_2_2::RootOf_2}, for algebraic numbers 
+of degree up to~2.} 
+\ccGlue
+\ccNestedType{Root_for_circles_2_2}{A model of 
+\ccc{AlgebraicKernelForCircles_2_2::RootForCircles_2_2}, for
+solutions of systems of two models of 
+\ccc{AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2}.} 
+
+\ccNestedType{Construct_polynomial_1_2}{A model of
+\ccc{AlgebraicKernelForCircles_2_2::ConstructPolynomial_1_2}.}
+ \ccGlue
+\ccNestedType{Construct_polynomial_for_circles_2_2}{A model of
+\ccc{AlgebraicKernelForCircles_2_2::ConstructPolynomialForCircles_2_2}.}
+
+\ccNestedType{Solve}{A model of the concept \ccc{AlgebraicKernelForCircles_2_2::Solve}.}
+
+\ccNestedType{Sign_at}{A model of the concept \ccc{AlgebraicKernelForCircles_2_2::SignAt}.}
+
+\footnote{no \_2 (or \_2\_2) for functors ????????? problem of compatibility 
+with CK and the current kernel. On the other hand, allows to have only
+one functor for several types of arguments}
+
+\ccNestedType{X_critical_points}{A model of the concept 
+\ccc{AlgebraicKernelForCircles_2_2::XCriticalPoints}.}
+\ccGlue
+\ccNestedType{Y_critical_points}{A model of the concept 
+\ccc{AlgebraicKernelForCircles_2_2::YCriticalPoints}.}
+
+\ccNestedType{Compare_x}{A model of the concept 
+\ccc{AlgebraicKernelForCircles_2_2::CompareX}.}
+\ccGlue
+\ccNestedType{Compare_y}{A model of the concept 
+\ccc{AlgebraicKernelForCircles_2_2::CompareY}.}
+\ccGlue
+\ccNestedType{Compare_xy}{A model of the concept 
+\ccc{AlgebraicKernelForCircles_2_2::CompareXY}.}
+
+\ccHasModels
+
+\ccc{Algebraic_kernel_for_circles_2_2}
+
+\ccSeeAlso
+
+\ccRefIdfierPage{CGAL::Curved_kernel<BasicGeometricKernel,AlgebraicKernel>}
+
+\end{ccRefConcept}

Algebraic_kernel_for_circles/doc_tex/Algebraic_kernel_for_circles_ref/AlgebraicKernel_4_2.tex

@@ -0,0 +1,37 @@
+\begin{ccRefConcept}{AlgebraicKernel_4_2} 
+
+\ccDefinition
+
+The \ccc{AlgebraicKernel_4_2} concept is meant to provide the
+curved kernel with all the algebraic functionalities required for the
+manipulation of conic arcs. 
+
+\ccTypes
+
+A model of \ccc{AlgebraicKernel_4_2} is supposed to provide
+
+\ccNestedType{RT}{A model of \ccc{RingNumberType}. In addition, the class \ccc{Root_of_traits_4<RT>} must be defined and provide a nested type \ccc{Type} which must be the same as \ccc{Root_of_4} and a function \ccc{make_root_of_4(RT,RT,RT,RT,RT,int)} whose return type is \ccc{Type}.}\footnote{\ccc{make_root_of_4} to be replaced by solve, since we don't always want a resultant...}
+\ccGlue
+\ccNestedType{FT}{A model of \ccc{FieldNumberType<RT>}.} \footnote{concept template...?}
+\ccc{RT} is supposed to be \ccc{Rational_traits<FT>}.
+
+\ccNestedType{Root_of_4}{A model of \ccc{RootOf_4}, for algebraic numbers 
+		of degree up to~4.}
+\ccGlue
+\ccNestedType{Polynomial_2_2}{A model of \ccc{Polynomial_2_2}, for bivariate polynomials of degree up to~2.}
+
+\ccNestedType{Construct_polynomial_2_2}{A model of \ccc{ConstructPolynomial_2_2.}}
+
+\ccNestedType{Solve}{A model of the concept \ccc{AlgebraicKernel_4_2::Solve}.}
+
+\ccHasModels
+
+\footnote{to be done}
+
+\ccSeeAlso
+
+\ccRefIdfierPage{Algebraic_kernel_4_2::RootOf_4}\\
+\ccRefIdfierPage{Algebraic_kernel_4_2::Polynomial_2_2}\\
+\ccRefIdfierPage{CGAL::Curved_kernel<BasicGeometricKernel,AlgebraicKernel>}
+
+\end{ccRefConcept}

