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00001 // Copyright (c) 2005-2006 INRIA Sophia-Antipolis (France). 00002 // All rights reserved. 00003 // 00004 // This file is part of CGAL (www.cgal.org); you may redistribute it under 00005 // the terms of the Q Public License version 1.0. 00006 // See the file LICENSE.QPL distributed with CGAL. 00007 // 00008 // Licensees holding a valid commercial license may use this file in 00009 // accordance with the commercial license agreement provided with the software. 00010 // 00011 // This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE 00012 // WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. 00013 // 00014 // Partially supported by the IST Programme of the EU as a Shared-cost 00015 // RTD (FET Open) Project under Contract No IST-2000-26473 00016 // (ECG - Effective Computational Geometry for Curves and Surfaces) 00017 // and a STREP (FET Open) Project under Contract No IST-006413 00018 // (ACS -- Algorithms for Complex Shapes) 00019 // 00020 // $URL: svn+ssh://scm.gforge.inria.fr/svn/cgal/branches/CGAL-3.5-branch/Algebraic_kernel_for_spheres/include/CGAL/Algebraic_kernel_for_spheres/internal_functions_on_roots_and_polynomials_2_3.h $ 00021 // $Id: internal_functions_on_roots_and_polynomials_2_3.h 50465 2009-07-08 13:26:57Z sloriot $ 00022 // 00023 // Author(s) : Monique Teillaud <Monique.Teillaud@sophia.inria.fr> 00024 // Sylvain Pion 00025 // Pedro Machado 00026 00027 #ifndef CGAL_ALGEBRAIC_KERNEL_FUNCTIONS_ON_ROOTS_AND_POLYNOMIALS_2_3_H 00028 #define CGAL_ALGEBRAIC_KERNEL_FUNCTIONS_ON_ROOTS_AND_POLYNOMIALS_2_3_H 00029 00030 #include <CGAL/Algebraic_kernel_for_spheres/internal_functions_on_roots_and_polynomial_1_3_and_2_3.h> 00031 00032 namespace CGAL { 00033 namespace AlgebraicSphereFunctors { 00034 00035 template < class AK, class OutputIterator > 00036 inline 00037 OutputIterator 00038 solve( const typename AK::Polynomial_for_spheres_2_3 &e1, 00039 const typename AK::Polynomial_for_spheres_2_3 &e2, 00040 const typename AK::Polynomial_for_spheres_2_3 &e3, 00041 OutputIterator res ) 00042 { 00043 CGAL_kernel_precondition(!((e1 == e2) && (e2 == e3))); 00044 // we put as a precondition that the polynomial for spheres represents 00045 // a sphere and not an isolated point or an empty_space 00046 CGAL_kernel_precondition(!(e1.empty_space() || e1.isolated_point())); 00047 CGAL_kernel_precondition(!(e2.empty_space() || e2.isolated_point())); 00048 CGAL_kernel_precondition(!(e3.empty_space() || e3.isolated_point())); 00049 typedef typename AK::Polynomial_1_3 Polynomial_1_3; 00050 // The degenerated cases are 2 tangent spheres 00051 // os 2 non-intersecting spheres 00052 // beacause we cannot have infinitely many solutions 00053 if(e1 == e2) { 00054 if(tangent<AK>(e1,e3)) { 00055 Polynomial_1_3 p = plane_from_2_spheres<AK>(e1,e3); 00056 return CGALi::solve_tangent<AK>(p,e1,res); 00057 } 00058 CGAL_kernel_precondition(!(intersect<AK>(e1,e3))); 00059 return res; 00060 } 00061 if((e1 == e3) || (e2 == e3)) { 00062 if(tangent<AK>(e1,e2)) { 00063 Polynomial_1_3 p = plane_from_2_spheres<AK>(e1,e2); 00064 return CGALi::solve_tangent<AK>(p,e1,res); 00065 } 00066 CGAL_kernel_precondition(!