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dust3d/thirdparty/cgal/CGAL-5.1/include/CGAL/predicates/kernel_ftC3.h

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// Copyright (c) 2000, 2016
// 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/v5.1/Cartesian_kernel/include/CGAL/predicates/kernel_ftC3.h $
// $Id: kernel_ftC3.h 0779373 2020-03-26T13:31:46+01:00 Sébastien Loriot
// SPDX-License-Identifier: LGPL-3.0-or-later OR LicenseRef-Commercial
//
//
// Author(s) : Herve Bronnimann, Sylvain Pion, Oliver Devillers, Mariette Yvinec
#ifndef CGAL_PREDICATES_KERNEL_FTC3_H
#define CGAL_PREDICATES_KERNEL_FTC3_H
#include <CGAL/predicates/sign_of_determinant.h>
#include <CGAL/predicates/kernel_ftC2.h>
#include <CGAL/constructions/kernel_ftC3.h>
namespace CGAL {
template < class FT >
inline
typename Equal_to<FT>::result_type
parallelC3(const FT &v1x, const FT &v1y, const FT &v1z,
const FT &v2x, const FT &v2y, const FT &v2z)
{
return CGAL_AND_3( sign_of_determinant(v1x, v2x, v1y, v2y) == ZERO ,
sign_of_determinant(v1x, v2x, v1z, v2z) == ZERO ,
sign_of_determinant(v1y, v2y, v1z, v2z) == ZERO );
}
template < class FT >
typename Equal_to<FT>::result_type
parallelC3(const FT &s1sx, const FT &s1sy, const FT &s1sz,
const FT &s1tx, const FT &s1ty, const FT &s1tz,
const FT &s2sx, const FT &s2sy, const FT &s2sz,
const FT &s2tx, const FT &s2ty, const FT &s2tz)
{
// NB : Could be made slightly more efficient by computing the z differences
// only when they are needed.
FT v1x = s1tx - s1sx;
FT v1y = s1ty - s1sy;
FT v1z = s1tz - s1sz;
FT v2x = s2tx - s2sx;
FT v2y = s2ty - s2sy;
FT v2z = s2tz - s2sz;
return parallelC3(v1x, v1y, v1z, v2x, v2y, v2z);
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Compare<FT>::result_type
compare_lexicographically_xyzC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz)
{
typedef typename Compare<FT>::result_type Cmp;
Cmp c = CGAL_NTS compare(px, qx);
if (c != EQUAL) return c;
c = CGAL_NTS compare(py, qy);
if (c != EQUAL) return c;
return CGAL_NTS compare(pz, qz);
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Equal_to<FT>::result_type
strict_dominanceC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz)
{
return CGAL_AND_3( CGAL_NTS compare(px, qx) == LARGER ,
CGAL_NTS compare(py, qy) == LARGER ,
CGAL_NTS compare(pz, qz) == LARGER );
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Equal_to<FT>::result_type
dominanceC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz)
{
return CGAL_AND_3( CGAL_NTS compare(px, qx) != SMALLER ,
CGAL_NTS compare(py, qy) != SMALLER ,
CGAL_NTS compare(pz, qz) != SMALLER );
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Equal_to<FT>::result_type
collinearC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz,
const FT &rx, const FT &ry, const FT &rz)
{
FT dpx = px-rx;
FT dqx = qx-rx;
FT dpy = py-ry;
FT dqy = qy-ry;
if (sign_of_determinant(dpx, dqx, dpy, dqy) != ZERO)
return false;
FT dpz = pz-rz;
FT dqz = qz-rz;
return CGAL_AND( sign_of_determinant(dpx, dqx, dpz, dqz) == ZERO ,
sign_of_determinant(dpy, dqy, dpz, dqz) == ZERO );
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Same_uncertainty_nt<Orientation, FT>::type
