// Copyright (c) 2000,2001 Utrecht University (The Netherlands),
// ETH Zurich (Switzerland), Freie Universitaet Berlin (Germany),
// INRIA Sophia-Antipolis (France), Martin-Luther-University Halle-Wittenberg
// (Germany), Max-Planck-Institute Saarbruecken (Germany), RISC Linz (Austria),
// 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; version 2.1 of the License.
// See the file LICENSE.LGPL distributed with CGAL.
//
// 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.
//
// $Source: /CVSROOT/CGAL/Packages/Conic_2/include/CGAL/ConicHPA2.h,v $
// $Revision: 1.7 $ $Date: 2004/09/14 09:32:08 $
// $Name: $
//
// Author(s) : Bernd Gaertner, Sven Schoenherr <sven@inf.ethz.ch>
#ifndef CGAL_CONICHPA2_H
#define CGAL_CONICHPA2_H
// includes
#include <CGAL/Conic_misc.h>
#include <CGAL/kernel_assertions.h>
CGAL_BEGIN_NAMESPACE
template < class PT, class DA>
class ConicHPA2;
template < class PT, class DA>
class _Min_ellipse_2_adapterH2__Ellipse;
template < class _PT, class _DA>
class ConicHPA2
{
public:
// types
typedef _PT PT;
typedef _DA DA;
typedef typename _DA::RT RT;
//private:
//friend class Conic_2< CGAL::Homogeneous<RT> >;
friend class _Min_ellipse_2_adapterH2__Ellipse<PT,DA>;
DA dao;
RT _r, _s, _t, _u, _v, _w;
Conic_type type;
CGAL::Orientation o;
bool empty, trivial, degenerate;
void
set_linear_combination (const RT& a1, const ConicHPA2<PT,DA>& c1,
const RT& a2, const ConicHPA2<PT,DA>& c2)
{
_r = a1 * c1.r() + a2 * c2.r();
_s = a1 * c1.s() + a2 * c2.s();
_t = a1 * c1.t() + a2 * c2.t();
_u = a1 * c1.u() + a2 * c2.u();
_v = a1 * c1.v() + a2 * c2.v();
_w = a1 * c1.w() + a2 * c2.w();
}
static void set_two_linepairs (const PT& p1,
const PT& p2,
const PT& p3,
const PT& p4,
ConicHPA2<PT,DA>& pair1,
ConicHPA2<PT,DA>& pair2)
{
RT x1, y1, h1, x2, y2, h2, x3, y3, h3, x4, y4, h4;
const DA& da = pair1.da();
da.get (p1, x1, y1, h1);
da.get (p2, x2, y2, h2);
da.get (p3, x3, y3, h3);
da.get (p4, x4, y4, h4);
CGAL::Orientation side1_24 = (CGAL::Orientation)(CGAL_NTS sign
(-h2*x1*y4+h1*x2*y4
+h2*x4*y1-h4*x2*y1
+h4*x1*y2-h1*x4*y2)),
side3_24 = (CGAL::Orientation)(CGAL_NTS sign
(-h2*x3*y4+h3*x2*y4
+h2*x4*y3-h4*x2*y3
+h4*x3*y2-h3*x4*y2));
if (side1_24 != side3_24) {
// (counter)clockwise order
pair1.set_linepair (p1, p2, p3, p4);
pair2.set_linepair (p2, p3, p4, p1);
} else {
CGAL::Orientation side1_32 = (CGAL::Orientation)(CGAL_NTS sign
(-h3*x1*y2+h1*x3*y2
+h3*x2*y1-h2*x3*y1
+h2*x1*y3-h1*x2*y3));
if (side1_32 != side3_24) {
// p1, p2 need to be swapped
pair1.set_linepair (p2, p1, p3, p4);
pair2.set_linepair (p1, p3, p4, p2);
} else {
