357 lines
13 KiB
C++
Executable File
357 lines
13 KiB
C++
Executable File
// Copyright (c) 2000,2001
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// Utrecht University (The Netherlands),
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// ETH Zurich (Switzerland),
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// INRIA Sophia-Antipolis (France),
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// Max-Planck-Institute Saarbruecken (Germany),
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// and Tel-Aviv University (Israel). All rights reserved.
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//
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// This file is part of CGAL (www.cgal.org); you can redistribute it and/or
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// modify it under the terms of the GNU Lesser General Public License as
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// published by the Free Software Foundation; either version 3 of the License,
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// or (at your option) any later version.
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//
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// Licensees holding a valid commercial license may use this file in
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// accordance with the commercial license agreement provided with the software.
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//
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// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
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// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
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//
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// $URL$
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// $Id$
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// SPDX-License-Identifier: LGPL-3.0+
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//
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// Author(s) : Michael Seel
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#ifndef CGAL_HYPERPLANEHD_H
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#define CGAL_HYPERPLANEHD_H
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#include <CGAL/basic.h>
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#include <CGAL/Quotient.h>
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#include <CGAL/Kernel_d/PointHd.h>
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#include <CGAL/Kernel_d/VectorHd.h>
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#include <CGAL/Kernel_d/Aff_transformationHd.h>
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namespace CGAL {
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#define PointHd PointHd2
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template <class RT, class LA>
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std::istream& operator>>(std::istream&, HyperplaneHd<RT,LA>&);
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template <class RT, class LA>
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std::ostream& operator<<(std::ostream&, const HyperplaneHd<RT,LA>&);
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/*{\Manpage{Hyperplane_d}{R}{Hyperplanes in d-space}{h}}*/
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/*{\Msubst
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Hd<RT,LA>#_d<R>
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HyperplaneHd#Hyperplane_d
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Quotient<RT>#FT
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}*/
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template <class _RT, class _LA>
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class HyperplaneHd : public Handle_for< Tuple_d<_RT,_LA> > {
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typedef Tuple_d<_RT,_LA> Tuple;
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typedef Handle_for<Tuple> Base;
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typedef HyperplaneHd<_RT,_LA> Self;
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using Base::ptr;
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/*{\Mdefinition An instance of data type |HyperplaneHd| is an
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oriented hyperplane in $d$ - dimensional space. A hyperplane $h$ is
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represented by coefficients $(c_0,c_1,\ldots,c_d)$ of type |RT|. At
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least one of $c_0$ to $c_{ d - 1 }$ must be non-zero. The plane
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equation is $\sum_{ 0 \le i < d } c_i x_i + c_d = 0$, where $x_0$ to
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$x_{d-1}$ are Cartesian point coordinates.
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For a particular $x$ the sign of $\sum_{ 0 \le i < d } c_i x_i +
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c_d$ determines the position of a point $x$ with respect to the
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hyperplane (on the hyperplane, on the negative side, or on the
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positive side).
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There are two equality predicates for hyperplanes. The (weak)
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equality predicate (|weak_equality|) declares two hyperplanes equal if
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they consist of the same set of points, the strong equality predicate
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(|operator==|) requires in addition that the negative halfspaces
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agree. In other words, two hyperplanes are strongly equal if their
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coefficient vectors are positive multiples of each other and they are
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(weakly) equal if their coefficient vectors are multiples of each
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other.}*/
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const typename _LA::Vector& vector_rep() const { return ptr()->v; }
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_RT& entry(int i) { return ptr()->v[i]; }
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const _RT& entry(int i) const { return ptr()->v[i]; }
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void invert_rep() { ptr()->invert(); }
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public:
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/*{\Mtypes 4}*/
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typedef _RT RT;
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/*{\Mtypemember the ring type.}*/
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typedef Quotient<_RT> FT;
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/*{\Mtypemember the field type.}*/
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typedef _LA LA;
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/*{\Mtypemember the linear algebra layer.}*/
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typedef typename Tuple::const_iterator Coefficient_const_iterator;
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/*{\Mtypemember a read-only iterator for the coefficients.}*/
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/*{\Mcreation h 4}*/
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/*{\Moptions nextwarning=no}*/
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HyperplaneHd(int d = 0) : Base( Tuple(d+1) ) {}
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/*{\Mcreate introduces a variable |\Mvar| of type |\Mname|
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initialized to some hyperplane in $d$ - dimensional space. }*/
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template <class InputIterator>
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HyperplaneHd(int d, InputIterator first, InputIterator last)
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: Base( Tuple(d+1,first,last) ) {}
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/*{\Mcreate introduces a variable |\Mvar| of type |\Mname|
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initialized to the hyperplane with coefficients |set [first,last)|.
