mirror of https://github.com/CGAL/cgal
184 lines
6.5 KiB
TeX
184 lines
6.5 KiB
TeX
\begin{ccRefClass} {Point_2<Kernel>}
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\ccDefinition
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An object of the class \ccRefName\ is a point in the two-dimensional
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Euclidean plane $\E^2$.
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Remember that \ccStyle{Kernel::RT} and \ccStyle{Kernel::FT} denote a
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\ccc{RingNumberType} and a \ccc{FieldNumberType}, respectively. For the kernel
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model \ccStyle{Cartesian<T>}, the two types are the same. For the
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kernel model \ccStyle{Homogeneous<T>}, \ccStyle{Kernel::RT} is equal
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to \ccStyle{T}, and \ccStyle{Kernel::FT} is equal to
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\ccStyle{Quotient<T>}.
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\ccTypes
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\ccThree{Cartesian_const_iterator}{Facet }{}
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\ccThreeToTwo
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\ccNestedType{Cartesian_const_iterator}{An iterator for enumerating the
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\ccHtmlNoLinksFrom{Cartesian} coordinates of a point.}
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\ccCreation
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\ccCreationVariable{p}
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\ccHidden \ccConstructor{Point_2();}
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{introduces an uninitialized variable \ccVar.}
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\ccHidden \ccConstructor{Point_2(const Point_2<Kernel> &q);}
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{copy constructor.}
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\ccConstructor{Point_2(const Origin &ORIGIN);}
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{introduces a variable \ccVar\ with \ccHtmlNoLinksFrom{Cartesian} coordinates
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$(0,0)$.}
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\ccConstructor{Point_2(int x, int y);}
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{introduces a point \ccVar\ initialized to $(x,y)$.}
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\ccConstructor{Point_2(double x, double y);}
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{introduces a point \ccVar\ initialized to $(x,y)$
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provided \ccc{RT} supports construction from \ccc{double}.}
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\ccConstructor{Point_2(const Kernel::RT &hx, const Kernel::RT &hy, const Kernel::RT &hw = RT(1));}
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{introduces a point \ccVar\ initialized to $(hx/hw,hy/hw)$.
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\ccPrecond \ccc{hw} $\neq$ \ccc{Kernel::RT(0)} }
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\ccConstructor{Point_2(const Kernel::FT &x, const Kernel::FT &y);}
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{introduces a point \ccVar\ initialized to $(x,y)$.}
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\ccOperations
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%\ccSetTwoOfThreeColumns{5cm}{4cm}
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\ccHidden \ccMethod{Point_2<Kernel> & operator=(const Point_2<Kernel> &q);}
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{Assignment.}
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\ccMethod{bool operator==(const Point_2<Kernel> &q) const;}
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{Test for equality. Two points are equal, iff their $x$ and $y$
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coordinates are equal. The point can be compared with
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\ccc{ORIGIN}.}
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\ccMethod{bool operator!=(const Point_2<Kernel> &q) const;}
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{Test for inequality. The point can be compared with \ccc{ORIGIN}.}
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There are two sets of coordinate access functions, namely to the
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homogeneous and to the \ccHtmlNoLinksFrom{Cartesian} coordinates. They can be used
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independently from the chosen kernel model.
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\ccMethod{Kernel::RT hx() const;}
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{returns the homogeneous $x$ coordinate.}
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\ccGlue
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\ccMethod{Kernel::RT hy() const;}
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{returns the homogeneous $y$ coordinate.}
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\ccGlue
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\ccMethod{Kernel::RT hw() const;}
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{returns the homogenizing coordinate.}
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Note that you do not loose information with the homogeneous
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representation, because the \ccc{FieldNumberType} is a quotient.
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\ccMethod{Kernel::FT x() const;}
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{returns the \ccHtmlNoLinksFrom{Cartesian} $x$ coordinate, that is $hx/hw$.}
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\ccGlue
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\ccMethod{Kernel::FT y() const;}
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{returns the \ccHtmlNoLinksFrom{Cartesian} $y$ coordinate, that is $hy/hw$.}
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The following operations are for convenience and for compatibility
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with higher dimensional points. Again they come in a
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\ccHtmlNoLinksFrom{Cartesian} and in a homogeneous flavor.