orientationC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz,
const FT &rx, const FT &ry, const FT &rz,
const FT &sx, const FT &sy, const FT &sz)
{
return sign_of_determinant<FT>(qx-px,rx-px,sx-px,
qy-py,ry-py,sy-py,
qz-pz,rz-pz,sz-pz);
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Same_uncertainty_nt<Orientation, FT>::type
orientationC3(const FT &ux, const FT &uy, const FT &uz,
const FT &vx, const FT &vy, const FT &vz,
const FT &wx, const FT &wy, const FT &wz)
{
return sign_of_determinant(ux, vx, wx,
uy, vy, wy,
uz, vz, wz);
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Same_uncertainty_nt<Angle, FT>::type
angleC3(const FT &ux, const FT &uy, const FT &uz,
const FT &vx, const FT &vy, const FT &vz)
{
return enum_cast<Angle>(CGAL_NTS sign(ux*vx + uy*vy + uz*vz));
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Same_uncertainty_nt<Angle, FT>::type
angleC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz,
const FT &rx, const FT &ry, const FT &rz)
{
return enum_cast<Angle>(CGAL_NTS sign((px-qx)*(rx-qx)+
(py-qy)*(ry-qy)+
(pz-qz)*(rz-qz)));
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Same_uncertainty_nt<Angle, FT>::type
angleC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz,
const FT &rx, const FT &ry, const FT &rz,
const FT &sx, const FT &sy, const FT &sz)
{
return enum_cast<Angle>(CGAL_NTS sign((px-qx)*(rx-sx)+
(py-qy)*(ry-sy)+
(pz-qz)*(rz-sz)));
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Same_uncertainty_nt<Orientation, FT>::type
coplanar_orientationC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz,
const FT &rx, const FT &ry, const FT &rz,
const FT &sx, const FT &sy, const FT &sz)
{
typedef typename Same_uncertainty_nt<Orientation, FT>::type Ori;
Ori oxy_pqr = orientationC2(px,py,qx,qy,rx,ry);
if (oxy_pqr != COLLINEAR)
return oxy_pqr * orientationC2(px,py,qx,qy,sx,sy);
Ori oyz_pqr = orientationC2(py,pz,qy,qz,ry,rz);
if (oyz_pqr != COLLINEAR)
return oyz_pqr * orientationC2(py,pz,qy,qz,sy,sz);
Ori oxz_pqr = orientationC2(px,pz,qx,qz,rx,rz);
CGAL_kernel_assertion(oxz_pqr != COLLINEAR);
return oxz_pqr * orientationC2(px,pz,qx,qz,sx,sz);
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Same_uncertainty_nt<Orientation, FT>::type
coplanar_orientationC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz,
const FT &rx, const FT &ry, const FT &rz)
{
typedef typename Same_uncertainty_nt<Orientation, FT>::type Ori;
Ori oxy_pqr = orientationC2(px,py,qx,qy,rx,ry);
if (oxy_pqr != COLLINEAR)
return oxy_pqr;
Ori oyz_pqr = orientationC2(py,pz,qy,qz,ry,rz);
if (oyz_pqr != COLLINEAR)
return oyz_pqr;
return orientationC2(px,pz,qx,qz,rx,rz);
}
template < class FT >
CGAL_KERNEL_LARGE_INLINE
typename Same_uncertainty_nt<Bounded_side, FT>::type
coplanar_side_of_bounded_circleC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz,
const FT &rx, const FT &ry, const FT &rz,
const FT &tx, const FT &ty, const FT &tz)
{
// The approach is to compute side_of_bounded_sphere(p,q,r,t+v,t),
// with v = pq ^ pr.
// Note : since the circle defines the orientation of the plane, it can not
// be considered oriented.