Algebraic_kernel_for_circles/doc_tex/Algebraic_kernel_for_circles_ref/Algebraic_kernel_for_circles_2_2.tex

@@ -0,0 +1,5 @@
+\begin{ccRefClass}{Algebraic_kernel_for_circles_2_2<RT>}
+
+Model of \ccc{AlgebraicKernelForCircles_2_2}
+
+\end{ccRefClass}

Algebraic_kernel_for_circles/doc_tex/Algebraic_kernel_for_circles_ref/Global_functions.tex

@@ -0,0 +1,183 @@
+%%%%%%%%%%%%%% predicates
+
+\begin{ccRefFunction}{compare_x}
+
+\ccFunction{template < class AlgebraicKernelForCircles_2_2>
+	Comparison_result compare_x
+	(const AlgebraicKernelForCircles_2_2::Root_for_circles_2_2 &r1,
+	const AlgebraicKernelForCircles_2_2::Root_for_circles_2_2 &r2);}
+	{Calls the operator() of \ccc{AlgebraicKernelForCircles_2_2::Compare_X}.}
+
+\ccSeeAlso
+
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::CompareX}
+
+\end{ccRefFunction}
+
+\begin{ccRefFunction}{compare_y}
+
+\ccFunction{template < class AlgebraicKernelForCircles_2_2>
+	Comparison_result compare_y
+	(const AlgebraicKernelForCircles_2_2::Root_for_circles_2_2 &r1,
+	const AlgebraicKernelForCircles_2_2::Root_for_circles_2_2 &r2);}
+	{Calls the operator() of \ccc{AlgebraicKernelForCircles_2_2::Compare_Y}.}
+
+\ccSeeAlso
+
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::CompareY}
+
+\end{ccRefFunction}
+
+\begin{ccRefFunction}{compare_xy}
+
+\ccFunction{template < class AlgebraicKernelForCircles_2_2>
+	Comparison_result compare_xy
+	(const AlgebraicKernelForCircles_2_2::Root_for_circles_2_2 &r1,
+	const AlgebraicKernelForCircles_2_2::Root_for_circles_2_2 &r2);}
+	{Calls the operator() of \ccc{AlgebraicKernelForCircles_2_2::Compare_XY}.}
+
+\ccSeeAlso
+
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::CompareXY}
+
+\end{ccRefFunction}
+
+\begin{ccRefFunction}{sign_at}
+
+\ccFunction{template < class AlgebraicKernelForCircles_2_2, class OutputIterator >
+	OutputIterator sign_at
+	(const AlgebraicKernelForCircles_2_2::Polynomial_for_circles_2_2 &p,
+	const AlgebraicKernelForCircles_2_2::Root_for_circles_2_2 &r);}
+	{Calls the operator() of \ccc{AlgebraicKernelForCircles_2_2::Sign_at}.}
+
+\ccSeeAlso
+
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::SignAt}
+
+\end{ccRefFunction}
+
+%%%%%%%%%%%%%%%%%%%%%%%% constructions
+\begin{ccRefFunction}{construct_polynomial_1_2}
+
+\ccDefinition
+
+\ccFunction{template < class AlgebraicKernelForCircles_2_2>
+	AlgebraicKernelForCircles_2_2::Polynomial_1_2
+	construct_polynomial_1_2
+	(AlgebraicKernelForCircles_2_2::RT a,
+	 AlgebraicKernelForCircles_2_2::RT b,
+	 AlgebraicKernelForCircles_2_2::RT c);}
+{Calls the operator() of \ccc{AlgebraicKernelForCircles_2_2::ConstructPolynomial_1_2}.}
+
+\end{ccRefFunction}
+
+\begin{ccRefFunction}{construct_polynomial_for_circles_2_2}
+
+\ccDefinition
+
+\ccFunction{template < class AlgebraicKernelForCircles_2_2>
+	AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2
+	construct_polynomial_for_circles_2_2
+	(AlgebraicKernelForCircles_2_2::RT a,
+	 AlgebraicKernelForCircles_2_2::RT b,
+	 AlgebraicKernelForCircles_2_2::RT c);}
+{Calls the operator() of \ccc{AlgebraicKernelForCircles_2_2::ConstructPolynomialForCircles_2_2}.}
+
+\end{ccRefFunction}
+
+\begin{ccRefFunction}{solve}
+
+\ccDefinition
+
+\ccFunction{template < class AlgebraicKernelForCircles_2_2, class OutputIterator >