// p2, p3 need to be swapped
pair1.set_linepair (p1, p3, p2, p4);
pair2.set_linepair (p3, p2, p4, p1);
}
}
}
void set_ellipse (const ConicHPA2<PT,DA>& pair1,
const ConicHPA2<PT,DA>& pair2)
{
RT b = RT(2) * (pair1.r() * pair2.s() + pair1.s() * pair2.r()) -
pair1.t() * pair2.t();
set_linear_combination (pair2.det()-b, pair1,
pair1.det()-b, pair2);
}
void set (const ConicHPA2<PT,DA>& c1,
const ConicHPA2<PT,DA>& c2,
const PT& p)
{
set_linear_combination (c2.evaluate(p), c1, -c1.evaluate(p), c2);
}
CGAL::Sign vol_derivative (RT dr, RT ds, RT dt,
RT du, RT dv, RT dw) const
{
RT a1 = RT(4)*r()*ds+RT(4)*dr*s()-RT(2)*t()*dt,
a0 = RT(4)*r()*s()-t()*t(),
b1 = (RT(4)*r()*s()-t()*t())*dw+(RT(4)*r()*ds+RT(4)*dr*s()-
RT(2)*t()*dt)*w()-u()*u()*ds -
RT(2)*u()*du*s()-v()*v()*dr-RT(2)*v()*dv*r()+u()*v()*dt+
(u()*dv+du*v())*t(),
b0 = (RT(4)*r()*s()-t()*t())*w()
-u()*u()*s()-v()*v()*r()+u()*v()*t(),
c0 = -RT(2)*a0*b1 + RT(3)*a1*b0;
return CGAL_NTS sign (-CGAL_NTS sign (c0)*o);
}
double vol_minimum (RT dr, RT ds, RT dt, RT du, RT dv, RT dw) const
{
RT a2 = RT(4)*dr*ds-dt*dt,
a1 = RT(4)*r()*ds+RT(4)*dr*s()-RT(2)*t()*dt,
a0 = RT(4)*r()*s()-t()*t(),
b3 = (RT(4)*dr*ds-dt*dt)*dw-du*du*ds-dv*dv*dr+du*dv*dt,
b2 = (RT(4)*r()*ds+RT(4)*dr*s()-RT(2)*t()*dt)*dw+
(RT(4)*dr*ds-dt*dt)*w()-RT(2)*u()*du*ds-du*du*s()-
RT(2)*v()*dv*dr-dv*dv*r()+(u()*dv+du*v())*dt+du*dv*t(),
b1 = (RT(4)*r()*s()-t()*t())*dw+(RT(4)*r()*ds+RT(4)*dr*s()-
RT(2)*t()*dt)*w()-u()*u()*ds -
RT(2)*u()*du*s()-v()*v()*dr-RT(2)*v()*dv*r()+u()*v()*dt+
(u()*dv+du*v())*t(),
b0 = (RT(4)*r()*s()-t()*t())*w()
-u()*u()*s()-v()*v()*r()+u()*v()*t(),
c3 = -RT(3)*a1*b3 + RT(2)*a2*b2,
c2 = -RT(6)*a0*b3 - a1*b2 + RT(4)*a2*b1,
c1 = -RT(4)*a0*b2 + a1*b1 + RT(6)*a2*b0,
c0 = -RT(2)*a0*b1 + RT(3)*a1*b0;
double roots[3];
int nr_roots = solve_cubic
(CGAL::to_double(c3), CGAL::to_double(c2),
CGAL::to_double(c1), CGAL::to_double(c0),
roots[0], roots[1], roots[2]);
CGAL_kernel_precondition (nr_roots > 0); // minimum exists
return best_value (roots, nr_roots,
CGAL::to_double(a2), CGAL::to_double(a1),
CGAL::to_double(a0), CGAL::to_double(b3),
CGAL::to_double(b2), CGAL::to_double(b1),
CGAL::to_double(b0));
}
protected:
RT det () const
{
return RT(4)*s()*r() - t()*t();
}
void analyse( )
{
RT d = det();
type = (Conic_type)(CGAL_NTS sign(d));
switch (type) {
case HYPERBOLA:
{
trivial = empty = false;
RT z_prime = d*w() - u()*u()*s() - v()*v()*r() + u()*v()*t();
o = (CGAL::Orientation)(CGAL_NTS sign (z_prime));
degenerate = (o == CGAL::ZERO);
}
break;
case PARABOLA:
{
if (!CGAL_NTS is_zero (r())) {
trivial = false;
degenerate = (t()*u() == RT(2)*r()*v());
