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// 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) // // $URL: https://github.com/CGAL/cgal/blob/v6.1/Kernel_d/include/CGAL/predicates_d.h $ // $Id: include/CGAL/predicates_d.h b26b07a1242 $ // SPDX-License-Identifier: LGPL-3.0-or-later OR LicenseRef-Commercial // // // Author(s) : Michael Seel #ifndef CGAL_PREDICATES_D_H #define CGAL_PREDICATES_D_H #include #include #include 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|. 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 OutputIterator barycentric_coordinates( ForwardIterator first, ForwardIterator last, const Point_d& 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|, |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 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|.}*/ { typedef typename std::iterator_traits::value_type PointD; typedef typename PointD::R R; typename R::Orientation_d or_; return or_(first,last); } template 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|.}*/ { typedef typename std::iterator_traits::value_type PointD; typedef typename PointD::R R; typename R::Coaffine_orientation_d or_; return or_(first,last); } template Oriented_side side_of_oriented_sphere( ForwardIterator first, ForwardIterator last, const Point_d& 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|.}*/ { typename R::Side_of_oriented_sphere_d side; return side(first,last,x); } template Bounded_side side_of_bounded_sphere( ForwardIterator first, ForwardIterator last, const Point_d &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| and $|orientation(first,last)| \neq |ZERO|$.}*/ { typename R::Side_of_bounded_sphere_d side; return side(first,last,p); } template bool contained_in_simplex( ForwardIterator first, ForwardIterator last, const Point_d& 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| and the points in |A| are affinely independent.}*/ { typename R::Contained_in_simplex_d contained; return contained(first,last,p); } template bool contained_in_affine_hull( ForwardIterator first, ForwardIterator last, const Point_d& 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|.}*/ { typename R::Contained_in_affine_hull_d contained; return contained(first,last,p); } template 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|.}*/ { typedef typename std::iterator_traits::value_type PointD; typedef typename PointD::R R; typename R::Affine_rank_d rank; return rank(first,last); } template 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|.}*/ { typedef typename std::iterator_traits::value_type PointD; typedef typename PointD::R R; typename R::Affinely_independent_d ind; return ind(first,last); } template Comparison_result compare_lexicographically( const Point_d& p1, const Point_d& p2) /*{\Mfunc compares the Cartesian coordinates of points |p1| and |p2| lexicographically.}*/ { typename R::Compare_lexicographically_d cmp; return cmp(p1,p2); } template bool lexicographically_smaller( const Point_d& p1, const Point_d& 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 bool lexicographically_smaller_or_equal( const Point_d& p1, const Point_d& 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 bool contained_in_linear_hull( ForwardIterator first, ForwardIterator last, const Vector_d& 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|.}*/ { typename R::Contained_in_linear_hull_d contained; return contained(first,last,x); } template 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|.}*/ { typedef typename std::iterator_traits:: value_type value_type; typedef typename value_type::R R; typename R::Linear_rank_d rank; return rank(first,last); } template 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|.}*/ { TUPLE_DIM_CHECK(first,last,linearly_independent); typedef typename std::iterator_traits:: 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