+	OutputIterator solve
+	(const AlgebraicKernelForCircles_2_2::Polynomial_1_2 &p1,
+	const AlgebraicKernelForCircles_2_2::Polynomial_1_2 &p2,
+	OutputIterator res);}
+	{Calls the operator() of \ccc{AlgebraicKernelForCircles_2_2::Solve}.}
+
+\ccFunction{template < class AlgebraicKernelForCircles_2_2, class OutputIterator >
+	OutputIterator solve
+	(const AlgebraicKernelForCircles_2_2::Polynomial_1_2 &p1,
+	const AlgebraicKernelForCircles_2_2::Polynomial_for_circles_2_2 &p2,
+	OutputIterator res);}
+	{Calls the operator() of \ccc{AlgebraicKernelForCircles_2_2::Solve}.}
+
+\ccFunction{template < class AlgebraicKernelForCircles_2_2, class OutputIterator >
+	OutputIterator solve
+	(const AlgebraicKernelForCircles_2_2::Polynomial_for_circles_2_2 &p1,
+	const AlgebraicKernelForCircles_2_2::Polynomial_for_circles_2_2 &p2,
+	OutputIterator res);}
+	{Calls the operator() of \ccc{AlgebraicKernelForCircles_2_2::Solve}.}
+\ccSeeAlso
+
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::Solve}
+
+\end{ccRefFunction}
+
+\begin{ccRefFunction}{x_critical_points}
+
+\ccFunction{template < class AlgebraicKernelForCircles_2_2, class OutputIterator >
+	OutputIterator
+	x_critical_points
+	(const typename AlgebraicKernelForCircles_2_2::Polynomial_for_circles_2_2 & c, 
+	           OutputIterator res);}
+	{Calls the operator() of \ccc{AlgebraicKernelForCircles_2_2::XCriticalPoints}.}
+
+\end{ccRefFunction}
+
+\begin{ccRefFunction}{x_critical_point}
+
+\ccFunction{template < class AlgebraicKernelForCircles_2_2 >
+	AlgebraicKernelForCircles_2_2::Root_for_circles_2_2
+	x_critical_points(const typename AK::Polynomial_for_circles_2_2 & c,
+	bool i);}
+	{Calls the operator() of \ccc{AlgebraicKernelForCircles_2_2::XCriticalPoints}.}
+
+\end{ccRefFunction}
+
+\begin{ccRefFunction}{y_critical_points}
+
+\ccFunction{template < class AlgebraicKernelForCircles_2_2, class OutputIterator >
+	OutputIterator
+	y_critical_points
+	(const typename AlgebraicKernelForCircles_2_2::Polynomial_for_circles_2_2 & c, 
+	           OutputIterator res);}
+	{Calls the operator() of \ccc{AlgebraicKernelForCircles_2_2::YCriticalPoints}.}
+
+\end{ccRefFunction}
+
+\begin{ccRefFunction}{y_critical_point}
+
+\ccFunction{template < class AlgebraicKernelForCircles_2_2 >
+	AlgebraicKernelForCircles_2_2::Root_for_circles_2_2
+	x_critical_point(const typename AK::Polynomial_for_circles_2_2 & c,
+	bool i);}
+	{Calls the operator() of \ccc{AlgebraicKernelForCircles_2_2::YCriticalPoints}.}
+
+\end{ccRefFunction}
+
+\begin{ccRefFunction}{make_root_of_2}
+
+\ccFunction{template < class RT >
+	Root_of_2<RT>
+	make_root_of_2(RT a, RT b, RT c, bool s);}
+	{Returns the smaller (resp. larger) root of equation $aX^2+bX+c=0$
+if \ccc{s} is true (resp. false).
+\ccc{RT} is supposed to be a \ccc{RingNumberType}, and \ccc{Root_of_2<RT>} is
+the type given by \ccc{Root_of_traits_2<RT>}.}
+
+\ccFunction{template < class RT >
+	Root_of_2<RT>
+	make_root_of_2(FT a, FT b, FT c, bool s);}
+	{.}
+
+\footnote{groumpf. \ccc{Root_of_taits} is templated by RT but we also need (do we?) make-root-of with FT. Relation between RT and FT...?}
+
+\ccSeeAlso
+
+\ccRefIdfierPage{CGAL::Root_of_2<RT>}\\
+\ccRefIdfierPage{CGAL::Root_of_traits_2<RT>}
+
+\end{ccRefFunction}
+