if (degenerate) {
CGAL::Sign discr = (CGAL::Sign)
CGAL_NTS sign(u()*u()-RT(4)*r()*w());
switch (discr) {
case CGAL::NEGATIVE:
empty = true;
o = (CGAL::Orientation)(CGAL_NTS sign (w()));
break;
case CGAL::ZERO:
empty = false;
o = (CGAL::Orientation)(CGAL_NTS sign (r()));
break;
case CGAL::POSITIVE:
empty = false;
o = CGAL::ZERO;
break;
}
} else {
empty = false;
o = (CGAL::Orientation)(-CGAL_NTS sign (r()));
}
} else if (!CGAL_NTS is_zero (s())) {
trivial = false;
degenerate = (t()*v() == RT(2)*s()*u());
if (degenerate) {
CGAL::Sign discr = (CGAL::Sign)
CGAL_NTS sign(v()*v()-RT(4)*s()*w());
switch (discr) {
case CGAL::NEGATIVE:
empty = true;
o = (CGAL::Orientation)(CGAL_NTS sign (w()));
break;
case CGAL::ZERO:
empty = false;
o = (CGAL::Orientation)(CGAL_NTS sign (s()));
break;
case CGAL::POSITIVE:
empty = false;
o = CGAL::ZERO;
break;
}
} else {
empty = false;
o = (CGAL::Orientation)(-CGAL_NTS sign (s()));
}
} else { // r=0, s=0
degenerate = true;
bool uv_zero = CGAL_NTS is_zero (u())
&& CGAL_NTS is_zero (v());
trivial = uv_zero && CGAL_NTS is_zero (w());
empty = uv_zero && !trivial;
if (empty)
o = (CGAL::Orientation)(CGAL_NTS sign (w()));
else if (trivial)
o = CGAL::POSITIVE;
else
o = CGAL::ZERO;
}
}
break;
case ELLIPSE:
{
trivial = false;
RT z_prime = d*w() - u()*u()*s() - v()*v()*r() + u()*v()*t();
if (CGAL_NTS is_positive (r())) {
empty = CGAL_NTS is_positive(CGAL_NTS sign (z_prime));
empty ? o = CGAL::POSITIVE : o = CGAL::NEGATIVE;
} else {
empty = CGAL_NTS is_negative(CGAL_NTS sign (z_prime));
empty ? o = CGAL::NEGATIVE : o = CGAL::POSITIVE;
}
degenerate = empty || CGAL_NTS is_zero (z_prime);
}
break;
}
}
RT evaluate (const PT& p) const
{
RT x, y, h;
dao.get (p, x, y, h);
return r()*x*x + s()*y*y + t()*x*y + u()*x*h + v()*y*h + w()*h*h;
}
public:
ConicHPA2 ( const DA& da = DA()) : dao( da) { }
ConicHPA2 (RT r, RT s, RT t, RT u, RT v, RT w, const DA& da = DA())
: dao( da), _r(r), _s(s), _t(t), _u(u), _v(v), _w(w)
{
analyse();
}
const DA& da() const
{
return dao;
}
RT r() const { return _r;}
RT s() const { return _s;}
RT t() const { return _t;}
RT u() const { return _u;}
RT v() const { return _v;}
RT w() const { return _w;}
PT center () const
{
CGAL_kernel_precondition (type != PARABOLA);
// PT p;
// replaced previous line by following hack (no idea
// why original version doesn't work)
typename DA::Point_2 p;
RT two = RT(2);
dao.set( p, two*s()*u() - t()*v(), two*r()*v() - t()*u(), -det());
return p;
}
Conic_type conic_type () const
{
return type;
}
bool is_hyperbola () const
{
return (type == HYPERBOLA);
}