(intersect<AK>(e1,e2))); 00067 return res; 00068 } 00069 00070 // non degenerated case 00071 if(intersect<AK>(e1,e2)) { 00072 Polynomial_1_3 p1 = plane_from_2_spheres<AK>(e1,e2); 00073 if(intersect<AK>(e2,e3)) { 00074 Polynomial_1_3 p2 = plane_from_2_spheres<AK>(e2,e3); 00075 return solve<AK>(p1,p2,e2,res); 00076 } return res; 00077 } return res; 00078 } 00079 00080 template <class AK> 00081 typename AK::Root_for_spheres_2_3 00082 x_critical_point(const typename AK::Polynomial_for_spheres_2_3 & s, bool i) 00083 { 00084 typedef typename AK::Root_of_2 Root_of_2; 00085 typedef typename AK::Root_for_spheres_2_3 Root_for_spheres_2_3; 00086 00087 return Root_for_spheres_2_3( 00088 make_root_of_2(s.a(),typename AK::FT(i?-1:1),s.r_sq()), 00089 Root_of_2(s.b()), 00090 Root_of_2(s.c())); 00091 } 00092 00093 template <class AK, class OutputIterator> 00094 OutputIterator 00095 x_critical_points(const typename AK::Polynomial_for_spheres_2_3 & s, OutputIterator res) 00096 { 00097 typedef typename AK::Root_of_2 Root_of_2; 00098 typedef typename AK::Root_for_spheres_2_3 Root_for_spheres_2_3; 00099 typedef typename AK::FT FT; 00100 00101 *res++ = Root_for_spheres_2_3(make_root_of_2(s.a(),FT(-1),s.r_sq()), 00102 Root_of_2(s.b()), 00103 Root_of_2(s.c())); 00104 *res++ = Root_for_spheres_2_3(make_root_of_2(s.a(),FT(1),s.r_sq()), 00105 Root_of_2(s.b()), 00106 Root_of_2(s.c())); 00107 return res; 00108 } 00109 00110 template <class AK> 00111 typename AK::Root_for_spheres_2_3 00112 y_critical_point(const typename AK::Polynomial_for_spheres_2_3 &s, bool i) 00113 { 00114 typedef typename AK::Root_of_2 Root_of_2; 00115 typedef typename AK::Root_for_spheres_2_3 Root_for_spheres_2_3; 00116 00117 return Root_for_spheres_2_3( 00118 Root_of_2(s.a()), 00119 make_root_of_2(s.b(),typename AK::FT(i?-1:1),s.r_sq()), 00120 Root_of_2(s.c())); 00121 } 00122 00123 template <class AK, class OutputIterator> 00124 OutputIterator 00125 y_critical_points(const typename AK::Polynomial_for_spheres_2_3 & s, OutputIterator res) 00126 { 00127 typedef typename AK::Root_of_2 Root_of_2; 00128 typedef typename AK::Root_for_spheres_2_3 Root_for_spheres_2_3; 00129 typedef typename AK::FT FT; 00130 00131 *res++ = Root_for_spheres_2_3(Root_of_2(s.a()), 00132 make_root_of_2(s.b(),FT(-1),s.r_sq()), 00133 Root_of_2(s.c())); 00134 *res++ = Root_for_spheres_2_3(Root_of_2(s.a()), 00135 make_root_of_2(s.b(),FT(1),s.r_sq()), 00136 Root_of_2(s.c())); 00137 return res; 00138 } 00139 00140 template <class AK> 00141 typename AK::Root_for_spheres_2_3 00142 z_critical_point(const typename AK::Polynomial_for_spheres_2_3 &s, bool i) 00143 { 00144 typedef typename AK::Root_of_2 Root_of_2; 00145 typedef typename AK::Root_for_spheres_2_3 Root_for_spheres_2_3; 00146 00147 return Root_for_spheres_2_3(Root_of_2(s.a()), 00148 Root_of_2(s.b()), 00149 make_root_of_2(s.c(),typename AK::FT(i?