Algebraic_kernel_for_circles/doc_tex/Algebraic_kernel_for_circles_ref/Polynomial_1.tex

@@ -0,0 +1,39 @@
+\begin{ccRefConcept}{AlgebraicKernelForCircles_2_2::Polynomial_1_2}
+
+\ccDefinition
+
+The \ccc{AlgebraicKernelForCircles_2_2::Polynomial_1_2} represents bivariate
+polynomials of degree 1 whose coefficients are of type \ccc{RT}, a
+model of \ccc{RingNumberType}.
+
+\ccTypes
+
+\ccCreation
+\ccCreationVariable{pol}
+
+\ccConstructor{Polynomial_1_2();}{Default constructor.}
+
+%\ccAccessFunctions
+
+%\ccFunction{const RT & a() const;}{}
+%\ccGlue
+%\ccFunction{const RT & b() const;}{}
+%\ccGlue
+%\ccFunction{const RT & c() const;}{}
+
+%\ccOperations
+
+%The comparison operator \ccc{==} must be provided. 
+
+%\ccFunction{bool operator ==(const AlgebraicKernelForCircles_2_2::Polynomial_1_2 & p,
+%	const AlgebraicKernelForCircles_2_2::Polynomial_1_2 & q);}{}
+
+\ccHasModels
+
+\ccc{Polynomial_1_2}
+
+\ccSeeAlso
+
+\ccRefIdfierPage{AlgebraicKernelForCircles_2_2}
+
+\end{ccRefConcept}

Algebraic_kernel_for_circles/doc_tex/Algebraic_kernel_for_circles_ref/Polynomial_1_2.tex

@@ -0,0 +1,5 @@
+\begin{ccRefClass}{Polynomial_1_2<RT>}
+
+Model of \ccc{AlgebraicKernelForCircles_2_2::Polynomial_1_2}
+
+\end{ccRefClass}

Algebraic_kernel_for_circles/doc_tex/Algebraic_kernel_for_circles_ref/Polynomial_2.tex

@@ -0,0 +1,53 @@
+\begin{ccRefConcept}{AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2}
+
+\ccDefinition
+
+The \ccc{AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2} represents
+bivariate polynomials of degree up to~2 capable of storing equations
+of circles, whose center has coordinates of type \ccc{FT}, a model of
+\ccc{FieldNumberType}, as well as the square of the radius.
+\footnote{Of course the name looks bad since it mixes geometry with algebra. 
+suggestions welcome}
+
+\ccCreation
+\ccCreationVariable{pol}
+
+\ccConstructor{PolynomialForCircles_2_2();}{Default constructor.}
+
+%\ccConstructor{PolynomialForCircles_2_2(const FT & a, const FT & b, const FT & rsq);}{Constructs polynomial \ccc{(x-a)^2 + (y-b)^2 - rsq}.}
+
+%\ccAccessFunctions
+
+%\ccMethod{const FT & a();}{\ccc{x}-coordinate of the center of the circle.}
+%\ccGlue
+%\ccMethod{const FT & b();}{\ccc{y}-coordinate of the center of the circle.}
+%\ccGlue
+%\ccMethod{const FT & r_sq();}{Square radius of the center of the circle.}
+
+\ccOperations
+
+The comparison operator \ccc{==} must be provided. 
+
+\ccFunction{bool operator ==
+(AlgebraicKernelForCircles_2_2::const PolynomialForCircles_2_2 & p,
+	const AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2 & q);}{}
+
+\ccHasModels
+
+\ccc{Polynomial_for_circles_2_2}
+
+\ccSeeAlso
+
+\ccRefIdfierPage{AlgebraicKernelForCircles_2_2}
+
+\end{ccRefConcept}
+
+\begin{ccRefConcept}{Algebraic_kernel_4_2::Polynomial_2_2}
+
+\ccDefinition
+
+The \ccc{Algebraic_kernel_4_2::Polynomial_2_2} represents bivariate
+polynomials of degree up to~2 whose coefficients are of type \ccc{RT},
+a model of \ccc{RingNumberType}.
+
+\end{ccRefConcept}