FT ptx = px - tx;
FT pty = py - ty;
FT ptz = pz - tz;
FT pt2 = CGAL_NTS square(ptx) + CGAL_NTS square(pty) + CGAL_NTS square(ptz);
FT qtx = qx - tx;
FT qty = qy - ty;
FT qtz = qz - tz;
FT qt2 = CGAL_NTS square(qtx) + CGAL_NTS square(qty) + CGAL_NTS square(qtz);
FT rtx = rx - tx;
FT rty = ry - ty;
FT rtz = rz - tz;
FT rt2 = CGAL_NTS square(rtx) + CGAL_NTS square(rty) + CGAL_NTS square(rtz);
FT pqx = qx - px;
FT pqy = qy - py;
FT pqz = qz - pz;
FT prx = rx - px;
FT pry = ry - py;
FT prz = rz - pz;
FT vx = pqy*prz - pqz*pry;
FT vy = pqz*prx - pqx*prz;
FT vz = pqx*pry - pqy*prx;
FT v2 = CGAL_NTS square(vx) + CGAL_NTS square(vy) + CGAL_NTS square(vz);
return enum_cast<Bounded_side>(sign_of_determinant(ptx,pty,ptz,pt2,
rtx,rty,rtz,rt2,
qtx,qty,qtz,qt2,
vx,vy,vz,v2));
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Equal_to<FT>::result_type
collinear_are_ordered_along_lineC3(
const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz,
const FT &rx, const FT &ry, const FT &rz)
{
if (px < qx) return !(rx < qx);
if (qx < px) return !(qx < rx);
if (py < qy) return !(ry < qy);
if (qy < py) return !(qy < ry);
if (pz < qz) return !(rz < qz);
if (qz < pz) return !(qz < rz);
return true; // p==q
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Equal_to<FT>::result_type
collinear_are_strictly_ordered_along_lineC3(
const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz,
const FT &rx, const FT &ry, const FT &rz)
{
if (px < qx) return (qx < rx);
if (qx < px) return (rx < qx);
if (py < qy) return (qy < ry);
if (qy < py) return (ry < qy);
if (pz < qz) return (qz < rz);
if (qz < pz) return (rz < qz);
return false; // p==q
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Equal_to<FT>::result_type
equal_directionC3(const FT &dx1, const FT &dy1, const FT &dz1,
const FT &dx2, const FT &dy2, const FT &dz2)
{
return sign_of_determinant(dx1, dy1, dx2, dy2) == ZERO
&& sign_of_determinant(dx1, dz1, dx2, dz2) == ZERO
&& sign_of_determinant(dy1, dz1, dy2, dz2) == ZERO
&& CGAL_NTS sign(dx1) == CGAL_NTS sign(dx2)
&& CGAL_NTS sign(dy1) == CGAL_NTS sign(dy2)
&& CGAL_NTS sign(dz1) == CGAL_NTS sign(dz2);
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Equal_to<FT>::result_type
equal_planeC3(const FT &ha, const FT &hb, const FT &hc, const FT &hd,
const FT &pa, const FT &pb, const FT &pc, const FT &pd)
{
typedef typename Sgn<FT>::result_type Sg;
if (!equal_directionC3(ha, hb, hc, pa, pb, pc))
return false; // Not parallel.
Sg s1a = CGAL_NTS sign(ha);
if (s1a != ZERO)
return CGAL_AND( s1a == CGAL_NTS sign(pa) ,
sign_of_determinant(pa, pd, ha, hd) == ZERO );
Sg s1b = CGAL_NTS sign(hb);
if (s1b != ZERO)
return s1b == CGAL_NTS sign(pb)
&& sign_of_determinant(pb, pd, hb, hd) == ZERO;
return CGAL_NTS sign(pc) == CGAL_NTS sign(hc)
&& sign_of_determinant(pc, pd, hc, hd) == ZERO;
}
template <class FT >
CGAL_KERNEL_LARGE_INLINE
typename Same_uncertainty_nt<Oriented_side, FT>::type
side_of_oriented_planeC3(const FT &a, const FT &b, const FT &c, const FT &d,
const FT &px, const FT &py, const FT &pz)
{
return CGAL_NTS sign(a*px + b*py + c*pz + d);
}
template <class FT >
CGAL_KERNEL_LARGE_INLINE
typename Same_uncertainty_nt<Oriented_side, FT>::type
side_of_oriented_sphereC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz,
const FT &rx, const FT &ry, const FT &rz,
const FT &sx, const FT &sy, const FT &sz,
const FT &tx, const FT &ty, const FT &tz)
{
FT ptx = px - tx;
FT pty = py - ty;
FT ptz = pz - tz;
FT pt2 = CGAL_NTS square(ptx) + CGAL_NTS square(pty) + CGAL_NTS square(ptz);
FT qtx = qx - tx;
FT qty = qy - ty;
FT qtz = qz - tz;
FT qt2 = CGAL_NTS square(qtx) + CGAL_NTS square(qty) + CGAL_NTS square(qtz);
FT rtx = rx - tx;
FT rty = ry - ty;
FT rtz = rz - tz;
FT rt2 = CGAL_NTS square(rtx) + CGAL_NTS square(rty) + CGAL_NTS square(rtz);
FT stx = sx - tx;
FT sty = sy - ty;
FT stz = sz - tz;
FT st2 = CGAL_NTS square(stx) + CGAL_NTS square(sty) + CGAL_NTS square(stz);
return sign_of_determinant(ptx,pty,ptz,pt2,
rtx,rty,rtz,rt2,
qtx,qty,qtz,qt2,
stx,sty,stz,st2);
// Note that the determinant above is det(P,R,Q,S) (Q and R are swapped)!
