# Ticket #10132: parametrized_surface3d.3.py

File parametrized_surface3d.3.py, 40.4 KB (added by , 11 years ago) |
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1 | """ |

2 | Basic invariants of parametrized surfaces. |

3 | |

4 | |

5 | AUTHORS: |

6 | - Mikhail Malakhaltsev (2010-09-25): initial version |

7 | |

8 | """ |

9 | #***************************************************************************** |

10 | # Copyright (C) 2010 Mikhail Malakhaltsev <mikarm@gmail.com> |

11 | # |

12 | # Distributed under the terms of the GNU General Public License (GPL) |

13 | # http://www.gnu.org/licenses/ |

14 | #***************************************************************************** |

15 | |

16 | |

17 | # The following is maybe too specific ... |

18 | #from vector_functions import * |

19 | |

20 | # mikarm: This is for the moment, at the final version will be replaced by import |

21 | attach('/home/prince/MY_SAGE_MODULES/vector_functions.py') |

22 | |

23 | from sage.structure.sage_object import SageObject |

24 | from sage.calculus.functional import diff |

25 | from sage.functions.other import sqrt |

26 | |

27 | from sage.misc.cachefunc import cached_method |

28 | |

29 | |

30 | |

31 | # TODO (distant future): update class... |

32 | class GenericManifold(SageObject): |

33 | pass |

34 | |

35 | |

36 | class ParametrizedSurface3D(GenericManifold): |

37 | """ |

38 | This is a class for finding basic invariants of surfaces. |

39 | |

40 | USAGE: |

41 | parametrized_surface3d(surface_equation,variables,name_of_surface) |

42 | where |

43 | surface_equation is a vector or list of three components |

44 | variables is a list of two variables |

45 | name_of_surface is a string (optional) |

46 | |

47 | Warning: The orientation on the surface is given by the parametrization, that is the positive frame is |

48 | $\partial_1 \\vec r$, $\partial_2 \\vec r$. |

49 | |

50 | This class includes the following methods: |

51 | natural_frame |

52 | normal_vector |

53 | first_fundamental_form_coefficients |

54 | first_fundamental_form |

55 | first_fundamental_form_inverse_coefficients |

56 | first_fundamental_form_inverse_coefficients |

57 | area_form_squared |

58 | area_form |

59 | rotation |

60 | orthonormal_frame |

61 | lie_bracket |

62 | frame_structure_functions |

63 | second_order_natural_frame |

64 | second_fundamental_form_coefficients |

65 | second_fundamental_form |

66 | gauss_curvature |

67 | mean_curvature |

68 | principal curvatures |

69 | principal directions |

70 | connection_coefficients |

71 | geodesics_numerical |

72 | parallel_translation_numerical |

73 | |

74 | |

75 | EXAMPLES:: |

76 | |

77 | sage: var('u,v') |

78 | sage: paraboloid = parametrized_surface3d([u,v,u^2+v^2],[u,v],'paraboloid') |

79 | sage: paraboloid = parametrized_surface3d([u,v,u^2+v^2],[u,v]) |

80 | |

81 | """ |

82 | |

83 | def __init__(self,equation,variables,*name): |

84 | |

85 | self.equation = equation |

86 | self.variables_list = variables |

87 | self.variables = {1:self.variables_list[0],2:self.variables_list[1]} |

88 | self.name = name |

89 | |

90 | # Callable version of the underlying equation |

91 | def eq_callable(u, v): |

92 | u1, u2 = self.variables_list |

93 | return [fun.subs({u1: u, u2: v}) for fun in self.equation] |

94 | |

95 | self.eq_callable = eq_callable |

96 | |

97 | # Various index tuples |

98 | self.index_list = [(i,j) for i in (1,2) for j in (1,2)] |

99 | self.index_list_3=[(i,j,k) for i in (1,2) for j in (1,2) for k in (1,2)] |

100 | |

101 | |

102 | @cached_method |

103 | def natural_frame(self,vector_number=0): |

104 | """ |

105 | This functions find the natural frame of the parametrized surface |

106 | INPUT: |

107 | the argument can be empty, or equal to 1, or to 2 |

108 | |

109 | OUTPUT: |

110 | if argument is equal to 1 or 2, the output is the corresponding vector of the natural frame |

111 | if argument is empty, the output is the dictionary of the vector fields of the natural frame |

112 | |

113 | EXAMPLES:: |

114 | |

115 | sage: eparaboloid = parametrized_surface3d([u,v,u^2+v^2],[u,v],'elliptic paraboloid') |

116 | sage: eparaboloid.natural_frame() |

117 | {1: (1, 0, 2*u), 2: (0, 1, 2*v)} |

118 | sage: eparaboloid.natural_frame(1) |

119 | (1, 0, 2*u) |

120 | sage: eparaboloid.natural_frame(2) |

121 | (0, 1, 2*v) |

122 | """ |

123 | |

124 | dr1 = vector([diff(f,self.variables[1]).simplify_full() for f in self.equation]) |

