248 lines
9.6 KiB
C++
Executable File
248 lines
9.6 KiB
C++
Executable File
// Copyright (c) 2000,2001
|
|
// Utrecht University (The Netherlands),
|
|
// ETH Zurich (Switzerland),
|
|
// INRIA Sophia-Antipolis (France),
|
|
// Max-Planck-Institute Saarbruecken (Germany),
|
|
// and Tel-Aviv University (Israel). All rights reserved.
|
|
//
|
|
// This file is part of CGAL (www.cgal.org); you can redistribute it and/or
|
|
// modify it under the terms of the GNU Lesser General Public License as
|
|
// published by the Free Software Foundation; either version 3 of the License,
|
|
// or (at your option) any later version.
|
|
//
|
|
// Licensees holding a valid commercial license may use this file in
|
|
// accordance with the commercial license agreement provided with the software.
|
|
//
|
|
// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
|
|
// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
|
|
//
|
|
// $URL$
|
|
// $Id$
|
|
// SPDX-License-Identifier: LGPL-3.0+
|
|
//
|
|
//
|
|
// Author(s) : Michael Seel
|
|
|
|
#ifndef CGAL_PREDICATES_D_H
|
|
#define CGAL_PREDICATES_D_H
|
|
|
|
#include <CGAL/basic.h>
|
|
#include <CGAL/enum.h>
|
|
#include <CGAL/Kernel_d/Point_d.h>
|
|
|
|
namespace CGAL {
|
|
|
|
/*{\Moptions outfile=predicates_d.man}*/
|
|
/*{\Mtext \setopdims{4cm}{2cm}\computewidths
|
|
\headerline{Linear and affine predicates}
|
|
|
|
For an iterator range |[first,last)| we define |T = tuple [first,last)|
|
|
as the ordered tuple $(|T[0]|,|T[1]|, \ldots |T[d-1]|)$ where
|
|
$|T[i]| = |*| |++|^{(i)}|first|$ (the element obtained by $i$ times
|
|
forwarding the iterator by operator |++| and then dereferencing it to
|
|
get the value to which it points). We write |d = size [first,last)|.
|
|
This extends the syntax of random access iterators to input iterators.
|
|
If we index the tuple as above then we require that
|
|
$|++|^{(d+1)}|first == last|$.
|
|
|
|
In the following we require the Iterators to be globally
|
|
of value type |Point_d<R>|. Also if we are handed over an iterator
|
|
range |[first,last)|, then all points in |T = tuple [first,last)|
|
|
have the same dimension |dim|.
|
|
}*/
|
|
|
|
template <class R, class ForwardIterator, class OutputIterator>
|
|
OutputIterator barycentric_coordinates(
|
|
ForwardIterator first, ForwardIterator last, const Point_d<R>& p,
|
|
OutputIterator result)
|
|
/*{\Xfunc returns the barycentric coordinates of a point $p \in R^d$ in a
|
|
affine space of dimension $k$ spanned by the points in |tuple [first,last)|.
|
|
\precond value type of |ForwardIterator| is |Point_d<R>|,
|
|
|affinely_independed(first,last)| and
|
|
|affine_rank(tuple [first,last),p)==k|.}*/
|
|
{ typename R::Barycentric_coordinates_d coords;
|
|
return coords(first,last,p,result);
|
|
}
|
|
|
|
template <class ForwardIterator>
|
|
Orientation
|
|
orientation(ForwardIterator first, ForwardIterator last)
|
|
/*{\Mfunc determines the orientation of the points in the tuple
|
|
|A = tuple [first,last)| where $A$ consists of $d + 1$ points in $d$ - space.
|
|
This is the sign of the determinant
|
|
\[ \left\Lvert \begin{array}{cccc}
|
|
1 & 1 & 1 & 1 \\
|
|
A[0] & A[1] & \dots & A[d]
|
|
\end{array} \right\Lvert \]
|
|
where |A[i]| denotes the cartesian coordinate vector of
|
|
the $i$-th point in $A$.
|
|
\precond |size [first,last) == d+1| and |A[i].dimension() == d|
|
|
$\forall 0 \leq i \leq d$
|
|
and the value type of |ForwardIterator| is |Point_d<R>|.}*/
|
|
{ typedef typename std::iterator_traits<ForwardIterator>::value_type PointD;
|
|
typedef typename PointD::R R;
|
|
typename R::Orientation_d or_; return or_(first,last); }
|
|
|
|
|
|
template <class ForwardIterator>
|
|
Orientation
|
|
coaffine_orientation(ForwardIterator first, ForwardIterator last)
|
|
/*{\Mfunc determines the orientation of the points in the tuple
|
|
|A = tuple [first,last)| where $A$ consists of $k + 1$ points in $d$ - space.