}
template <class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Same_uncertainty_nt<Bounded_side, FT>::type
side_of_bounded_sphereC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz,
const FT &rx, const FT &ry, const FT &rz,
const FT &sx, const FT &sy, const FT &sz,
const FT &tx, const FT &ty, const FT &tz)
{
return enum_cast<Bounded_side>( side_of_oriented_sphereC3(px, py, pz,
qx, qy, qz,
rx, ry, rz,
sx, sy, sz,
tx, ty, tz)
* orientationC3(px, py, pz,
qx, qy, qz,
rx, ry, rz,
sx, sy, sz) );
}
template <class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Same_uncertainty_nt<Bounded_side, FT>::type
side_of_bounded_sphereC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz,
const FT &tx, const FT &ty, const FT &tz)
{
// Returns whether T lies inside or outside the sphere which diameter is PQ.
return enum_cast<Bounded_side>( CGAL_NTS sign((tx-px)*(qx-tx)
+ (ty-py)*(qy-ty)
+ (tz-pz)*(qz-tz)) );
}
template < class FT >
CGAL_KERNEL_INLINE
typename Compare<FT>::result_type
cmp_dist_to_pointC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz,
const FT &rx, const FT &ry, const FT &rz)
{
return CGAL_NTS compare(squared_distanceC3(px,py,pz,qx,qy,qz),
squared_distanceC3(px,py,pz,rx,ry,rz));
}
// Because of the way the filtered predicates generator script works,
// cmp_dist_to_pointC3() must be defined _before_ ths following one.
template <class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Same_uncertainty_nt<Bounded_side, FT>::type
side_of_bounded_sphereC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz,
const FT &sx, const FT &sy, const FT &sz,
const FT &tx, const FT &ty, const FT &tz)
{
// Returns whether T lies inside or outside the sphere which equatorial
// circle is PQR.
// This code is inspired by the one of circumcenterC3(3 points).
FT psx = px-sx;
FT psy = py-sy;
FT psz = pz-sz;
FT ps2 = CGAL_NTS square(psx) + CGAL_NTS square(psy) + CGAL_NTS square(psz);
FT qsx = qx-sx;
FT qsy = qy-sy;
FT qsz = qz-sz;
FT qs2 = CGAL_NTS square(qsx) + CGAL_NTS square(qsy) + CGAL_NTS square(qsz);
FT rsx = psy*qsz-psz*qsy;
FT rsy = psz*qsx-psx*qsz;
FT rsz = psx*qsy-psy*qsx;
FT tsx = tx-sx;
FT tsy = ty-sy;
FT tsz = tz-sz;
FT num_x = ps2 * determinant(qsy,qsz,rsy,rsz)
- qs2 * determinant(psy,psz,rsy,rsz);
FT num_y = ps2 * determinant(qsx,qsz,rsx,rsz)
- qs2 * determinant(psx,psz,rsx,rsz);
FT num_z = ps2 * determinant(qsx,qsy,rsx,rsy)
- qs2 * determinant(psx,psy,rsx,rsy);
FT den2 = 2 * determinant(psx,psy,psz,
qsx,qsy,qsz,
rsx,rsy,rsz);
// The following could be simplified a bit.
