mirror of https://github.com/CGAL/cgal
138 lines
4.6 KiB
TeX
138 lines
4.6 KiB
TeX
\begin{ccRefClass} {Point_3<R>}
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\ccDefinition
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An object of the class \ccRefName\ is a point in the three-dimensional
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Euclidean space $\E^3$.
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%%
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%% \cgal\ defines a symbolic constant
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%% \ccStyle{ORIGIN} which denotes the point at the origin. It can be used
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%% wherever a point can be used, with the only exception that you can not
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%% access its dimension as it is dimensionless.
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%%
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Remember that \ccStyle{R::RT} and \ccStyle{R::FT} denote a ring type
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and a field type.
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For the representation class \ccStyle{Cartesian<T>}, the two types are
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equivalent.
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However, for the representation class \ccStyle{Homogeneous<T>} the ring
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type is \ccStyle{R::RT} is equal to \ccStyle{T} and the field type is
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\ccStyle{R::FT} is equal to \ccStyle{Quotient<T>}.
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\ccCreation
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\ccCreationVariable{p}
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\ccHidden \ccConstructor{Point_3();}
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{introduces an uninitialized variable \ccVar.}
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\ccHidden \ccConstructor{Point_3(const Point_3<R> &q);}
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{copy constructor.}
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\ccConstructor{Point_3(const Origin &ORIGIN);}
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{introduces a point with \ccHtmlNoLinks{Cartesian} coordinates$(0,0,0)$.}
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\ccConstructor{Point_3(const R::RT &hx, const R::RT &hy, const R::RT &hz, const R::RT &hw = R::RT(1));}
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{introduces a point \ccVar\ initialized to $(hx/hw,hy/hw, hz/hw)$.
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If the third argument is not explicitly given it defaults
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to \ccStyle{R::RT(1)}.
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\ccPrecond \ccc{hw} $\neq$ \ccc{Kernel::RT(0)}.}
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\ccOperations
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%\ccSetTwoOfThreeColumns{5cm}{4cm}
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\ccHidden \ccMethod{Point_3<R> & operator=(const Point_3<R> &q);}
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{Assignment.}
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\ccMethod{bool operator==(const Point_3<R> &q) const;}
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{Test for equality: Two points are equal, iff their $x$, $y$ and $z$
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coordinates are equal.}
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\ccMethod{bool operator!=(const Point_3<R> &q) const;}
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{Test for inequality.}
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There are two sets of coordinate access functions, namely to the
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homogeneous and to the \ccHtmlNoLinks{Cartesian} coordinates. They can be used
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independently from the chosen representation type \ccStyle{R}.
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\ccMethod{R::RT hx() const;}
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{returns the homogeneous $x$ coordinate.}
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\ccGlue
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\ccMethod{R::RT hy() const;}
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{returns the homogeneous $y$ coordinate.}
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\ccGlue
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\ccMethod{R::RT hz() const;}
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{returns the homogeneous $z$ coordinate.}
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\ccGlue
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\ccMethod{R::RT hw() const;}
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{returns the homogenizing coordinate.}
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Here come the \ccHtmlNoLinks{Cartesian} access functions. Note that you do not loose
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information with the homogeneous representation, because then the field
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type is a quotient.
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\ccMethod{R::FT x() const;}
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{returns the \ccHtmlNoLinks{Cartesian} $x$ coordinate, that is $hx/hw$.}
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\ccGlue
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\ccMethod{R::FT y() const;}
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{returns the \ccHtmlNoLinks{Cartesian} $y$ coordinate, that is $hy/hw$.}
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\ccGlue
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\ccMethod{R::FT z() const;}
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{returns the \ccHtmlNoLinks{Cartesian} $z$ coordinate, that is $hz/hw$.}
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The following operations are for convenience and for making this
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point class compatible with code for higher dimensional points.
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Again they come in a \ccHtmlNoLinks{Cartesian} and homogeneous flavor.
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\ccMethod{R::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 $0\leq i \leq 3$.}
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\ccMethod{R::FT cartesian(int i) const;}
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{returns the i'th \ccHtmlNoLinks{Cartesian} coordinate of \ccVar, starting with 0.
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\ccPrecond $0\leq i \leq 2$.}
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\ccMethod{R::FT operator[](int i) const;}
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{returns \ccStyle{cartesian(i)}.
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\ccPrecond $0\leq i \leq 2$.}
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\ccMethod{int dimension() const;}
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{returns the dimension (the constant 3).}
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\ccMethod{Bbox_3 bbox() const;}
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{returns a bounding box containing \ccVar.}
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\ccMethod{Point_3<R> transform(const Aff_transformation_3<R> &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{Vector_3<R> operator-(const Point_3<R> &p,
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const Point_3<R> &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_3<R> operator+(const Point_3<R> &p,
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const Vector_3<R> &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_3<R> operator-(const Point_3<R> &p,
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const Vector_3<R> &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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\ccSeeAlso
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\ccRefConceptPage{Kernel::Point_3}
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\end{ccRefClass}
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