-1:1),s.r_sq())); 00150 } 00151 00152 template <class AK, class OutputIterator> 00153 OutputIterator 00154 z_critical_points(const typename AK::Polynomial_for_spheres_2_3 & s, OutputIterator res) 00155 { 00156 typedef typename AK::Root_of_2 Root_of_2; 00157 typedef typename AK::Root_for_spheres_2_3 Root_for_spheres_2_3; 00158 typedef typename AK::FT FT; 00159 00160 *res++ = Root_for_spheres_2_3(Root_of_2(s.a()), 00161 Root_of_2(s.b()), 00162 make_root_of_2(s.c(),FT(-1),s.r_sq())); 00163 *res++ = Root_for_spheres_2_3(Root_of_2(s.a()), 00164 Root_of_2(s.b()), 00165 make_root_of_2(s.c(),FT(1),s.r_sq())); 00166 return res; 00167 } 00168 00169 template <class AK> 00170 typename AK::Root_for_spheres_2_3 00171 x_critical_point( const std::pair<typename AK::Polynomial_for_spheres_2_3, 00172 typename AK::Polynomial_1_3 > &c, bool i) 00173 { 00174 typedef typename AK::FT FT; 00175 typedef typename AK::Root_of_2 Root_of_2; 00176 typedef typename AK::Root_for_spheres_2_3 Root_for_spheres_2_3; 00177 typedef typename AK::Polynomial_for_spheres_2_3 Polynomial_for_spheres_2_3; 00178 typedef typename AK::Polynomial_1_3 Polynomial_1_3; 00179 00180 const Polynomial_for_spheres_2_3 &s = c.first; 00181 const Polynomial_1_3 &p = c.second; 00182 00183 // It has to be the equation of a diametral circle 00184 CGAL_kernel_precondition((intersect<AK>(p,s))); 00185 CGAL_kernel_precondition(CGAL_NTS sign(p.a() * s.a() + p.b() * s.b() + 00186 p.c() * s.c() + p.d()) == ZERO); 00187 CGAL_kernel_precondition(!(is_zero(p.b()) && is_zero(p.c()))); 00188 00189 const FT sqbc = CGAL::square(p.b()) + CGAL::square(p.c()); 00190 const FT sq_sum = sqbc + CGAL::square(p.a()); 00191 const FT delta = (sqbc * s.r_sq())/sq_sum; 00192 00193 const FT cy = (p.a()*p.b())/sqbc; 00194 const FT cz = (p.a()*p.c())/sqbc; 00195 00196 const Root_of_2 x = make_root_of_2(s.a(),FT(i?-1:1),delta); 00197 const Root_of_2 y = make_root_of_2(s.b(),FT(i?(cy):(-cy)),delta); 00198 const Root_of_2 z = make_root_of_2(s.c(),FT(i?(cz):(-cz)),delta); 00199 00200 return Root_for_spheres_2_3(x,y,z); 00201 } 00202 00203 template <class AK, class OutputIterator> 00204 OutputIterator 00205 x_critical_points( const std::pair<typename AK::Polynomial_for_spheres_2_3, 00206 typename AK::Polynomial_1_3 > &c, 00207 OutputIterator res) 00208 { 00209 typedef typename AK::FT FT; 00210 typedef typename AK::Root_of_2 Root_of_2; 00211 typedef typename AK::Root_for_spheres_2_3 Root_for_spheres_2_3; 00212 typedef typename AK::Polynomial_for_spheres_2_3 Polynomial_for_spheres_2_3; 00213 typedef typename AK::Polynomial_1_3 Polynomial_1_3; 00214 00215 const Polynomial_for_spheres_2_3 &s = c.first; 00216 const Polynomial_1_3 &p = c.second; 00217 00218 // It has to be the equation of a diametral circle 00219 CGAL_kernel_precondition((intersect<AK>(p,s))); 00220 CGAL_kernel_precondition(CGAL_NTS sign(p.a() * s.a() + p.b() * s.b() + 00221 p.c() * s.c() + p.d()) == ZERO); 00222 CGAL_kernel_precondition(!