return enum_cast<Bounded_side>(
cmp_dist_to_pointC3<FT>(num_x, - num_y, num_z,
psx*den2, psy*den2, psz*den2,
tsx*den2, tsy*den2, tsz*den2) );
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Equal_to<FT>::result_type
has_larger_dist_to_pointC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz,
const FT &rx, const FT &ry, const FT &rz)
{
return cmp_dist_to_pointC3(px,py,pz,qx,qy,qz,rx,ry,rz) == LARGER;
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Equal_to<FT>::result_type
has_smaller_dist_to_pointC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz,
const FT &rx, const FT &ry, const FT &rz)
{
return cmp_dist_to_pointC3(px,py,pz,qx,qy,qz,rx,ry,rz) == SMALLER;
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Compare<FT>::result_type
cmp_signed_dist_to_directionC3( const FT &pa, const FT &pb, const FT &pc,
const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz)
{
return CGAL_NTS compare(scaled_distance_to_directionC3(pa,pb,pc,px,py,pz),
scaled_distance_to_directionC3(pa,pb,pc,qx,qy,qz));
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Equal_to<FT>::result_type
has_larger_signed_dist_to_directionC3(
const FT &pa, const FT &pb, const FT &pc,
const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz)
{
return cmp_signed_dist_to_directionC3(pa,pb,pc,px,py,pz,qx,qy,qz) == LARGER;
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Equal_to<FT>::result_type
has_smaller_signed_dist_to_directionC3(
const FT &pa, const FT &pb, const FT &pc,
const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz)
{
return cmp_signed_dist_to_directionC3(pa,pb,pc,px,py,pz,qx,qy,qz) == SMALLER;
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Compare<FT>::result_type
cmp_signed_dist_to_planeC3(
const FT &ppx, const FT &ppy, const FT &ppz,
const FT &pqx, const FT &pqy, const FT &pqz,
const FT &prx, const FT &pry, const FT &prz,
const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz)
{
return sign_of_determinant<FT>( pqx-ppx, pqy-ppy, pqz-ppz,
prx-ppx, pry-ppy, prz-ppz,
px-qx, py-qy, pz-qz);
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Equal_to<FT>::result_type
has_larger_signed_dist_to_planeC3(
const FT &ppx, const FT &ppy, const FT &ppz,
const FT &pqx, const FT &pqy, const FT &pqz,
const FT &prx, const FT &pry, const FT &prz,
const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz)
{
return cmp_signed_dist_to_planeC3(ppx, ppy, ppz, pqx, pqy, pqz,
prx, pry, prz, px, py, pz, qx, qy, qz) == LARGER;
}
template < class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Equal_to<FT>::result_type
has_smaller_signed_dist_to_planeC3(
const FT &ppx, const FT &ppy, const FT &ppz,
const FT &pqx, const FT &pqy, const FT &pqz,
const FT &prx, const FT &pry, const FT &prz,
const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz)
{
return cmp_signed_dist_to_planeC3(ppx, ppy, ppz, pqx, pqy, pqz,
prx, pry, prz, px, py, pz, qx, qy, qz) == SMALLER;
}
// return minus the sign of the 5x5 determinant [P,Q,R,S,T]
// where column [P] = transpose[px,py,pz,p^2 -wp,1]
template <class FT>
CGAL_KERNEL_MEDIUM_INLINE
typename Same_uncertainty_nt<Oriented_side, FT>::type
power_side_of_oriented_power_sphereC3(
const FT &px, const FT &py, const FT &pz, const FT &pwt,
const FT &qx, const FT &qy, const FT &qz, const FT &qwt,
const FT &rx, const FT &ry, const FT &rz, const FT &rwt,
const FT &sx, const FT &sy, const FT &sz, const FT &swt,
const FT &tx, const FT &ty, const FT &tz, const FT &twt)
{
// We translate the points so that T becomes the origin.
FT dpx = px - tx;
FT dpy = py - ty;
FT dpz = pz - tz;
FT dpt = CGAL_NTS square(dpx) + CGAL_NTS square(dpy) +
CGAL_NTS square(dpz) + (twt - pwt);
FT dqx = qx - tx;
FT dqy = qy - ty;
FT dqz = qz - tz;
FT dqt = CGAL_NTS square(dqx) + CGAL_NTS square(dqy) +
CGAL_NTS square(dqz) + (twt - qwt);
FT drx = rx - tx;
FT dry = ry - ty;
FT drz = rz - tz;
FT drt = CGAL_NTS square(drx) + CGAL_NTS square(dry) +
CGAL_NTS square(drz) + (twt - rwt);
FT dsx = sx - tx;
FT dsy = sy - ty;
FT dsz = sz - tz;
FT dst = CGAL_NTS square(dsx) + CGAL_NTS square(dsy) +
CGAL_NTS square(dsz) + (twt - swt);
return - sign_of_determinant(dpx, dpy, dpz, dpt,
dqx, dqy, dqz, dqt,
drx, dry, drz, drt,
dsx, dsy, dsz, dst);
}
template <class FT>
CGAL_KERNEL_MEDIUM_INLINE
typename Same_uncertainty_nt<Oriented_side, FT>::type
power_side_of_oriented_power_sphereC3(
const FT &px, const FT &py, const FT &pz, const FT &pwt,
const FT &qx, const FT &qy, const FT &qz, const FT &qwt,
const FT &rx, const FT &ry, const FT &rz, const FT &rwt,
const FT &tx, const FT &ty, const FT &tz, const FT &twt)
{
// Same translation as above.