|
|
This is the sign of the determinant
|
|
\[ \left\Lvert \begin{array}{cccc}
|
|
1 & 1 & 1 & 1 \\
|
|
A[0] & A[1] & \dots & A[k]
|
|
\end{array} \right\Lvert \]
|
|
where |A[a_i]| denotes the cartesian coordinate vector of
|
|
the $i$-th point in $A$, after projection on an axis-aligned |k|-affine subspace.
|
|
\precond |size [first,last) <= d| and |A[i].dimension() == d|
|
|
$\forall 0 \leq i \leq d$
|
|
and the value type of |ForwardIterator| is |Point_d<R>|.}*/
|
|
{
|
|
typedef typename std::iterator_traits<ForwardIterator>::value_type PointD;
|
|
typedef typename PointD::R R;
|
|
typename R::Coaffine_orientation_d or_;
|
|
return or_(first,last);
|
|
}
|
|
|
|
|
|
template <class R, class ForwardIterator>
|
|
Oriented_side side_of_oriented_sphere(
|
|
ForwardIterator first, ForwardIterator last,
|
|
const Point_d<R>& x)
|
|
/*{\Mfuncl determines to which side of the sphere $S$ defined by the
|
|
points in |A = tuple [first,last)| the point $x$ belongs,
|
|
where $A$ consists of $d + 1$ points in $d$ - space.
|
|
The positive side is determined by the positive sign
|
|
of the determinant \[
|
|
\left\Lvert \begin{array}{ccccc}
|
|
1 & 1 & 1 & 1 & 1\\
|
|
|lift(A[0])| & |lift(A[1])| & \dots & |lift(A[d])| & |lift(x)|
|
|
\end{array} \right\Lvert \]
|
|
where for a point $p$ with cartesian coordinates $p_i$ we use
|
|
|lift(p)| to denote the $d + 1$-dimensional point with cartesian
|
|
coordinate vector $(p_0, \ldots,p_{d-1},\sum_{0 \le i < d}p_i^2)$. If
|
|
the points in $A$ are positively oriented then the positive
|
|
side is the inside of the sphere and the negative side is
|
|
the outside of the sphere.
|
|
\precond value type of |ForwardIterator| is |Point_d<R>|.}*/
|
|
{ typename R::Side_of_oriented_sphere_d side;
|
|
return side(first,last,x); }
|
|
|
|
template <class R, class ForwardIterator>
|
|
Bounded_side side_of_bounded_sphere(
|
|
ForwardIterator first, ForwardIterator last,
|
|
const Point_d<R> &p)
|
|
/*{\Mfunc determines whether the point |p| lies |ON_BOUNDED_SIDE|,
|
|
|ON_BOUNDARY|, or |ON_UNBOUNDED_SIDE| of the sphere defined by
|
|
the points in |A = tuple [first,last)| where $A$ consists of $d+1$
|
|
points in $d$-space.
|
|
\precond value type of |ForwardIterator| is |Point_d<R>| and
|
|
$|orientation(first,last)| \neq |ZERO|$.}*/
|
|
{ typename R::Side_of_bounded_sphere_d side;
|
|
return side(first,last,p); }
|
|
|
|
template <class R, class ForwardIterator>
|
|
bool contained_in_simplex(
|
|
ForwardIterator first, ForwardIterator last, const Point_d<R>& p)
|
|
/*{\Mfunc determines whether |p| is contained in the simplex spanned
|
|
by the points in |A = tuple [first,last)|. |A| may consists of up to
|
|
$d + 1$ points.
|
|
\precond value type of |ForwardIterator| is |Point_d<R>| and
|
|
the points in |A| are affinely independent.}*/
|
|
{ typename R::Contained_in_simplex_d contained;
|
|
return contained(first,last,p); }
|
|
|
|
template <class R, class ForwardIterator>
|
|
bool contained_in_affine_hull(
|
|
ForwardIterator first, ForwardIterator last,
|
|
const Point_d<R>& p)
|
|
/*{\Mfunc determines whether $p$ is contained in the affine hull
|
|
of the points in |A = tuple [first,last)|.