bool is_parabola () const
{
return (type == PARABOLA);
}
bool is_ellipse () const
{
return (type == ELLIPSE);
}
bool is_circle () const
{
return (type == ELLIPSE && (r()==s()) && CGAL_NTS is_zero (t()));
}
bool is_empty () const
{
return empty;
}
bool is_trivial () const
{
return trivial;
}
bool is_degenerate () const
{
return degenerate;
}
CGAL::Orientation orientation () const
{
return o;
}
CGAL::Oriented_side oriented_side (const PT& p) const
{
return (CGAL::Oriented_side)(CGAL_NTS sign (evaluate (p)));
}
bool has_on_positive_side (const PT& p) const
{
return (CGAL_NTS is_positive (evaluate(p)));
}
bool has_on_negative_side (const PT& p) const
{
return (CGAL_NTS is_negative (evaluate(p)));
}
bool has_on_boundary (const PT& p) const
{
return (CGAL_NTS is_zero (evaluate(p)));
}
bool has_on (const PT& p) const
{
return (CGAL_NTS is_zero (evaluate(p)));
}
Convex_side convex_side (const PT& p) const
{
switch (o) {
case CGAL::POSITIVE:
return (Convex_side)( CGAL_NTS sign (evaluate (p)));
case CGAL::NEGATIVE:
return (Convex_side)(-CGAL_NTS sign (evaluate (p)));
case CGAL::ZERO:
return (Convex_side)( CGAL_NTS sign (CGAL_NTS abs (evaluate(p))));
}
// keeps g++ happy
return( Convex_side( 0));
}
bool has_on_convex_side (const PT& p) const
{
return (convex_side (p) == ON_CONVEX_SIDE);
}
bool has_on_nonconvex_side (const PT& p) const
{
return (convex_side (p) == ON_NONCONVEX_SIDE);
}
bool operator == ( const ConicHPA2<_PT,_DA>& c) const
{
// find coefficient != 0
RT factor1(0);
if ( ! CGAL_NTS is_zero( r())) factor1 = r(); else
if ( ! CGAL_NTS is_zero( s())) factor1 = s(); else
if ( ! CGAL_NTS is_zero( t())) factor1 = t(); else
if ( ! CGAL_NTS is_zero( u())) factor1 = u(); else
if ( ! CGAL_NTS is_zero( v())) factor1 = v(); else
if ( ! CGAL_NTS is_zero( w())) factor1 = w(); else
CGAL_kernel_assertion_msg( false, "all coefficients zero");
// find coefficient != 0
RT factor2(0);
if ( ! CGAL_NTS is_zero( c.r())) factor2 = c.r(); else
if ( ! CGAL_NTS is_zero( c.s())) factor2 = c.s(); else
if ( ! CGAL_NTS is_zero( c.t())) factor2 = c.t(); else
if ( ! CGAL_NTS is_zero( c.u())) factor2 = c.u(); else
if ( ! CGAL_NTS is_zero( c.v())) factor2 = c.v(); else
if ( ! CGAL_NTS is_zero( c.w())) factor2 = c.w(); else
CGAL_kernel_assertion_msg( false, "all coefficients zero");
return( ( r()*factor2 == c.r()*factor1)
&& ( s()*factor2 == c.s()*factor1)
&& ( t()*factor2 == c.t()*factor1)
&& ( u()*factor2 == c.u()*factor1)
&& ( v()*factor2 == c.v()*factor1)
&& ( w()*factor2 == c.w()*factor1));
}
void set (RT r_, RT s_, RT t_, RT u_, RT v_, RT w_)
{