125 | dr2 = vector([diff(f,self.variables[2]).simplify_full() for f in self.equation]) |

126 | |

127 | return {1:dr1, 2:dr2} |

128 | |

129 | |

130 | @cached_method |

131 | def normal_vector(self, normalized=False): |

132 | """ |

133 | This functions find the normal vector of the parametrized surface |

134 | |

135 | INPUT: |

136 | empty argument, of a nonzero real number l |

137 | |

138 | OUTPUT: |

139 | without arguments gives the normal vector which is the vector product of the |

140 | natural frame vectors; |

141 | |

142 | with the argument l gives the normal vector of length l (the orientation determined |

143 | by the sign of l) |

144 | |

145 | EXAMPLES:: |

146 | |

147 | sage: eparaboloid = parametrized_surface3d([u,v,u^2+v^2],[u,v],'elliptic paraboloid') |

148 | sage: eparaboloid.normal_vector() |

149 | (-2*u,-2*v,1) |

150 | sage: eparaboloid.normal_vector(1) |

151 | (-2*u/sqrt(4*u^2 + 4*v^2 + 1), -2*v/sqrt(4*u^2 + 4*v^2 + 1),1/sqrt(4*u^2 + 4*v^2 + 1)) |

152 | |

153 | """ |

154 | |

155 | dr = self.natural_frame() |

156 | normal = dr[1].cross_product(dr[2]) |

157 | |

158 | if normalized: |

159 | normal /= normal.norm() |

160 | return simplify_vector(normal) |

161 | |

162 | |

163 | |

164 | # The following can become part of the docstring at some point... |

165 | |

166 | # I've defined a cached (private) method which does not check its arguments and |

167 | # computes one component at a time. Note that caching is just a matter of |

168 | # prepending "@cached_method" to the method signature. For this to work, the |

169 | # arguments of the method must be hashable (e.g tuple, string, ...) and the method |

170 | # must not change the internal state of the object. For all of the methods in this |

171 | # class, this is the case. |

172 | # |

173 | # I then added a method that does simple argument checking, converting the arguments to |

174 | # tuples, and invokes the private method to do the work. Note that this method should |

175 | # not be cached, as the user might specify the index of the component as a list, which is |

176 | # not hashable. |

177 | # |

178 | # A second method returns a dictionary of all the components, again invoking the hidden |

179 | # helper method. |

180 | |

181 | @cached_method |

182 | def _compute_first_fundamental_form_coefficient(self, index): |

183 | dr = self.natural_frame() |

184 | return (dr[index[0]]*dr[index[1]]).simplify_full() |

185 | |

186 | def first_fundamental_form_coefficient(self, index): |

187 | index = tuple(sorted(index)) |

188 | if index not in self.index_list: |

189 | raise ValueError, "Index %s out of bounds." % str(index) |

190 | return self._compute_first_fundamental_form_coefficient(index) |

191 | |

192 | @cached_method |

193 | def first_fundamental_form_coefficients(self): |

194 | |

195 | """ |

196 | This function gives the coefficients of the first fundamental form |

197 | |

198 | INPUT: |

199 | empty argument, or one of the the following pair of indices: 1,1; 1,2; 2,1; 2,2. |

200 | |

201 | OUTPUT: |

202 | with empty argument the output is the dictionary of coefficients of the first fundamental form |

203 | |

204 | with given indices the output is the corresponding coefficient of the first fundamental form, |

205 | for example index1 = 1, index2 = 2 gives the coefficient $g_{12}$ |

206 | |

207 | EXAMPLES:: |

208 | |

209 | sage: var('u,v') |

210 | sage: sphere = parametrized_surface3d([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |

211 | sage: sphere.first_fundamental_form_coefficients() |

212 | {(1, 2): 0, (1, 1): cos(v)^2, (2, 1): 0, (2, 2): 1} |

213 | |

214 | sage: sphere.first_fundamental_form_coefficients(1,1) |

215 | cos(v)^2 |

216 | |

217 | sage: sphere.first_fundamental_form_coefficients(2,1) |

218 | 0 |

219 | |

220 | sage: sphere.first_fundamental_form_coefficients(3,2) |

221 | 'The argument is not appropriate. Read doc' |

222 | """ |

223 | coefficients = {} |

224 | for index in self.index_list: |

225 | sorted_index = list(sorted(index)) |

226 | coefficients[index] = \ |

227 | self._compute_first_fundamental_form_coefficient(index) |

228 | return coefficients |

229 | |

230 | |

231 | |

232 | def first_fundamental_form(self,vector1,vector2): |

233 | """ |

234 | Finds the value of first fundamental form on two vectors |

235 | |

236 | INPUT: |

237 | Two vectors $v=(v^1,v^2)$ and $w=(w^1,w^2)$ |

238 | |

239 | OUTPUT: |

240 | $g_{11} v^1 w^1 + g_{12}(v^1 w^2 + v^2 w^1) + g_{22} v^2 w^2$ |

241 | |

242 | #EXAMPLES:: |

243 | |

244 | sage: var('u,v') |

245 | sage: var ('v1,v2,w1,w2') |

246 | sage: sphere = parametrized_surface3d([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |

