% begin cgal manual page

\begin{ccRefClass}{Line_d<Kernel>}\ccCreationVariable{l}

\ccDefinition


An instance of data type \ccc{Line_d} is an oriented line in
$d$-dimensional Euclidean space. 

\ccSetOneOfTwoColumns{5cm}

\ccTypes

% \ccNestedType{R}{the representation type. 
% }

% \ccNestedType{RT}{the ring type. 
% }

% \ccNestedType{FT}{the field type. 
% }

\ccNestedType{LA}{the linear algebra layer. 
}

\ccSetOneOfTwoColumns{3cm}

\ccCreation

\ccConstructor{Line_d<Kernel>()}{introduces a variable \ccc{l} of
type \ccc{Line_d<Kernel>}.}

\ccConstructor{Line_d<Kernel>(Point_d<Kernel> p, Point_d<Kernel> q)}{introduces a
line through \ccc{p} and \ccc{q} and oriented from \ccc{p} to
\ccc{q}. \ccPrecond $p$ and $q$ are distinct and have the same
dimension.}

\ccConstructor{Line_d<Kernel>(Point_d<Kernel> p, Direction_d<Kernel> dir)}{introduces
a line through \ccc{p} with direction \ccc{dir}.  \ccPrecond
\ccc{p.dimension()==dir.dimension()}, \ccc{dir} is not degenerate.}

\ccConstructor{Line_d<Kernel>(Segment_d<Kernel> s)}{introduces a variable
\ccc{l} of type \ccc{Line_d<Kernel>} and initializes it to the line through
\ccc{s.source()} and \ccc{s.target()} with direction from
\ccc{s.source()} to \ccc{s.target()}. \ccPrecond $s$ is not
degenerate.  }

\ccConstructor{Line_d<Kernel>(Ray_d<Kernel> r)}{introduces a variable \ccc{l} of
type \ccc{Line_d<Kernel>} and initializes it to the line through
\ccc{r.point(1)} and \ccc{r.point(2)}.  }

\ccSetTwoOfThreeColumns{3cm}{3cm}

\ccOperations

\ccMethod{int dimension();}{returns the dimension of the ambient
space.}

\ccMethod{Point_d<Kernel> point(int i) ;}{returns an arbitrary point on
\ccc{l}.  It holds that \ccc{point(i) == point(j)}, iff
\ccc{i==j}. Furthermore, \ccc{l} is directed from \ccc{point(i)} to
\ccc{point(j)}, for all \ccc{i < j}.  }

\ccMethod{Line_d<Kernel> opposite() ;}{returns the line
\ccc{(point(2),point(1))} of opposite direction.}

\ccMethod{Direction_d<Kernel> direction();}{returns the direction of
\ccc{l}.  }

\ccMethod{Line_d<Kernel> transform(const Aff_transformation_d<Kernel> &
t);}{returns $t(l)$. \ccPrecond \ccc{l.dimension()==t.dimension()}. }

\ccMethod{Line_d<Kernel> operator+(const Vector_d<Kernel>& v);}{returns
\ccc{l+v}, i.e., \ccc{l} translated by vector $v$. 
\ccPrecond \ccc{l.dimension()==v.dimension()}. }

\ccMethod{Point_d<Kernel> projection(const Point_d<Kernel>& p) ;}{returns the
point of intersection of \ccc{l} with the hyperplane that is
orthogonal to \ccc{l} and that contains \ccc{p}. \ccPrecond
\ccc{l.dimension()==p.dimension()}. }

\ccMethod{bool has_on(const Point_d<Kernel>& p) ;}{returns true if $p$ lies
on \ccc{l} and false otherwise. \ccPrecond
\ccc{l.dimension()==p.dimension()}. }

\ccHeading{Non-Member Functions}  

\ccFunction{bool weak_equality(const Line_d<Kernel>& l1, const Line_d<Kernel>&
l2) ;}{Test for equality as unoriented lines.\ccPrecond
\ccc{l1.dimension()==l2.dimension()}.  }

\ccFunction{bool parallel(const Line_d<Kernel>& l1, const Line_d<Kernel>& l2)
;}{returns true if \ccc{l1} and \ccc{l2} are parallel as unoriented
lines and false otherwise. \ccPrecond
\ccc{l1.dimension()==l2.dimension()}.}

\ccImplementation

Lines are implemented by a pair of points as an item type.  All
operations like creation, initialization, tests, direction
calculation, input and output on a line $l$ take time
$O(\ccc{l.dimension()})$. \ccc{dimension()}, coordinate and point
access, and identity test take constant time.  The operations for
intersection calculation also take time $O(\ccc{l.dimension()})$. The
space requirement is $O(\ccc{l.dimension()})$.

\end{ccRefClass}