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\precond |size [first,last) == d+1| and the value type of
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InputIterator is |RT|.}*/
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template <class InputIterator>
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HyperplaneHd(int d, InputIterator first, InputIterator last, const RT& D)
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: Base( Tuple(d+1,first,last,D) ) {}
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/*{\Mcreate introduces a variable |\Mvar| of type |\Mname|
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initialized to the hyperplane with coefficients |set [first,last)| and
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|D|. \precond |size [first,last) == d| and the value type of
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InputIterator is |RT|.}*/
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/* We want to construct a hyperplane that passes through a set |P =
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set [first,last)| of points in $d$-dimensional space and has a
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specified point $o$ on a specified side. We simply have to find a
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vector $x$ such that $P^T \cdot x = 0$ for every point in $P$. This
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amounts to solving a homogeneous linear system. If the system has only
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a trivial solution the task at hand is unsolvable and we report an
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error. So assume that the system has a non-trivial solution. Let
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vectors $s_1, \ldots, s_k$ span the solution space. if |side == ZERO|
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we may take any $s_j$ as the normal vector of our hyperplane. if
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$|side| \neq 0$ and the task at hand is solvable there must be a $j$
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such that $o^T \cdot s_j \neq 0$. We take $s_j$ as the normal vector
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of our hyperplane and use |o| to normalize the hyperplane equation. */
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template <class ForwardIterator>
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void
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construct_from_points(ForwardIterator first, ForwardIterator last,
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const PointHd<RT,LA>& o, Oriented_side side)
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{
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TUPLE_DIM_CHECK(first,last,hyperplane::construction);
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CGAL_assertion_msg((first->dimension()==o.dimension()),
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"hyperplane::construction: dimensions disagree.");
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int d = first->dimension(); // we are in $d$ - dimensional space
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int m = static_cast<int>(std::distance(first,last)); // |P| has $m$ points
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typename LA::Matrix A(m,d + 1);
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for (int i = 0; i < m; i++) { /* define $i$-th equation */
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for (int j = 0; j <= d; j++)
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A(i,j) = first->homogeneous(j); // $j$ - th coord of $i$-th point
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++first;
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}
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typename LA::Matrix spanning_vecs; // columns span solution
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int dim = LA::homogeneous_linear_solver(A,spanning_vecs);
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if (dim == 0)
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CGAL_error_msg("HyperplaneHd::constructor: \
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set P is full dimensional.");
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if (side == ON_ORIENTED_BOUNDARY) {
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ptr()->v = spanning_vecs.column(0);
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return;
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}
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RT sum = 0;
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int j;
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for (j = 0; j < dim; j++) {
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for (int i = 0; i <= d; i++)
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sum += spanning_vecs(i,j)*o.homogeneous(i);
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if (sum != 0) break;
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}
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if (j == dim)
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CGAL_error_msg("HyperplaneHd::constructor: \
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cannot use o to determine side.");
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ptr()->v = spanning_vecs.column(j);
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if ( ( CGAL_NTS sign(sum) > 0 && side == ON_NEGATIVE_SIDE ) ||
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( CGAL_NTS sign(sum) < 0 && side == ON_POSITIVE_SIDE ) )
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invert_rep();
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}
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template <class ForwardIterator>
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HyperplaneHd(ForwardIterator first, ForwardIterator last,
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const PointHd<RT,LA>& o,
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Oriented_side side = ON_ORIENTED_BOUNDARY)
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/*{\Mcreate constructs some hyperplane that passes through the points
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in |set [first,last)|. If |side| is |ON_POSITIVE_SIDE| or
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|ON_NEGATIVE_SIDE| then |o| is on that side of the constructed
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hyperplane. \precond A hyperplane with the stated properties must