247 | |

248 | sage: sphere.first_fundamental_form(vector([v1,v2]),vector([w1,w2])) |

249 | v1*w1*cos(v)^2 + v2*w2 |

250 | |

251 | sage: vv = vector([1,2]) |

252 | sage: sphere.first_fundamental_form(vv,vv) |

253 | cos(v)^2 + 4 |

254 | |

255 | sage: sphere.first_fundamental_form([1,1],[2,1]) |

256 | 2*cos(v)^2 + 1 |

257 | """ |

258 | gg = self.first_fundamental_form_coefficients() |

259 | return sum(gg[ind]*vector1[ind[0]-1]*vector2[ind[1]-1] for ind in self.index_list) |

260 | |

261 | |

262 | @cached_method |

263 | def area_form_squared(self): |

264 | """ |

265 | Gives $g_{11}g_{22} - g_{12}^2$, which is the square of the coefficient of the area form |

266 | |

267 | INPUT: |

268 | No arguments |

269 | |

270 | OUTPUT: |

271 | $g_{11}g_{22} - g_{12}^2$ |

272 | |

273 | EXAMPLES:: |

274 | |

275 | sage: var('u,v') |

276 | sage: sphere = parametrized_surface3d([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |

277 | sage: sphere.area_form_squared() |

278 | cos(v)^2 |

279 | |

280 | """ |

281 | gg = self.first_fundamental_form_coefficients() |

282 | return (gg[(1,1)]*gg[(2,2)]-gg[(1,2)]**2).simplify_full() |

283 | |

284 | |

285 | @cached_method |

286 | def area_form(self): |

287 | """ |

288 | Gives $\sqrt{g_{11}g_{22} - g_{12}^2}$, which is the coefficient of the area form |

289 | |

290 | INPUT: |

291 | No arguments |

292 | |

293 | OUTPUT: |

294 | $\sqrt{g_{11}g_{22} - g_{12}^2}$ |

295 | |

296 | EXAMPLES:: |

297 | |

298 | sage: var('u,v') |

299 | sage: sphere = parametrized_surface3d([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |

300 | sage: sphere.area_form_squared() |

301 | abs(cos(v)) |

302 | |

303 | """ |

304 | return sqrt(self.area_form_squared()).simplify_full() |

305 | |

306 | |

307 | @cached_method |

308 | def first_fundamental_form_inverse_coefficients(self): |

309 | """ |

310 | Gives $g^{ij}$ |

311 | |

312 | INPUT: |

313 | No arguments |

314 | |

315 | OUTPUT: |

316 | dictionary {(1,1):$g^{11}$, (1,2):$g^{12}$, (2,1):$g^{21}$, (2,2):$g^{22}$ |

317 | |

318 | EXAMPLES:: |

319 | |

320 | sage: var('u,v') |

321 | sage: sphere = parametrized_surface3d([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |

322 | sage: sphere.first_fundamental_form_inverse_coefficients() |

323 | {(1, 2): 0, (1, 1): cos(v)^(-2), (2, 1): 0, (2, 2): 1} |

324 | """ |

325 | |

326 | gg = self.first_fundamental_form_coefficients() |

327 | DD = gg[(1,1)]*gg[(2,2)]-gg[(1,2)]**2 |

328 | |

329 | gi11 = (gg[(2,2)]/DD).simplify_full() |

330 | gi12 = (-gg[(1,2)]/DD).simplify_full() |

331 | gi21 = gi12 |

332 | gi22 = (gg[(1,1)]/DD).simplify_full() |

333 | |

334 | return {(1,1):gi11,(1,2):gi12,(2,1):gi21,(2,2):gi22} |

335 | |

336 | def first_fundamental_form_inverse_coefficient(self, index): |

337 | index = tuple(sorted(index)) |

338 | if index not in self.index_list: |

339 | raise ValueError, "Index %s out of bounds." % str(index) |

340 | return self._compute_first_fundamental_form_inverse_coefficients()[index] |

341 | |

342 | |

343 | @cached_method |

344 | def rotation(self,theta): |

345 | """ |

346 | Gives the matrix of the operator of rotation on the given angle $\\theta$ with respect to the natural frame |

347 | |

348 | INPUT: |

349 | given angle $\\theta$ |

350 | |

351 | OUTPUT: |

352 | matrix of the operator of rotation on $\\theta$ with respect to the natural frame |

353 | |

354 | ALGORITHM: |

355 | The operator of rotation on $\pi/2$ is $J^i_j = g^{ik}\omega_{jk}$, where $\omega$ is the area form |

356 | The operator of rotation on angle $\\theta$ is $\cos(\\theta) I + sin(\\theta) J$ |