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\ccMethod{Kernel::RT homogeneous(int i) const;}
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{returns the i'th homogeneous coordinate of \ccVar, starting with 0.
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\ccPrecond $0\leq i \leq 2$.}
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\ccMethod{Kernel::FT cartesian(int i) const;}
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{returns the i'th \ccHtmlNoLinksFrom{Cartesian} coordinate of \ccVar, starting with 0.
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\ccPrecond $0\leq i \leq 1$.}
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\ccMethod{Kernel::FT operator[](int i) const;}
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{returns \ccStyle{cartesian(i)}.
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\ccPrecond $0\leq i \leq 1$.}
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\ccMethod{Cartesian_const_iterator cartesian_begin() const;}
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{returns an iterator to the \ccHtmlNoLinksFrom{Cartesian} coordinates
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of \ccVar, starting with the 0th coordinate.}
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\ccMethod{Cartesian_const_iterator cartesian_end() const;}
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{returns an off the end iterator to the \ccHtmlNoLinksFrom{Cartesian}
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coordinates of \ccVar.}
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\ccMethod{int dimension() const;}
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{returns the dimension (the constant 2).}
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\ccMethod{Bbox_2 bbox() const;}
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{returns a bounding box containing \ccVar. Note that bounding boxes
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are not parameterized with whatsoever. }
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\ccMethod{Point_2<Kernel> transform(const Aff_transformation_2<Kernel> &t) const;}
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{returns the point obtained by applying $t$ on \ccVar.}
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\ccHeading{Operators}
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The following operations can be applied on points:
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\ccFunction{bool operator<(const Point_2<Kernel> &p,
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const Point_2<Kernel> &q);}
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{returns true iff \ccc{p} is lexicographically smaller than \ccc{q},
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i.e. either if \ccc{p.x() < q.x()} or if \ccc{p.x() == q.x()} and
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\ccc{p.y() < q.y()}.}
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\ccFunction{bool operator>(const Point_2<Kernel> &p,
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const Point_2<Kernel> &q);}
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{returns true iff \ccc{p} is lexicographically greater than \ccc{q}.}
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\ccFunction{bool operator<=(const Point_2<Kernel> &p,
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const Point_2<Kernel> &q);}
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{returns true iff \ccc{p} is lexicographically smaller or equal to \ccc{q}.}
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\ccFunction{bool operator>=(const Point_2<Kernel> &p,
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const Point_2<Kernel> &q);}
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{returns true iff \ccc{p} is lexicographically greater or equal to \ccc{q}.}
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\ccFunction{Vector_2<Kernel> operator-(const Point_2<Kernel> &p,
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const Point_2<Kernel> &q);}
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{returns the difference vector between \ccStyle{q} and \ccStyle{p}.
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You can substitute \ccc{ORIGIN} for either \ccc{p} or \ccc{q},
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but not for both.}
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\ccFunction{Point_2<Kernel> operator+(const Point_2<Kernel> &p,
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const Vector_2<Kernel> &v);}
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{returns the point obtained by translating \ccStyle{p} by the
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vector \ccStyle{v}.}
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\ccFunction{Point_2<Kernel> operator-(const Point_2<Kernel> &p,
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const Vector_2<Kernel> &v);}
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{returns the point obtained by translating \ccStyle{p} by the
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vector -\ccStyle{v}.}
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\ccExample
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The following declaration creates two points with
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\ccHtmlNoLinksFrom{Cartesian} double coordinates.
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\begin{cprog}
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Point_2< Cartesian<double> > p, q(1.0, 2.0);
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\end{cprog}
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The variable {\tt p} is uninitialized and should first be used on
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the left hand side of an assignment.
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\begin{cprog}
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p = q;
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std::cout << p.x() << " " << p.y() << std::endl;
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\end{cprog}
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\ccSeeAlso
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\ccRefConceptPage{Kernel::Point_2}
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\end{ccRefClass}
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