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exist. The value type of |ForwardIterator| is |PointHd<RT,LA>|. }*/
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: Base( Tuple(o.dimension()+1) )
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{ construct_from_points(first,last,o,side); }
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HyperplaneHd(const PointHd<RT,LA>& p, const DirectionHd<RT,LA>& dir)
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/*{\Mcreate constructs the hyperplane with normal direction |dir|
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that passes through $p$. The direction |dir| points into the positive
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side. \precond |dir| is not the trivial direction.}*/
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: Base( Tuple(p.dimension()+1) ) {
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int d = p.dimension();
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CGAL_assertion_msg((dir.dimension() == d), "HyperplaneHd::constructor: \
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parameter dimensions disagree.");
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CGAL_assertion_msg((dir.dimension() == d), "HyperplaneHd::constructor: \
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parameter dimensions disagree.");
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RT sum = 0;
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for (int i = 0; i < d; i++) {
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sum += dir.delta(i)*p.homogeneous(i);
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entry(i) = dir.delta(i)*p.homogeneous(d);
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}
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entry(d) = -sum;
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}
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HyperplaneHd(const RT& a, const RT& b, const RT& c) :
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Base( Tuple(a,b,c,MatchHelper()) ) {}
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/*{\Mcreate introduces a variable |\Mvar| of type |\Mname| in
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$2$-dimensional space with equation $ax+by+c=0$. }*/
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HyperplaneHd(int a, int b, int c) :
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Base( Tuple(RT(a),RT(b),RT(c),MatchHelper()) ) {}
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HyperplaneHd(const RT& a, const RT& b, const RT& c, const RT& d) :
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Base( Tuple(a,b,c,d) ) {}
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/*{\Mcreate introduces a variable |\Mvar| of type |\Mname| in
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$3$-dimensional space with equation $ax+by+cz+d=0$. }*/
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HyperplaneHd(int a, int b, int c, int d) :
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Base( Tuple(RT(a),RT(b),RT(c),RT(d)) ) {}
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HyperplaneHd(const HyperplaneHd<RT,LA>& h) : Base(h) {}
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~HyperplaneHd() {}
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/*{\Moperations 4 2}*/
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int dimension() const { return ptr()->size()-1; }
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/*{\Mop returns the dimension of |\Mvar|. }*/
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RT operator[](int i) const
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/*{\Marrop returns the $i$-th coefficient of |\Mvar|.
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\precond $0 \leq i \leq d$.}*/
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{ CGAL_assertion_msg((0<=i && i<=(dimension())), "HyperplaneHd::op[]:\
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index out of range."); return entry(i); }
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RT coefficient(int i) const { return entry(i); }
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/*{\Mop returns the $i$-th coefficient of |\Mvar|.
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\precond $0 \leq i \leq d$.}*/
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const typename LA::Vector& coefficient_vector() const
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/*{\Xop returns the coefficient vector $(c_0,\ldots,c_d)$ of |\Mvar|. }*/
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{ return vector_rep(); }
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Coefficient_const_iterator coefficients_begin() const
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/*{\Mop returns an iterator pointing to the first coefficient.}*/
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{ return ptr()->begin(); }
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Coefficient_const_iterator coefficients_end() const
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/*{\Mop returns an iterator pointing beyond the last coefficient.}*/
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{ return ptr()->end(); }
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VectorHd<RT,LA> orthogonal_vector() const;
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/*{\Mop returns the orthogonal vector of |\Mvar|. It points from the
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negative halfspace into the positive halfspace and its
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homogeneous coordinates are $(c_0, \ldots, c_{d - 1},1)$. }*/
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DirectionHd<RT,LA> orthogonal_direction() const
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/*{\Mop returns the orthogonal direction of |\Mvar|. It points from the
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negative halfspace into the positive halfspace. }*/
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{ return orthogonal_vector().direction(); }
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RT value_at(const PointHd<RT,LA>& p) const
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/*{\Xop returns the value of |\Mvar| at the point |p|, i.e.,
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$\sum_{ 0 \le i \le d } h_i p_i$.\\
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Warning: this value depends on the particular representation
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of |\Mvar| and |p|. }*/