357 | |

358 | EXAMPLES:: |

359 | |

360 | sage: var('u,v') |

361 | sage: assume(cos(v)>0) |

362 | sage: sphere = parametrized_surface3d([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |

363 | sage: rotation = sphere.rotation(pi/3) |

364 | sage: rotation^3 |

365 | [-1 0] |

366 | [ 0 -1] |

367 | # it is true because the rotation at $\pi/3$ applied three times gives rotation a $\pi$ |

368 | """ |

369 | |

370 | from sage.functions.trig import sin, cos |

371 | |

372 | gi = self.first_fundamental_form_inverse_coefficients() |

373 | w12 = self.area_form() |

374 | RR11 = (cos(theta) + sin(theta)*gi[1,2]*w12).simplify_full() |

375 | RR12 = (- sin(theta)*gi[1,1]*w12).simplify_full() |

376 | RR21 = (sin(theta)*gi[2,2]*w12).simplify_full() |

377 | RR22 = (cos(theta) - sin(theta)*gi[2,1]*w12).simplify_full() |

378 | rotation = matrix([[RR11,RR12],[RR21,RR22]]) |

379 | return rotation |

380 | |

381 | |

382 | # XXXXXX |

383 | |

384 | |

385 | |

386 | |

387 | |

388 | @cached_method |

389 | def orthonormal_frame_vector(self, vector_number=0, coordinates='ext'): |

390 | """ |

391 | Gives the orthonormal frame field of the surface |

392 | |

393 | INPUT: |

394 | (vector_number,coordinates), where |

395 | vector number is 0 (default), 1 or 2 |

396 | coordinates is 'ext'(default) or 'int' |

397 | |

398 | |

399 | OUTPUT: |

400 | If vector_number is equal to 1 or 2, the output is the corresponding vector of an orthonormal frame. |

401 | If argument vector_number is empty, the output is the dictionary of the vector fields of an orthonormal frame. |

402 | |

403 | If coordinates='ext', output is coordinates in $\mathbb{R}^3$ |

404 | If coordinates='int', output is coordinates with respect to the natural frame |

405 | |

406 | ALGORITHM: |

407 | We normalize the first vector $\\vec e_1$ of the natural frame and then get the second frame vector |

408 | $\\vec e_2 = [\\vec n, \\vec e_1]$. |

409 | |

410 | EXAMPLES:: |

411 | |

412 | sage: var('u,v') |

413 | sage: assume(cos(v)>0) |

414 | sage: sphere = parametrized_surface3d([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |

415 | sage: EE = sphere.orthonormal_frame() |

416 | sage: (EE[1]*EE[1]).simplify_full() |

417 | 1 |

418 | sage: (EE[1]*EE[2]).simplify_full() |

419 | 0 |

420 | sage: simplify_vector(vector_product(EE[1],EE[2])-sphere.normal_vector(1)) |

421 | [0,0,0] |

422 | |

423 | sage: EE_int = sphere.orthonormal_frame(coordinates='int') |

424 | sage: EE_int |

425 | {1: (1/cos(v), 0), 2: (0, 1)} |

426 | sage: sphere.first_fundamental_form(EE_int[1],EE_int[1]) |

427 | 1 |

428 | sage: sphere.first_fundamental_form(EE_int[1],EE_int[2]) |

429 | 0 |

430 | sage: sphere.first_fundamental_form(EE_int[2],EE_int[2]) |

431 | 1 |

432 | |

433 | """ |

434 | |

435 | if vector_number not in [1, 2]: |

436 | raise ValueError, "Basis vector number out of range." |

437 | return self.orthonormal_frame(coordinates)[vector_number] |

438 | |

439 | |

440 | @cached_method |

441 | def orthonormal_frame(self, coordinates='ext'): |

442 | |

443 | from sage.symbolic.constants import pi |

444 | |

445 | if coordinates not in ['ext', 'int']: |

446 | raise ValueError, \ |

447 | r"Coordinate system must be exterior ('ext') or interior ('int')." |

448 | |

449 | c = self.first_fundamental_form_coefficient([1,1]) |

450 | if coordinates == 'ext': |

451 | f1 = self.natural_frame()[1] |

452 | |

453 | E1 = simplify_vector(f1/sqrt(c)) |

454 | E2 = simplify_vector(self.normal_vector(normalized=True).cross_product(E1)) |

455 | else: |

456 | E1 = vector([(1/sqrt(c)).simplify_full(), 0]) |

457 | E2 = simplify_vector(self.rotation(pi/2)*E1) |

458 | return {1:E1, 2:E2} |

459 | |

460 | |

461 | def lie_bracket(self,v,w): |

462 | """ |

463 | Gives the Lie bracket of two vector fields which are given by coordinates with respect to the natural frame |