FT dpx = px - tx;
FT dpy = py - ty;
FT dpz = pz - tz;
FT dpt = CGAL_NTS square(dpx) + CGAL_NTS square(dpy) +
CGAL_NTS square(dpz) + (twt - pwt);
FT dqx = qx - tx;
FT dqy = qy - ty;
FT dqz = qz - tz;
FT dqt = CGAL_NTS square(dqx) + CGAL_NTS square(dqy) +
CGAL_NTS square(dqz) + (twt - qwt);
FT drx = rx - tx;
FT dry = ry - ty;
FT drz = rz - tz;
FT drt = CGAL_NTS square(drx) + CGAL_NTS square(dry) +
CGAL_NTS square(drz) + (twt - rwt);
Sign cmp;
// Projection on the (xy) plane.
cmp = sign_of_determinant(dpx, dpy, dpt,
dqx, dqy, dqt,
drx, dry, drt);
if (cmp != ZERO)
return cmp * sign_of_determinant(px-rx, py-ry,
qx-rx, qy-ry);
// Projection on the (xz) plane.
cmp = sign_of_determinant(dpx, dpz, dpt,
dqx, dqz, dqt,
drx, drz, drt);
if (cmp != ZERO)
return cmp * sign_of_determinant(px-rx, pz-rz,
qx-rx, qz-rz);
// Projection on the (yz) plane.
cmp = sign_of_determinant(dpy, dpz, dpt,
dqy, dqz, dqt,
dry, drz, drt);
return cmp * sign_of_determinant(py-ry, pz-rz,
qy-ry, qz-rz);
}
template <class FT>
CGAL_KERNEL_MEDIUM_INLINE
typename Same_uncertainty_nt<Oriented_side, FT>::type
power_side_of_oriented_power_sphereC3(
const FT &px, const FT &py, const FT &pz, const FT &pwt,
const FT &qx, const FT &qy, const FT &qz, const FT &qwt,
const FT &tx, const FT &ty, const FT &tz, const FT &twt)
{
// Same translation as above.
FT dpx = px - tx;
FT dpy = py - ty;
FT dpz = pz - tz;
FT dpt = CGAL_NTS square(dpx) + CGAL_NTS square(dpy) +
CGAL_NTS square(dpz) + (twt - pwt);
FT dqx = qx - tx;
FT dqy = qy - ty;
FT dqz = qz - tz;
FT dqt = CGAL_NTS square(dqx) + CGAL_NTS square(dqy) +
CGAL_NTS square (dqz) + (twt - qwt);
Comparison_result cmp;
// We do an orthogonal projection on the (x) axis, if possible.
cmp = CGAL_NTS compare(px, qx);
if (cmp != EQUAL)
return cmp * sign_of_determinant(dpx, dpt, dqx, dqt);
// We do an orthogonal projection on the (y) axis, if possible.
cmp = CGAL_NTS compare(py, qy);
if (cmp != EQUAL)
return cmp * sign_of_determinant(dpy, dpt, dqy, dqt);
// We do an orthogonal projection on the (z) axis.