(is_zero(p.b()) && is_zero(p.c()))); 00223 00224 const FT sqbc = CGAL::square(p.b()) + CGAL::square(p.c()); 00225 const FT sq_sum = sqbc + CGAL::square(p.a()); 00226 const FT delta = (sqbc * s.r_sq())/sq_sum; 00227 00228 const FT cy = (p.a()*p.b())/sqbc; 00229 const FT cz = (p.a()*p.c())/sqbc; 00230 00231 const Root_of_2 x1 = make_root_of_2(s.a(),-1,delta); 00232 const Root_of_2 y1 = make_root_of_2(s.b(),cy,delta); 00233 const Root_of_2 z1 = make_root_of_2(s.c(),cz,delta); 00234 const Root_of_2 x2 = make_root_of_2(s.a(),1,delta); 00235 const Root_of_2 y2 = make_root_of_2(s.b(),-cy,delta); 00236 const Root_of_2 z2 = make_root_of_2(s.c(),-cz,delta); 00237 00238 *res++ = Root_for_spheres_2_3(x1,y1,z1); 00239 *res++ = Root_for_spheres_2_3(x2,y2,z2); 00240 return res; 00241 } 00242 00243 template <class AK> 00244 typename AK::Root_for_spheres_2_3 00245 y_critical_point( const std::pair<typename AK::Polynomial_for_spheres_2_3, 00246 typename AK::Polynomial_1_3 > &c, bool i) 00247 { 00248 typedef typename AK::FT FT; 00249 typedef typename AK::Root_of_2 Root_of_2; 00250 typedef typename AK::Root_for_spheres_2_3 Root_for_spheres_2_3; 00251 typedef typename AK::Polynomial_for_spheres_2_3 Polynomial_for_spheres_2_3; 00252 typedef typename AK::Polynomial_1_3 Polynomial_1_3; 00253 00254 const Polynomial_for_spheres_2_3 &s = c.first; 00255 const Polynomial_1_3 &p = c.second; 00256 00257 // It has to be the equation of a diametral circle 00258 CGAL_kernel_precondition((intersect<AK>(p,s))); 00259 CGAL_kernel_precondition(CGAL_NTS sign(p.a() * s.a() + p.b() * s.b() + 00260 p.c() * s.c() + p.d()) == ZERO); 00261 CGAL_kernel_precondition(!(is_zero(p.a()) && is_zero(p.c()))); 00262 00263 const FT sqac = CGAL::square(p.a()) + CGAL::square(p.c()); 00264 const FT sq_sum = sqac + CGAL::square(p.b()); 00265 const FT delta = (sqac * s.r_sq())/sq_sum; 00266 00267 const FT cx = (p.a()*p.b())/sqac; 00268 const FT cz = (p.c()*p.b())/sqac; 00269 00270 if(!is_positive(cx)) { 00271 const Root_of_2 x = make_root_of_2(s.a(),FT(i?(cx):(-cx)),delta); 00272 const Root_of_2 y = make_root_of_2(s.b(),FT(i?-1:1),delta); 00273 const Root_of_2 z = make_root_of_2(s.c(),FT(i?(cz):(-cz)),delta); 00274 return Root_for_spheres_2_3(x,y,z); 00275 } else { 00276 const Root_of_2 x = make_root_of_2(s.a(),FT(i?(-cx):(cx)),delta); 00277 const Root_of_2 y = make_root_of_2(s.b(),FT(i?1:-1),delta); 00278 const Root_of_2 z = make_root_of_2(s.c(),FT(i?(-cz):(cz)),delta); 00279 return Root_for_spheres_2_3(x,y,z); 00280 } 00281 } 00282 00283 template <class AK, class OutputIterator> 00284 OutputIterator 00285 y_critical_points( const std::pair<typename AK::Polynomial_for_spheres_2_3, 00286 typename AK::Polynomial_1_3 > &c, 00287 OutputIterator res) 00288 { 00289 typedef typename AK::FT FT; 00290 typedef typename AK::Root_of_2 Root_of_2; 00291 typedef typename AK::Root_for_spheres_2_3 Root_for_spheres_2_3; 00292 typedef typename AK::Polynomial_for_spheres_2_3 Polynomial_for_spheres_2_3; 00293 typedef typename AK::Polynomial_1_3 Polynomial_1_3; 00294 00295 const Polynomial_for_spheres_2_3 &s = c.first; 00296 const Polynomial_1_3 &p = c.second; 00297 00298 // It has to be the equation of a diametral circle 00299 CGAL_kernel_precondition((intersect<AK>(p,s))); 00300 CGAL_kernel_precondition(CGAL_NTS sign(p.a() * s.a() + p.b() * s.b() + 00301 p.c() * s.c() + p.d()) == ZERO); 00302 CGAL_kernel_precondition(!