464 | |

465 | |

466 | INPUT: |

467 | v and w are vectors |

468 | |

469 | |

470 | OUTPUT: |

471 | vector [v,w] |

472 | |

473 | |

474 | EXAMPLES:: |

475 | |

476 | sage: var('u,v') |

477 | sage: assume(cos(v)>0) |

478 | sage: sphere = parametrized_surface3d([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |

479 | sage: sphere.lie_bracket([u,v],[-v,u]) |

480 | (0, 0) |

481 | |

482 | sage: EE_int = sphere.orthonormal_frame(coordinates='int') |

483 | sage: sphere.lie_bracket(EE_int[1],EE_int[2]) |

484 | (sin(v)/cos(v)^2, 0) |

485 | """ |

486 | vv = vector([xx for xx in v]) |

487 | ww = vector([xx for xx in w]) |

488 | uuu = [self.variables[1],self.variables[2]] |

489 | Dvv = matrix([ [ diff(xxx,uu).simplify_full() for uu in uuu ] for xxx in vv]) |

490 | Dww = matrix([ [ diff(xxx,uu).simplify_full() for uu in uuu ] for xxx in ww]) |

491 | return simplify_vector(vector(Dvv*ww - Dww*vv)) |

492 | |

493 | |

494 | |

495 | def frame_structure_functions(self,e1,e2): |

496 | """ |

497 | Gives the structure functions $c^k_{ij}$ for a frame field $e_1$, $e_2$, where |

498 | $[e_i,e_j] = c^k_{ij}e_k$ |

499 | |

500 | |

501 | INPUT: |

502 | Frame vectors $e_1$, $e_2$ |

503 | |

504 | |

505 | OUTPUT: |

506 | The dictionary of $c^k_{ij}$. |

507 | Warning: the tuple (i,j,k) corresponds to $c^k_{ij}$ |

508 | |

509 | |

510 | EXAMPLES:: |

511 | |

512 | sage: var('u,v') |

513 | sage: assume(cos(v)>0) |

514 | sage: sphere = parametrized_surface3d([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |

515 | sage: sphere.frame_structure_functions([u,v],[-v,u]) |

516 | {(1, 2, 1): 0, (2, 1, 2): 0, (2, 2, 2): 0, (1, 2, 2): 0, (1, 1, 1): 0, (2, 1, 1): 0, (2, 2, 1): 0, (1, 1, 2): 0} |

517 | sage: structfun = sphere.frame_structure_functions(EE_int[1],EE_int[2]) |

518 | sage: structfun |

519 | {(1, 2, 1): sin(v)/cos(v), (2, 1, 2): 0, (2, 2, 2): 0, (1, 2, 2): 0, (1, 1, 1): 0, (2, 1, 1): -sin(v)/cos(v), (2, 2, 1): 0, (1, 1, 2): 0} |

520 | sage: sphere.lie_bracket(EE_int[1],EE_int[2]) - CC[(1,2,1)]*EE_int[1] - CC[(1,2,2)]*EE_int[2] |

521 | (0, 0) |

522 | """ |

523 | ee1 = vector([xx for xx in e1]) |

524 | ee2 = vector([xx for xx in e2]) |

525 | |

526 | lb = simplify_vector(self.lie_bracket(ee1,ee2)) |

527 | trans_matrix = matrix([[ee1[0],ee2[0]],[ee1[1],ee2[1]]]) |

528 | zz = simplify_vector(trans_matrix.inverse()*lb) |

529 | return {(1,1,1):0, (1,1,2):0, (1,2,1):zz[0], (1,2,2):zz[1], (2,1,1):-zz[0], (2,1,2):-zz[1],(2,2,1):0, (2,2,2):0} |

530 | |

531 | |

532 | @cached_method |

533 | def _compute_second_order_frame_element(self, index): |

534 | variables = [self.variables[i] for i in index] |

535 | ddr_element = vector([diff(f, variables).simplify_full() for f in self.equation]) |

536 | |

537 | return ddr_element |

538 | |

539 | @cached_method |

540 | def second_order_natural_frame(self): |

541 | |

542 | """ |

543 | Gives the second derivatives of the equation $\\vec r = \\vec r(u^1,u^2)$ of parametrized surface |

544 | |

545 | |

546 | INPUT: |

547 | empty argument, or one of the the following pair of indices: 1,1; 1,2; 2,1; 2,2. |

548 | |

549 | |

550 | OUTPUT: |

551 | With empty argument the output is the dictionary of $\partial_{ij}\\vec r(u^1,u^2)$. |

552 | |

553 | With given indices the output is the corresponding second derivative of the surface parametric equation, |

554 | For example, index1 = 1, index2 = 2 gives $\partial_{12}\\vec r(u^1,u^2)$. |

555 | |

556 | |

557 | EXAMPLES:: |

558 | |

559 | sage: var('u,v') |

560 | sage: sphere = parametrized_surface3d([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |

561 | sage: sphere.second_order_natural_frame() |

562 | {(1, 2): (sin(u)*sin(v), -sin(v)*cos(u), 0), (1, 1): (-cos(u)*cos(v), |

563 | -sin(u)*cos(v), 0), (2, 1): (sin(u)*sin(v), -sin(v)*cos(u), 0), (2, 2): |

564 | (-cos(u)*cos(v), -sin(u)*cos(v), -sin(v))} |

565 | |

566 | sage: sphere.second_order_natural_frame(1,1) |

567 | (-cos(u)*cos(v), -sin(u)*cos(v), 0) |

568 | sage: sphere.second_order_natural_frame(1,2) |

569 | (sin(u)*sin(v), -sin(v)*cos(u), 0) |

570 | sage: sphere.second_order_natural_frame(2,2) |

571 | (-cos(u)*cos(v), -sin(u)*cos(v), -sin(v) |

572 | """ |

573 | |

574 | vectors = {} |

575 | for index in self.index_list: |

576 | sorted_index = tuple(sorted(index)) |

577 | vectors[index] = \ |

578 | self._compute_second_order_frame_element(sorted_index) |

579 | return vectors |

580 | |

581 | def second_order_natural_frame_element(self, index): |

582 | index = tuple(sorted(index)) |

583 | if index not in self.index_list: |

584 | raise ValueError, "Index %s out of bounds." % str(index) |

585 | return self._compute_second_order_frame_element(index) |

586 | |

587 | |

588 | @cached_method |

589 | def _compute_second_fundamental_form_coefficient(self, index): |

590 | NN = self.normal_vector(normalized=True) |

591 | # v = self.second_order_natural_frame_element_new(index) |

592 | v = self.second_order_natural_frame_element(index) |

593 | return (v*NN).simplify_full() |

594 | |

595 | |

596 | def second_fundamental_form_coefficient(self, index): |

597 | index = tuple(index) |

598 | if index not in self.index_list: |

599 | raise ValueError, "Index %s out of bounds." % str(index) |

600 | return self._compute_second_fundamental_form_coefficient(index) |

601 | |

602 | |

603 | @cached_method |

604 | def second_fundamental_form_coefficients(self): |

605 | """ |

606 | This function gives the coefficients $h_{ij}$ of the second fundamental form. |

607 | |

608 | INPUT: |

609 | empty argument, or one of the the following pair of indices: 1,1; 1,2; 2,1; 2,2. |

610 | |

611 | OUTPUT: |

612 | with empty argument the output is the dictionary of coefficients of the second fundamental form |

613 | |

614 | with given indices the output is the corresponding coefficient of the second fundamental form, |

615 | for example index1 = 1, index2 = 2 gives the coefficient $h_{12}$ |

616 | |

617 | EXAMPLES:: |

618 | |

619 | sage: var('u,v') |

620 | sage: assume(cos(v)>0) |

621 | sage: sphere = parametrized_surface3d([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |

622 | |

623 | sage: sphere.second_fundamental_form_coefficients() |

624 | {(1, 2): 0, (1, 1): -cos(v)^2, (2, 1): 0, (2, 2): -1} |

625 | sage: sphere.second_fundamental_form_coefficients(1,1) |

626 | -cos(v)^2 |

627 | sage: sphere.second_fundamental_form_coefficients(2,1) |

628 | 0 |

629 | |

630 | sage: sphere.second_fundamental_form_coefficients(3,2) |

631 | 'The argument is not appropriate. Read doc' |

632 | """ |

633 | |

634 | coefficients = {} |

635 | for index in self.index_list: |

636 | coefficients[index] = \ |

637 | self._compute_second_fundamental_form_coefficient(index) |

638 | return coefficients |

639 | |

640 | |

641 | def second_fundamental_form(self,vector1,vector2): |

642 | """ |

643 | Finds the value of second fundamental form on two vectors |

644 | |

645 | INPUT: |

646 | Two vectors $v=(v^1,v^2)$ and $w=(w^1,w^2)$ |

647 | |

648 | OUTPUT: |

649 | $h_{11} v^1 w^1 + h_{12}(v^1 w^2 + v^2 w^1) + h_{22} v^2 w^2$ |

650 | |

651 | EXAMPLES:: |

652 | |

653 | sage: var('u,v') |

654 | sage: var ('v1,v2,w1,w2') |

655 | sage: assume(cos(v) > 0) |

656 | sage: sphere = parametrized_surface3d([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |

657 | sage: sphere.second_fundamental_form(vector([v1,v2]),vector([w1,w2])) |

658 | -v1*w1*cos(v)^2 - v2*w2 |

659 | sage: vv = vector([1,2]) |

660 | sage: sphere.second_fundamental_form(vv,vv) |

661 | -cos(v)^2 - 4 |

662 | sage: sphere.second_fundamental_form([1,1],[2,1]) |

663 | -2*cos(v)^2 - 1 |

664 | |

665 | """ |

666 | hh = self.second_fundamental_form_coefficients() |

667 | return sum(hh[ind]*vector1[ind[0]-1]*vector2[ind[1]-1] for ind in self.index_list) |