_r = r_; _s = s_; _t = t_; _u = u_; _v = v_; _w = w_;
analyse();
}
void set_opposite ()
{
_r = -r(); _s = -s(); _t = -t(); _u = -u(); _v = -v(); _w = -w();
o = CGAL::opposite(orientation());
}
void set_circle (const PT& p1, const PT& p2, const PT& p3)
{
// the circle will have r = s = det*h1*h2*h3, t=0
RT x1, y1, h1, x2, y2, h2, x3, y3, h3;
dao.get (p1, x1, y1, h1);
dao.get (p2, x2, y2, h2);
dao.get (p3, x3, y3, h3);
// precondition: p1, p2, p3 not collinear
RT det = -h1*x3*y2+h3*x1*y2+h1*x2*y3-h2*x1*y3+h2*x3*y1-h3*x2*y1;
CGAL_kernel_precondition (!CGAL_NTS is_zero (det));
// Cramer's rule
RT sqr1 = (-x1*x1 - y1*y1)*h2*h3;
RT sqr2 = (-x2*x2 - y2*y2)*h1*h3;
RT sqr3 = (-x3*x3 - y3*y3)*h1*h2;
_u = -h1*sqr3*y2+h3*sqr1*y2+h1*sqr2*y3-h2*sqr1*y3+h2*sqr3*y1-h3*sqr2*y1;
_v = -h1*x3*sqr2+h3*x1*sqr2+h1*x2*sqr3-h2*x1*sqr3+h2*x3*sqr1-h3*x2*sqr1;
_w = -sqr1*x3*y2+sqr3*x1*y2+sqr1*x2*y3-sqr2*x1*y3+sqr2*x3*y1-sqr3*x2*y1;
_r = det*h1*h2*h3;
_s = _r;
_t = RT(0);
analyse();
CGAL_kernel_postcondition(is_circle());
CGAL_kernel_postcondition(has_on_boundary(p1));
CGAL_kernel_postcondition(has_on_boundary(p2));
CGAL_kernel_postcondition(has_on_boundary(p3));
}
void set_linepair (const PT& p1, const PT& p2, const PT& p3,
const PT& p4, const DA& da = DA())
{
RT x1, y1, h1, x2, y2, h2, x3, y3, h3, x4, y4, h4;
da.get (p1, x1, y1, h1);
da.get (p2, x2, y2, h2);
da.get (p3, x3, y3, h3);
da.get (p4, x4, y4, h4);
// precondition: p1 != p2, p3 != p4
CGAL_kernel_precondition
( ((x1*h2 != x2*h1) || (y1*h2 != y2*h1)) &&
((x3*h4 != x4*h3) || (y3*h4 != y4*h3)) );
RT h1x2_x1h2 = h1*x2-x1*h2;
RT h3x4_x3h4 = h3*x4-x3*h4;
RT y1h2_h1y2 = y1*h2-h1*y2;
RT y3h4_h3y4 = y3*h4-h3*y4;
RT x1y2_y1x2 = x1*y2-y1*x2;
RT x3y4_y3x4 = x3*y4-y3*x4;
_r = y1h2_h1y2 * y3h4_h3y4;
_s = h1x2_x1h2 * h3x4_x3h4;
_t = h1x2_x1h2 * y3h4_h3y4 + y1h2_h1y2 * h3x4_x3h4;
_u = x1y2_y1x2 * y3h4_h3y4 + y1h2_h1y2 * x3y4_y3x4;
_v = x1y2_y1x2 * h3x4_x3h4 + h1x2_x1h2 * x3y4_y3x4;
_w = x1y2_y1x2 * x3y4_y3x4;
analyse();
}
void set_ellipse (const PT& p1, const PT& p2, const PT& p3)
{
RT x1, y1, h1, x2, y2, h2, x3, y3, h3;
dao.get (p1, x1, y1, h1);
dao.get (p2, x2, y2, h2);
dao.get (p3, x3, y3, h3);
// precondition: p1, p2, p3 not collinear
RT det = -h1*x3*y2+h3*x1*y2+h1*x2*y3-h2*x1*y3+h2*x3*y1-h3*x2*y1;
CGAL_kernel_precondition (!CGAL_NTS is_zero (det));
RT x1x1 = x1*x1, y1y1 = y1*y1,
x2x2 = x2*x2, y2y2 = y2*y2,
x3x3 = x3*x3, y3y3 = y3*y3, // x_i^2, y_i^2
x1h1 = x1*h1, y1h1 = y1*h1,
x2h2 = x2*h2, y2h2 = y2*h2,
x3h3 = x3*h3, y3h3 = y3*h3, // x_i h_i, y_i h_i
h1h1 = h1*h1,
h2h2 = h2*h2,
h3h3 = h3*h3, // h_i^2
two = RT(2); // 2