(is_zero(p.a()) && is_zero(p.c()))); 00303 00304 const FT sqac = CGAL::square(p.a()) + CGAL::square(p.c()); 00305 const FT sq_sum = sqac + CGAL::square(p.b()); 00306 const FT delta = (sqac * s.r_sq())/sq_sum; 00307 00308 const FT cx = (p.a()*p.b())/sqac; 00309 const FT cz = (p.c()*p.b())/sqac; 00310 00311 const Root_of_2 x1 = make_root_of_2(s.a(),cx,delta); 00312 const Root_of_2 y1 = make_root_of_2(s.b(),FT(-1),delta); 00313 const Root_of_2 z1 = make_root_of_2(s.c(),cz,delta); 00314 const Root_of_2 x2 = make_root_of_2(s.a(),-cx,delta); 00315 const Root_of_2 y2 = make_root_of_2(s.b(),FT(1),delta); 00316 const Root_of_2 z2 = make_root_of_2(s.c(),-cz,delta); 00317 00318 if(!is_positive(cx)) { 00319 *res++ = Root_for_spheres_2_3(x1,y1,z1); 00320 *res++ = Root_for_spheres_2_3(x2,y2,z2); 00321 } else { 00322 *res++ = Root_for_spheres_2_3(x2,y2,z2); 00323 *res++ = Root_for_spheres_2_3(x1,y1,z1); 00324 } 00325 return res; 00326 } 00327 00328 template <class AK> 00329 typename AK::Root_for_spheres_2_3 00330 z_critical_point( const std::pair<typename AK::Polynomial_for_spheres_2_3, 00331 typename AK::Polynomial_1_3 > &c, bool i) 00332 { 00333 typedef typename AK::FT FT; 00334 typedef typename AK::Root_of_2 Root_of_2; 00335 typedef typename AK::Root_for_spheres_2_3 Root_for_spheres_2_3; 00336 typedef typename AK::Polynomial_for_spheres_2_3 Polynomial_for_spheres_2_3; 00337 typedef typename AK::Polynomial_1_3 Polynomial_1_3; 00338 00339 const Polynomial_for_spheres_2_3 &s = c.first; 00340 const Polynomial_1_3 &p = c.second; 00341 00342 // It has to be the equation of a diametral circle 00343 CGAL_kernel_precondition((intersect<AK>(p,s))); 00344 CGAL_kernel_precondition(CGAL_NTS sign(p.a() * s.a() + p.b() * s.b() + 00345 p.c() * s.c() + p.d()) == ZERO); 00346 CGAL_kernel_precondition(!(is_zero(p.a()) && is_zero(p.b()))); 00347 00348 const FT sqab = CGAL::square(p.a()) + CGAL::square(p.b()); 00349 const FT sq_sum = sqab + CGAL::square(p.c()); 00350 const FT delta = (sqab * s.r_sq())/sq_sum; 00351 00352 const FT cx = (p.a()*p.c())/sqab; 00353 const FT cy = (p.c()*p.b())/sqab; 00354 00355 if(is_negative(cx)) { 00356 const Root_of_2 x = make_root_of_2(s.a(),FT(i?(cx):(-cx)),delta); 00357 const Root_of_2 y = make_root_of_2(s.b(),FT(i?(cy):(-cy)),delta); 00358 const Root_of_2 z = make_root_of_2(s.c(),FT(i?-1:1),delta); 00359 return Root_for_spheres_2_3(x,y,z); 00360 } else if(is_zero(cx)) { 00361 if(!is_positive(cy)) { 00362 const Root_of_2 x = s.a(); 00363 const Root_of_2 y = make_root_of_2(s.b(),FT(i?(cy):(-cy)),delta); 00364 const Root_of_2 z = make_root_of_2(s.c(),FT(i?-1:1),delta); 00365 return Root_for_spheres_2_3(x,y,z); 00366 } else { 00367 const Root_of_2 x = s.a(); 00368 const Root_of_2 y = make_root_of_2(s.b(),FT(i?(-cy):(cy)),delta); 00369 const Root_of_2 z = make_root_of_2(s.c(),FT(i?1:-1),delta); 00370 return Root_for_spheres_2_3(x,y,z); 00371 } 00372 } else { 00373 const Root_of_2 x = make_root_of_2(s.a(),FT(i?(-cx):(cx)),delta); 00374 const Root_of_2 y = make_root_of_2(s.b(),FT(i?