668 | |

669 | |

670 | @cached_method |

671 | def gauss_curvature(self): |

672 | """ |

673 | Finds the gaussian curvature $K = \\frac{h_{11}h_{22} - h_{12}^2}{g_{11}g_{22} - g_{12}^2}$. |

674 | |

675 | INPUT: |

676 | No arguments |

677 | |

678 | OUTPUT: |

679 | $K = \\frac{h_{11}h_{22} - h_{12}^2}{g_{11}g_{22} - g_{12}^2}$ |

680 | |

681 | EXAMPLES:: |

682 | |

683 | sage: var('R') |

684 | sage: assume(R>0) |

685 | sage: var('u,v') |

686 | sage: assume(cos(v)>0) |

687 | sage: sphere = parametrized_surface3d([R*cos(u)*cos(v),R*sin(u)*cos(v),R*sin(v)],[u,v],'sphere') |

688 | sage: sphere.gauss_curvature() |

689 | R^(-2) |

690 | |

691 | """ |

692 | hh = self.second_fundamental_form_coefficients() |

693 | return ((hh[(1,1)]*hh[(2,2)]-hh[(1,2)]**2)/self.area_form_squared()).simplify_full() |

694 | |

695 | |

696 | @cached_method |

697 | def mean_curvature(self): |

698 | """ |

699 | Finds the mean curvature $H = \\frac{1}{2}\\frac{g_{22}h_{11} - 2g_{12}h_{12} + g_{11}h_{22}}{g_{11}g_{22} - g_{12}^2}$. |

700 | |

701 | INPUT: |

702 | No arguments |

703 | |

704 | OUTPUT: |

705 | $H = \\frac{1}{2}\\frac{g_{22}h_{11} - 2g_{12}h_{12} + g_{11}h_{22}}{g_{11}g_{22} - g_{12}^2}$. |

706 | |

707 | EXAMPLES:: |

708 | |

709 | sage: var('R') |

710 | sage: assume(R>0) |

711 | sage: var('u,v') |

712 | sage: assume(cos(v)>0) |

713 | sage: sphere = parametrized_surface3d([R*cos(u)*cos(v),R*sin(u)*cos(v),R*sin(v)],[u,v],'sphere') |

714 | sage: sphere.mean_curvature() |

715 | -1/R |

716 | |

717 | """ |

718 | gg = self.first_fundamental_form_coefficients() |

719 | hh = self.second_fundamental_form_coefficients() |

720 | denom = 2*self.area_form_squared() |

721 | enum =(gg[(2,2)]*hh[(1,1)]-2*gg[(1,2)]*hh[(1,2)]+gg[(1,1)]*hh[(2,2)]).simplify_full() |

722 | return (enum/denom).simplify_full() |

723 | |

724 | |

725 | @cached_method |

726 | def principal_curvatures(self): |

727 | """ |

728 | Finds the principal curvatures of the surface |

729 | |

730 | INPUT: |

731 | No arguments |

732 | |

733 | OUTPUT: |

734 | The dictionary of principal curvatures |

735 | |

736 | EXAMPLES:: |

737 | |

738 | sage: var('R') |

739 | sage: assume(R>0) |

740 | sage: var('u,v') |

741 | sage: assume(cos(v)>0) |

742 | sage: sphere = parametrized_surface3d([R*cos(u)*cos(v),R*sin(u)*cos(v),R*sin(v)],[u,v],'sphere') |

743 | sage: sphere.principal_curvatures() |

744 | {1: -1/R, 2: -1/R} |

745 | |

746 | sage: var('u,v') |

747 | sage: var('R,r') |

748 | sage: assume(R>r,r>0) |

749 | sage: torus = parametrized_surface3d([(R+r*cos(v))*cos(u),(R+r*cos(v))*sin(u),r*sin(v)],[u,v],'torus') |

750 | sage: torus.principal_curvatures() |

751 | {1: -cos(v)/(r*cos(v) + R), 2: -1/r} |

752 | |

753 | """ |

754 | |

755 | from sage.symbolic.assumptions import assume |

756 | from sage.symbolic.relation import solve |

757 | from sage.calculus.var import var |

758 | |

759 | KK = self.gauss_curvature() |

760 | HH = self.mean_curvature() |

761 | |

762 | # jvkersch: when this assumption is uncommented, Sage raises an error stating that the assumption |

763 | # is redundant... Can we safely omit this, based on some geometric reasoning? |

764 | |

765 | # assume(HH**2-KK>=0) |

766 | # mikarm: This is a problem, I had a lot of trobles here. Of course, this inequality always hold true. |

767 | # I insert this assumption because sage sometimes, in simplification, uses the complex numbers, though the roots |

768 | # are, for sure, real. |

769 | # This, in turn, causes problems when we substitute coordinates into the expression of principal curvatures. |