_r = y1y1*h2h2*h3h3 - y1h1*y2h2*h3h3 - y1h1*h2h2*y3h3 +
h1h1*y2y2*h3h3 - h1h1*y2h2*y3h3 + h1h1*h2h2*y3y3;
_s = x1x1*h2h2*h3h3 - x1h1*x2h2*h3h3 - x1h1*h2h2*x3h3 +
h1h1*x2x2*h3h3 - h1h1*x2h2*x3h3 + h1h1*h2h2*x3x3;
_t = -two*x1*y1*h2h2*h3h3 + x1h1*y2h2*h3h3 + x1h1*h2h2*y3h3 +
y1h1*x2h2*h3h3 -two*h1h1*x2*y2*h3h3 + h1h1*x2h2*y3h3 +
y1h1*h2h2*x3h3 + h1h1*y2h2*x3h3 -two*h1h1*h2h2*x3*y3;
_u = -(h1h1*y2y2*x3h3 - h1h1*x2*y2*y3h3 - h1h1*y2h2*x3*y3 +
x1h1*h2h2*y3y3 + h1h1*x2h2*y3y3 +y1y1*x2h2*h3h3 +
y1y1*h2h2*x3h3 - x1*y1*y2h2*h3h3 - y1h1*x2*y2*h3h3 -
x1*y1*h2h2*y3h3 - y1h1*h2h2*x3*y3 + x1h1*y2y2*h3h3);
_v = -(h1h1*x2x2*y3h3 - h1h1*x2*y2*x3h3 - h1h1*x2h2*x3*y3 +
y1h1*h2h2*x3x3 + h1h1*y2h2*x3x3 +x1x1*y2h2*h3h3 +
x1x1*h2h2*y3h3 - x1*y1*x2h2*h3h3 - x1h1*x2*y2*h3h3 -
x1*y1*h2h2*x3h3 - x1h1*h2h2*x3*y3 + y1h1*x2x2*h3h3);
_w = y1y1*x2h2*x3h3 - x1*y1*y2h2*x3h3 - y1h1*x2*y2*x3h3 +
y1h1*y2h2*x3x3 - x1*y1*x2h2*y3h3 + y1h1*x2x2*y3h3 -
y1h1*x2h2*x3*y3 + x1h1*y2y2*x3h3 + x1x1*y2h2*y3h3 -
x1h1*x2*y2*y3h3 - x1h1*y2h2*x3*y3 + x1h1*x2h2*y3y3;
type = ELLIPSE;
degenerate = trivial = empty = false;
o = CGAL::NEGATIVE;
if (CGAL_NTS is_positive (det)) set_opposite ();
}
void set_ellipse (const PT& p1, const PT& p2,
const PT& p3, const PT& p4,
CGAL::Orientation _o = POSITIVE)
{
ConicHPA2<PT,DA> pair1, pair2;
set_two_linepairs (p1, p2, p3, p4, pair1, pair2);
set_ellipse (pair1, pair2);
analyse();
if (o != _o) set_opposite();
}
void set (const PT& p1, const PT& p2, const PT& p3, const PT& p4,
const PT& p5, CGAL::Orientation _o = POSITIVE)
{
ConicHPA2<PT,DA> c1; c1.set_linepair (p1, p2, p3, p4);
ConicHPA2<PT,DA> c2; c2.set_linepair (p1, p4, p2, p3);
set_linear_combination (c2.evaluate (p5), c1,
-c1.evaluate (p5), c2);
analyse();
// precondition: all points distinct <=> conic nontrivial
CGAL_kernel_precondition (!is_trivial());
if (o != _o) set_opposite();
}
};
#ifndef CGAL_NO_OSTREAM_INSERT_CONICHPA2
template< class _PT, class _DA>
std::ostream& operator << ( std::ostream& os, const ConicHPA2<_PT,_DA>& c)
{
return( os << c.r() << ' ' << c.s() << ' ' << c.t() << ' '
<< c.u() << ' ' << c.v() << ' ' << c.w());
}
template< class _PT, class _DA>
std::istream& operator >> ( std::istream& is, ConicHPA2<_PT,_DA>& c)
{
typedef ConicHPA2<_PT,_DA> Conic;
typedef typename _DA::RT RT;
RT r, s, t, u, v, w;
is >> r >> s >> t >> u >> v >> w;
c.set( r, s, t, u, v, w);
return( is);
}
#endif // CGAL_NO_OSTREAM_INSERT_CONICHPA2
CGAL_END_NAMESPACE
#endif // CGAL_CONICHPA2_H
// ===== EOF ==================================================================
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