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{ CGAL_assertion_msg((dimension()==p.dimension()),"HyperplaneHd::value_at:\
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dimensions disagree.");
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return vector_rep()*p.vector_rep();
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}
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Oriented_side oriented_side(const PointHd<RT,LA>& p) const
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/*{\Mop returns the side of the hyperplane |\Mvar| containing $p$. }*/
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/*{\Mtext \setopdims{2cm}{2cm}}*/
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{
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CGAL_assertion_msg((dimension()==p.dimension()),
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"HyperplaneHd::oriented_side: dimensions do not agree.");
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return CGAL_NTS sign(value_at(p));
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}
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bool has_on(const PointHd<RT,LA>& p) const
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/*{\Mop returns true iff point |p| lies on the hyperplane |\Mvar|. }*/
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{ return (oriented_side(p) == ON_ORIENTED_BOUNDARY); }
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bool has_on_boundary(const PointHd<RT,LA>& p) const
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/*{\Mop returns true iff point |p| lies on the boundary of
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hyperplane |\Mvar|. }*/
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{ return (oriented_side(p) == ON_ORIENTED_BOUNDARY); }
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bool has_on_positive_side(const PointHd<RT,LA>& p) const
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/*{\Mop returns true iff point |p| lies on the positive side of
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hyperplane |\Mvar|. }*/
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{ return (oriented_side(p) == ON_POSITIVE_SIDE); }
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bool has_on_negative_side(const PointHd<RT,LA>& p) const
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/*{\Mop returns true iff point |p| lies on the negative side of
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hyperplane |\Mvar|. }*/
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{ return (oriented_side(p) == ON_NEGATIVE_SIDE); }
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/*{\Mtext \restoreopdims }*/
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HyperplaneHd<RT,LA> transform(const Aff_transformationHd<RT,LA>& t) const
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/*{\Mop returns $t(h)$.}*/
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{ Aff_transformationHd<RT,LA> t_inv = t.inverse();
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typename LA::Vector res = LA::transpose(t_inv.matrix())*vector_rep();
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if ( t_inv.is_odd() ) res = -res;
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return HyperplaneHd<RT,LA>(dimension(),res.begin(),res.end()); }
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/*{\Mtext \headerline{Non-Member Functions}}*/
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static Comparison_result weak_cmp(
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const HyperplaneHd<RT,LA>&, const HyperplaneHd<RT,LA>&);
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static Comparison_result strong_cmp(
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const HyperplaneHd<RT,LA>&, const HyperplaneHd<RT,LA>&);
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bool operator==(const HyperplaneHd<RT,LA>& h2) const
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{ if (this->identical(h2)) return true;
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if (dimension()!=h2.dimension()) return false;
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return HyperplaneHd<RT,LA>::strong_cmp(*this,h2) == EQUAL;
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}
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bool operator!=(const HyperplaneHd<RT,LA>& h2) const
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{ return !operator==(h2); }
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friend std::istream& operator>> <>
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(std::istream&, HyperplaneHd<RT,LA>&);
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friend std::ostream& operator<< <>
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(std::ostream&, const HyperplaneHd<RT,LA>&);
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}; // end of class HyperplaneHd
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template <class RT, class LA>
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bool weak_equality(const HyperplaneHd<RT,LA>& h1,
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const HyperplaneHd<RT,LA>& h2)
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/*{\Mfunc test for weak equality. }*/
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{ if (h1.identical(h2)) return true;
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if (h1.dimension()!=h2.dimension()) return false;
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return HyperplaneHd<RT,LA>::weak_cmp(h1,h2) == EQUAL;
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}
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/*{\Mimplementation
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Hyperplanes are implemented by arrays of integers as an item type.
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All operations like creation, initialization, tests, vector
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arithmetic, input and output on a hyperplane $h$ take time
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$O(|h.dimension()|)$. coordinate access and |dimension()| take
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constant time. The space requirement is $O(|h.dimension()|)$. }*/
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#undef PointHd
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} //namespace CGAL
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#endif // CGAL_HYPERPLANEHD_H
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//----------------------- end of file ----------------------------------
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