(-cy):(cy)),delta); 00375 const Root_of_2 z = make_root_of_2(s.c(),FT(i?1:-1),delta); 00376 return Root_for_spheres_2_3(x,y,z); 00377 } 00378 } 00379 00380 template <class AK, class OutputIterator> 00381 OutputIterator 00382 z_critical_points( const std::pair<typename AK::Polynomial_for_spheres_2_3, 00383 typename AK::Polynomial_1_3 > &c, 00384 OutputIterator res) 00385 { 00386 typedef typename AK::FT FT; 00387 typedef typename AK::Root_of_2 Root_of_2; 00388 typedef typename AK::Root_for_spheres_2_3 Root_for_spheres_2_3; 00389 typedef typename AK::Polynomial_for_spheres_2_3 Polynomial_for_spheres_2_3; 00390 typedef typename AK::Polynomial_1_3 Polynomial_1_3; 00391 00392 const Polynomial_for_spheres_2_3 &s = c.first; 00393 const Polynomial_1_3 &p = c.second; 00394 00395 // It has to be the equation of a diametral circle 00396 CGAL_kernel_precondition((intersect<AK>(p,s))); 00397 CGAL_kernel_precondition(CGAL_NTS sign(p.a() * s.a() + p.b() * s.b() + 00398 p.c() * s.c() + p.d()) == ZERO); 00399 CGAL_kernel_precondition(!(is_zero(p.a()) && is_zero(p.b()))); 00400 00401 const FT sqab = CGAL::square(p.a()) + CGAL::square(p.b()); 00402 const FT sq_sum = sqab + CGAL::square(p.c()); 00403 const FT delta = (sqab * s.r_sq())/sq_sum; 00404 00405 const FT cx = (p.a()*p.c())/sqab; 00406 const FT cy = (p.c()*p.b())/sqab; 00407 00408 if(is_negative(cx)) { 00409 const Root_of_2 x1 = make_root_of_2(s.a(),(cx),delta); 00410 const Root_of_2 y1 = make_root_of_2(s.b(),(cy),delta); 00411 const Root_of_2 z1 = make_root_of_2(s.c(),-1,delta); 00412 const Root_of_2 x2 = make_root_of_2(s.a(),(-cx),delta); 00413 const Root_of_2 y2 = make_root_of_2(s.b(),(-cy),delta); 00414 const Root_of_2 z2 = make_root_of_2(s.c(),1,delta); 00415 *res++ = Root_for_spheres_2_3(x1,y1,z1); 00416 *res++ = Root_for_spheres_2_3(x2,y2,z2); 00417 } else if(is_zero(cx)) { 00418 if(!is_positive(cy)) { 00419 const Root_of_2 x1 = s.a(); 00420 const Root_of_2 y1 = make_root_of_2(s.b(),(cy),delta); 00421 const Root_of_2 z1 = make_root_of_2(s.c(),FT(-1),delta); 00422 const Root_of_2 y2 = make_root_of_2(s.b(),(-cy),delta); 00423 const Root_of_2 z2 = make_root_of_2(s.c(),FT(1),delta); 00424 *res++ = Root_for_spheres_2_3(x1,y1,z1); 00425 *res++ = Root_for_spheres_2_3(x1,y2,z2); 00426 } else { 00427 const Root_of_2 x1 = s.a(); 00428 const Root_of_2 y1 = make_root_of_2(s.b(),(-cy),delta); 00429 const Root_of_2 z1 = make_root_of_2(s.c(),FT(1),delta); 00430 const Root_of_2 y2 = make_root_of_2(s.b(),(cy),delta); 00431 const Root_of_2 z2 = make_root_of_2(s.c(),FT(-1),delta); 00432 *res++ = Root_for_spheres_2_3(x1,y1,z1); 00433 *res++ = Root_for_spheres_2_3(x1,y2,z2); 00434 } 00435 } else { 00436 const Root_of_2 x1 = make_root_of_2(s.a(),(-cx),delta); 00437 const Root_of_2 y1 = make_root_of_2(s.b(),(-cy),delta); 00438 const Root_of_2 z1 = make_root_of_2(s.c(),FT(1),delta); 00439 const Root_of_2 x2 = make_root_of_2(s.a(),(cx),delta); 00440 const Root_of_2 y2 = make_root_of_2(s.b(),(cy),delta); 00441 const Root_of_2 z2 = make_root_of_2(s.c(),FT(-1),delta); 00442 *res++ = Root_for_spheres_2_3(x1,y1,z1); 00443 *res++ = Root_for_spheres_2_3(x2,y2,z2); 00444 } 00445 return res; 00446 } 00447 00448 } // namespace AlgebraicSphereFunctors 00449 } // namespace CGAL 00450 00451 #endif // CGAL_ALGEBRAIC_KERNEL_FUNCTIONS_ON_ROOTS_AND_POLYNOMIALS_2_3_H
1.7.6.1