cmp = CGAL_NTS compare(pz, qz);
return cmp * sign_of_determinant(dpz, dpt, dqz, dqt);
}
template <class FT>
CGAL_KERNEL_MEDIUM_INLINE
typename Same_uncertainty_nt<Oriented_side, FT>::type
power_side_of_oriented_power_sphereC3(const FT &pwt, const FT &qwt)
{
return CGAL_NTS compare(qwt, pwt);
}
template < class FT >
Comparison_result
compare_power_distanceC3(const FT &px, const FT &py, const FT &pz,
const FT &qx, const FT &qy, const FT &qz, const FT &qw,
const FT &rx, const FT &ry, const FT &rz, const FT &rw)
{
FT dqx = qx - px;
FT dqy = qy - py;
FT dqz = qz - pz;
FT drx = rx - px;
FT dry = ry - py;
FT drz = rz - pz;
return CGAL_NTS compare(dqx*dqx + dqy*dqy + dqz*dqz - qw,
drx*drx + dry*dry + drz*drz - rw);
}
//return the sign of the power test of weighted point (sx,sy,sz,sw)
//with respect to the smallest sphere orthogonal to
//p,q,r
template< class FT >
CGAL_KERNEL_MEDIUM_INLINE
typename Same_uncertainty_nt<Bounded_side, FT>::type
power_side_of_bounded_power_sphereC3(
const FT &px, const FT &py, const FT &pz, const FT &pw,
const FT &qx, const FT &qy, const FT &qz, const FT &qw,
const FT &rx, const FT &ry, const FT &rz, const FT &rw,
const FT &sx, const FT &sy, const FT &sz, const FT &sw)
{
// Translate p to origin and compute determinants
FT qpx = qx-px;
FT qpy = qy-py;
FT qpz = qz-pz;
FT rpx = rx-px;
FT rpy = ry-py;
FT rpz = rz-pz;
FT qq = CGAL_NTS square(qpx) + CGAL_NTS square(qpy) + CGAL_NTS square(qpz);
FT rr = CGAL_NTS square(rpx) + CGAL_NTS square(rpy) + CGAL_NTS square(rpz);
FT qr = qpx*rpx + qpy*rpy + qpz*rpz;
FT qpw = qq - qw + pw ;
FT rpw = rr - rw + pw ;
FT den = determinant(qq,qr,
qr,rr);
FT detq = determinant(qpw,qr,
rpw,rr);
FT detr = determinant(qq,qpw,
qr,rpw);
// Smallest smallest orthogonal sphere center
// c = detq/2*den q + detr/2*den r (origin at p)
// square radius c^2 - pw
FT spx = sx-px;
FT spy = sy-py;
FT spz = sz-pz;
FT ss = CGAL_NTS square(spx) + CGAL_NTS square(spy) + CGAL_NTS square(spz);
FT sq = spx*qpx + spy*qpy + spz*qpz;
FT sr = spx*rpx + spy*rpy + spz*rpz;
CGAL_assertion( ! CGAL_NTS is_zero(den) );
// return - sign of (c- s)^2 - (c^2 - pw) - sw note that den >= 0 -
return enum_cast<Bounded_side>(
- CGAL_NTS sign( den*(ss - sw + pw)- detq*sq - detr*sr));
}
// return the sign of the power test of weighted point (rx,ry,rz,rw)
// with respect to the smallest sphere orthogoanal to
// p,q
template< class FT >
typename Same_uncertainty_nt<Bounded_side, FT>::type
power_side_of_bounded_power_sphereC3(
const FT &px, const FT &py, const FT &pz, const FT &pw,
const FT &qx, const FT &qy, const FT &qz, const FT &qw,
const FT &rx, const FT &ry, const FT &rz, const FT &rw)
{
FT FT2(2);
FT FT4(4);
FT dpx = px - qx;
FT dpy = py - qy;
FT dpz = pz - qz;
FT dpw = pw - qw;
FT dp2 = CGAL_NTS square(dpx) + CGAL_NTS square(dpy) + CGAL_NTS square(dpz);
FT drx = rx - (px + qx)/FT2;
FT dry = ry - (py + qy)/FT2;
FT drz = rz - (pz + qz)/FT2;
FT drw = rw - (pw + qw)/FT2;
FT dr2 = CGAL_NTS square(drx) + CGAL_NTS square(dry) + CGAL_NTS square(drz);
FT dpr = dpx*drx + dpy*dry +dpz*drz;
return enum_cast<Bounded_side>(
- CGAL_NTS sign (dr2 - dp2/FT4 + dpr*dpw/dp2 - drw ));
}
} // namespace CGAL
#endif // CGAL_PREDICATES_KERNEL_FTC3_H
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