|
|
\precond value type of |ForwardIterator| is |Point_d<R>|.}*/
|
|
{ typename R::Contained_in_affine_hull_d contained;
|
|
return contained(first,last,p); }
|
|
|
|
template <class ForwardIterator>
|
|
int affine_rank(ForwardIterator first, ForwardIterator last)
|
|
/*{\Mfunc computes the affine rank of the points in |A = tuple [first,last)|.
|
|
\precond value type of |ForwardIterator| is |Point_d<R>|.}*/
|
|
{ typedef typename std::iterator_traits<ForwardIterator>::value_type PointD;
|
|
typedef typename PointD::R R;
|
|
typename R::Affine_rank_d rank;
|
|
return rank(first,last); }
|
|
|
|
template <class ForwardIterator>
|
|
bool affinely_independent(ForwardIterator first, ForwardIterator last)
|
|
/*{\Mfunc decides whether the points in |A = tuple [first,last)| are
|
|
affinely independent.
|
|
\precond value type of |ForwardIterator| is |Point_d<R>|.}*/
|
|
{ typedef typename std::iterator_traits<ForwardIterator>::value_type PointD;
|
|
typedef typename PointD::R R;
|
|
typename R::Affinely_independent_d ind;
|
|
return ind(first,last); }
|
|
|
|
template <class R>
|
|
Comparison_result compare_lexicographically(
|
|
const Point_d<R>& p1, const Point_d<R>& p2)
|
|
/*{\Mfunc compares the Cartesian coordiantes of points |p1| and |p2|
|
|
lexicographically.}*/
|
|
{ typename R::Compare_lexicographically_d cmp;
|
|
return cmp(p1,p2); }
|
|
|
|
template <class R>
|
|
bool lexicographically_smaller(
|
|
const Point_d<R>& p1, const Point_d<R>& p2)
|
|
/*{\Mfunc returns true iff $|p1| < |p2|$ in the Cartesian lexicographic
|
|
order of points.}*/
|
|
{ typename R::Less_lexicographically_d lt;
|
|
return lt(p1,p2); }
|
|
|
|
template <class R>
|
|
bool lexicographically_smaller_or_equal(
|
|
const Point_d<R>& p1, const Point_d<R>& p2)
|
|
/*{\Mfunc returns true iff $|p1| \leq |p2|$ in the Cartesian lexicographic
|
|
order of points.}*/
|
|
{ typename R::Less_or_equal_lexicographically_d le;
|
|
return le(p1,p2); }
|
|
|
|
template <class R, class ForwardIterator>
|
|
bool contained_in_linear_hull(
|
|
ForwardIterator first, ForwardIterator last, const Vector_d<R>& x)
|
|
/*{\Mfunc determines whether $x$ is contained in the linear hull
|
|
of the vectors in |A = tuple [first,last)|.
|
|
\precond value type of |ForwardIterator| is |Vector_d<R>|.}*/
|
|
{ typename R::Contained_in_linear_hull_d contained;
|
|
return contained(first,last,x); }
|
|
|
|
template <class ForwardIterator>
|
|
int linear_rank(
|
|
ForwardIterator first, ForwardIterator last)
|
|
/*{\Mfunc computes the linear rank of the vectors in
|
|
|A = tuple [first,last)|.
|
|
\precond value type of |ForwardIterator| is |Vector_d<R>|.}*/
|
|
{ typedef typename std::iterator_traits<ForwardIterator>::
|
|
value_type value_type;
|
|
typedef typename value_type::R R;
|
|
typename R::Linear_rank_d rank;
|
|
return rank(first,last);
|
|
}
|
|
|
|
template <class ForwardIterator>
|
|
bool linearly_independent(
|
|
ForwardIterator first, ForwardIterator last)
|
|
/*{\Mfunc decides whether the vectors in $A$ are linearly independent.
|
|
\precond value type of |ForwardIterator| is |Vector_d<R>|.}*/
|
|
{ TUPLE_DIM_CHECK(first,last,linearly_independent);
|
|
typedef typename std::iterator_traits<ForwardIterator>::
|
|
value_type value_type;
|
|
typedef typename value_type::R R;
|
|
typename R::Linearly_independent_d independent;
|
|
return independent(first,last);
|
|
}
|
|
|
|
|
|
} //namespace CGAL
|
|
#endif // CGAL_PREDICATES_D_H
|