Algebraic_kernel_for_circles/doc_tex/Algebraic_kernel_for_circles_ref/Polynomial_for_circles_2_2.tex

@@ -0,0 +1,5 @@
+\begin{ccRefClass}{Polynomial_for_circles_2_2<FT>}
+
+Model of \ccc{AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2}
+
+\end{ccRefClass}

Algebraic_kernel_for_circles/doc_tex/Algebraic_kernel_for_circles_ref/RootCircle_2_2.tex

@@ -0,0 +1,32 @@
+\begin{ccRefConcept}{AlgebraicKernelForCircles_2_2::RootForCircles_2_2}
+
+\ccDefinition
+
+The \ccc{AlgebraicKernelForCircles_2_2::RootForCircles_2_2} concept represents
+algebraic numbers that can store the roots of systems of two
+\ccc{AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2}
+equations of degree 2 in two variables \ccc{x} and \ccc{y}. 
+
+%\ccAccessFunctions
+
+%\ccFunction{const AlgebraicKernelForCircles_2_2::RootOf_2 & x();}{}
+%\ccGlue
+%\ccFunction{const AlgebraicKernelForCircles_2_2::RootOf_2 & y();}{}
+
+\ccOperations
+
+The comparison operator \ccc{==} must be provided. 
+
+\ccFunction{bool operator ==(const AlgebraicKernelForCircles_2_2::RootForCircles_2_2 & p,
+	const AlgebraicKernelForCircles_2_2::RootForCircles_2_2 & q);}{}
+
+\ccHasModels
+
+\ccc{Root_for_circles_2_2}
+
+\ccSeeAlso
+
+\ccRefIdfierPage{AlgebraicKernelForCircles_2_2}
+
+\end{ccRefConcept}
+

Algebraic_kernel_for_circles/doc_tex/Algebraic_kernel_for_circles_ref/RootOf.tex

@@ -0,0 +1,59 @@
+\begin{ccRefConcept}{AlgebraicKernelForCircles_2_2::RootOf_2}
+
+\ccDefinition
+
+The \ccc{AlgebraicKernelForCircles_2_2::RootOf_2} concept represents algebraic numbers of
+degree up to~2 over a \ccc{RingNumberType} denoted as \ccc{RT}.
+
+\ccc{FT} denotes the \ccc{FieldNumberType} constructed from \ccc{RT}.
+\footnote{see \ccc{make_rational}, we need something to get FT from RT in a clean way.}
+
+\ccCreation
+
+\footnote{How to get RT...?}
+
+\ccFunction{RootOf_2 make_root_of_2(RT a, RT b, RT c, bool s);}{Returns 
+the smaller (resp. larger) root of equation $aX^2+bX+c=0$ if \ccc{s} is true
+(resp. false).}
+\footnote{numbering of roots from 0 or 1? to be checked}
+\ccGlue
+\ccFunction{RootOf_2 make_root_of_2(FT a, FT b, FT c, bool s);}{Returns
+the smaller (resp. larger) root of equation $aX^2+bX+c=0$ if \ccc{s} is true
+(resp. false).}
+
+\ccOperations
+
+The comparison operators \ccc{==, !=, <, >, <=, >=} as well as the \ccc{sign}
+and \ccc{compare} functions need to be
+provided to compare elements of types \ccc{RootOf_2, RT} and \ccc{FT}. 
+
+In addition, the following operations must be provided:
+
+\def\ccTagRmEigenClassName{\ccFalse}
+
+\ccFunction{RootOf_2 operator+(const RT &a, const RootOf_2 &r);}{}
+\ccGlue
+\ccFunction{RootOf_2 operator+(const FT &a, const RootOf_2 &r);}{}
+\ccGlue
+\ccFunction{RootOf_2 operator+(const RootOf_2 &r, const RT &a);}{}
+\ccGlue
+\ccFunction{RootOf_2 operator+(const RootOf_2 &r, const FT &a);}{}
+
+and similarly for operators -, * and /.
+
+\ccFunction{RootOf_2 square(const RootOf_2 & r);}{}
+
+\def\ccTagRmEigenClassName{\ccTrue}
+
+\ccHasModels
+
+\ccc{double}, \ccc{Root_of_2<RT>}, etc \footnote{to be precised}
+
+\ccSeeAlso
+
+\ccRefIdfierPage{CGAL::Root_of_2<RT>}\\
+\ccRefIdfierPage{CGAL::Root_of_traits_2<RT>}\\
+\ccRefIdfierPage{AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2}\\
+\ccRefIdfierPage{AlgebraicKernelForCircles_2_2}
+
+\end{ccRefConcept}

Algebraic_kernel_for_circles/doc_tex/Algebraic_kernel_for_circles_ref/Root_for_circles_2_2.tex

@@ -0,0 +1,5 @@
+\begin{ccRefClass}{Root_for_circles_2_2<FT>}
+
+Model for \ccc{AlgebraicKernelForCircles_2_2::RootForCircles_2_2}
+
+\end{ccRefClass}

Algebraic_kernel_for_circles/doc_tex/Algebraic_kernel_for_circles_ref/Root_of_2.tex

@@ -0,0 +1,5 @@
+\begin{ccRefClass}{Root_of_2<RT>}
+
+Model for \ccc{AlgebraicKernelForCircles_2_2::RootOf_2}
+
+\end{ccRefClass}

Algebraic_kernel_for_circles/doc_tex/Algebraic_kernel_for_circles_ref/Root_of_traits_2.tex

@@ -0,0 +1,13 @@
+\begin{ccRefClass}{Root_of_traits_2<RT>}
+
+\footnote{name should be changed because it is used ''the other way round'' here, compared with usual use. Find a name with \textit{associative} inside, or/and \textit{extended/extension}...}
+
+\ccDefinition
+
+Associates types for algebraic numbers to \ccc{RT}, supposed to be a \ccc{RingNumberType}. 
+
+\ccTypes
+
+\ccNestedType{Root_of_2;}{Model of \ccc{RootOf_2.}}
+
+\end{ccRefClass}

Algebraic_kernel_for_circles/doc_tex/Algebraic_kernel_for_circles_ref/intro.tex

@@ -0,0 +1,121 @@
+\ccRefChapter{Algebraic Kernel}
+
+\textbf{Submission - Monique - with Sylvain's help...}
+
+\begin{ccAdvanced}
+As in Curved-kernel, I use the ``Advanced'' environment in this
+document to distinguish between my current submission to the CGAL
+editorial board and plans for the future, related to ACS. The
+``Advanced'' parts will disappear if/when this is released.
+\end{ccAdvanced}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section*{Concepts}
+
+\footnote{Same big question as for curved-kernel. Should the kernel here
+be described as a concept or a class?}
+
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2}\\
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::Polynomial_1_2}\\
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::PolynomialForCircles_2_2}\\
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::RootOf_2}\\
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::RootForCircles_2_2}
+
+\begin{ccAdvanced}
+\ccRefConceptPage{AlgebraicKernel_4_2}\\
+\ccRefConceptPage{Algebraic_kernel_4_2::Polynomial_2_2}\\
+\ccRefConceptPage{AlgebraicKernel_4_2::RootOf_2_2}\\
+\ccRefConceptPage{AlgebraicKernel_4_2::RootOf_4}\\
+
+(The indices may look strange but there is indeed a logic: solve on two
+Polynomial-2-2's gives Root-of-4's...)
+
+General remark about the suffix \ccc{_d_v}: \ccc{_d} stands 
+for the degree of the polynomials and the algebraic numbers, and
+\ccc{_v} stands for the number of variables, which is analogous to the
+dimension for CGAL geometric objects. I had already mentioned this in an
+earlier version of this document (presented at the CGAL meeting at
+INRIA in march 05), and it is consistent with what Menelaos proposed
+later (CGAL meeting in Pisa in june 05) for a hierarchy of algebraic kernels.
+\end{ccAdvanced}
+
+	\subsubsection*{Functors} 
+
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::ConstructPolynomial_1_2}\\
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::ConstructPolynomialForCircles_2_2}
+
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::ComputeA}\\
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::ComputeB}\\
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::ComputeC}\\
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::ComputeRsq}\\
+(to be documented)
+
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::ComputeX}\\
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::ComputeY}\\
+(to be documented)
+
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::Solve}
+
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::SignAt}
+
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::XCriticalPoints}\\
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::YCriticalPoints}
+
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::CompareX}\\
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::CompareY}\\
+\ccRefConceptPage{AlgebraicKernelForCircles_2_2::CompareXY}
+
+\begin{ccAdvanced}
+\ccRefConceptPage{AlgebraicKernel_4_2::ConstructPolynomial_2_2}\\
+\ccRefConceptPage{AlgebraicKernel_4_2::Solve}\\
+etc
+\end{ccAdvanced}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section*{Classes}
+
+\footnote{how does the linking work when concepts and classes have the same name...? example \ccc{Polynomial_2_2}}
+
+\ccRefIdfierPage{CGAL::Algebraic_kernel_for_circles_2_2<RT>}
+
+\ccRefIdfierPage{CGAL::Polynomial_1_2<RT>}\\
+\ccRefIdfierPage{CGAL::Polynomial_for_circles_2_2<FT>}
+
+\ccRefIdfierPage{CGAL::Root_of_2<RT>}\\
+\ccRefIdfierPage{CGAL::Root_for_circles_2_2<FT>}
+
+\ccRefIdfierPage{CGAL::Root_of_traits_2<RT>}
+
+\begin{ccAdvanced}
+\ccRefIdfierPage{CGAL::Root_of_4<RT>}\\
+\ccRefIdfierPage{CGAL::Polynomial_2_2<RT>}\\
+\ccRefIdfierPage{CGAL::Root_of_2_2<RT>}\\
+\ccRefIdfierPage{CGAL::Root_of_traits_4<RT>}
+\end{ccAdvanced}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section*{Functions}
+
+	\subsubsection*{Predicates}
+\ccRefIdfierPage{CGAL::sign_at}\\
+
+\ccRefIdfierPage{CGAL::compare_x}\\
+\ccRefIdfierPage{CGAL::compare_y}\\
+\ccRefIdfierPage{CGAL::compare_xy}
+
+	\subsubsection*{Constructions}
+
+\ccRefIdfierPage{CGAL::construct_polynomial_1_2}\\
+\ccRefIdfierPage{CGAL::construct_polynomial_for_circles_2_2}\\
+
+\ccRefIdfierPage{CGAL::make_root_of_2}
+
+\ccRefIdfierPage{CGAL::solve}
+
+\ccRefIdfierPage{CGAL::x_critical_points}\\
+\ccRefIdfierPage{CGAL::y_critical_points}
+
+\begin{ccAdvanced}
+\ccRefIdfierPage{CGAL::make_root_of_4}\\
+etc
+\end{ccAdvanced}

Algebraic_kernel_for_circles/doc_tex/Algebraic_kernel_for_circles_ref/main.tex

@@ -0,0 +1,23 @@
+\input{Algebraic_kernel_for_circles_ref/intro}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\input{Algebraic_kernel_for_circles_ref/AlgebraicKernelForCircles_2_2} %concept
+\input{Algebraic_kernel_for_circles_ref/Algebraic_kernel_for_circles_2_2} 
+\input{Algebraic_kernel_for_circles_ref/AlgebraicKernel_4_2} %concept
+
+\input{Algebraic_kernel_for_circles_ref/RootOf} %concept
+\input{Algebraic_kernel_for_circles_ref/Root_of_2}
+\input{Algebraic_kernel_for_circles_ref/RootCircle_2_2} %concept
+\input{Algebraic_kernel_for_circles_ref/Root_for_circles_2_2} 
+\input{Algebraic_kernel_for_circles_ref/Root_of_traits_2} 
+\input{Algebraic_kernel_for_circles_ref/Polynomial_1} %concept
+\input{Algebraic_kernel_for_circles_ref/Polynomial_1_2} 
+\input{Algebraic_kernel_for_circles_ref/Polynomial_2} %concept
+\input{Algebraic_kernel_for_circles_ref/Polynomial_for_circles_2_2} 
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\input{Algebraic_kernel_for_circles_ref/AlgFunctorsPredicates} % concepts
+\input{Algebraic_kernel_for_circles_ref/AlgFunctorsConstruct} % concepts
+
+\input{Algebraic_kernel_for_circles_ref/Global_functions.tex}