770 | # Unfortunately, I did not manage to tell Sage that they are real (to declare the variables as real). |

771 | # Moreover, in a neighborhood of an umbilic point we even cannot "smoothly" order the set of principal curvatures. |

772 | # So, in general, at present this method is far from the final form. |

773 | |

774 | |

775 | |

776 | x = var('x') |

777 | sol = solve(x**2 -2*HH*x + KK == 0,x) |

778 | |

779 | #k1=var('k1') |

780 | #k2=var('k2') |

781 | |

782 | # jvkersch: when I tried to run the previous version of the code, I ran into the problem that if the equation for the principal curvatures had a double root (as in the case of the sphere example in the worksheet), solve returned only one root. Maybe this is a difference due to having different versions of sage. |

783 | |

784 | k1 = (x.substitute(sol[0])).simplify_full() |

785 | if len(sol) == 1: |

786 | k2 = k1 |

787 | else: |

788 | k2 = (x.substitute(sol[1])).simplify_full() |

789 | |

790 | return {1:k1,2: k2} |

791 | |

792 | #mikarm: add method for finding shape operator |

793 | @cached_method |

794 | def shape_operator_coefficients(self): |

795 | |

796 | gi = self.first_fundamental_form_inverse_coefficients() |

797 | hh = self.second_fundamental_form_coefficients() |

798 | |

799 | shop11 = (gi[(1,1)]*hh[(1,1)] + gi[(1,2)]*hh[(1,2)]).simplify_full() |

800 | shop12 = (gi[(1,1)]*hh[(2,1)] + gi[(1,2)]*hh[(2,2)]).simplify_full() |

801 | shop21 = (gi[(2,1)]*hh[(1,1)] + gi[(2,2)]*hh[(1,2)]).simplify_full() |

802 | shop22 = (gi[(2,1)]*hh[(2,1)] + gi[(2,2)]*hh[(2,2)]).simplify_full() |

803 | |

804 | return {(1,1):shop11,(1,2):shop12,(2,1):shop21,(2,2):shop22} |

805 | |

806 | def shape_operator(self,v): |

807 | vv = vector([xx for xx in v]) |

808 | shop = self.shape_operator_coefficients() |

809 | shop_matrix=matrix([[shop[(1,1)],shop[(1,2)]],[shop[(2,1)],shop[(2,2)]]]) |

810 | return simplify_vector(shop_matrix*vv) |

811 | |

812 | @cached_method |

813 | def principal_directions(self): |

814 | """ |

815 | Finds the principal curvatures and principal directions of the surface |

816 | |

817 | INPUT: |

818 | No arguments |

819 | |

820 | OUTPUT: |

821 | The dictionary of lists [a principal curvature, the corresponding principal direction] |

822 | |

823 | If principal curvatures coincide, gives the warning that the surface is a sphere. |

824 | |

825 | EXAMPLES:: |

826 | |

827 | sage: var('u,v') |

828 | sage: var('R,r') |

829 | sage: assume(R>r,r>0) |

830 | sage: torus = parametrized_surface3d([(R+r*cos(v))*cos(u),(R+r*cos(v))*sin(u),r*sin(v)],[u,v],'torus') |

831 | sage: pdd = torus.principal_directions() |

832 | sage: pdd[1] |

833 | [-cos(v)/(r*cos(v) + R), (1, 0)] |

834 | sage: pdd[2] |

835 | [-1/r, (0, -(R*r*cos(v) + R^2)/r)] |

836 | |

837 | sage: var('RR') |

838 | sage: assume(RR>0) |

839 | sage: var('u,v') |

840 | sage: assume(cos(v)>0) |

841 | sage: sphere = parametrized_surface3d([RR*cos(u)*cos(v),RR*sin(u)*cos(v),RR*sin(v)],[u,v],'sphere') |

842 | sage: sphere.principal_directions() |

843 | 'This is a sphere, so any direction is principal' |

844 | |

845 | sage: var('aa') |

846 | sage: assume(aa>0) |

847 | sage: catenoid = parametrized_surface3d([aa*cosh(v)*cos(u),aa*cosh(v)*sin(u),v],[u,v],'catenoid') |

848 | sage: pd = catenoid.principal_directions() |

849 | sage: pd[1][1] |

850 | (0, 1/2*(2*aa^3*sinh(v)^2*cosh(v) + sqrt(4*aa^4*sinh(v)^2*cosh(v)^2 + aa^4 + 4*aa^2*cosh(v)^2 - 2*aa^2 + 1)*aa*cosh(v) + (aa^3 + aa)*cosh(v))/(aa^2*sinh(v)^2 + 1)^(3/2)) |

851 | sage: pd[2][1] |

852 | (1, 0) |

853 | sage: pd[1][1]*pd[2][1] |

854 | 0 |

855 | """ |

856 | gg = self.first_fundamental_form_coefficients() |

857 | hh = self.second_fundamental_form_coefficients() |

858 | kk = self.principal_curvatures() |

859 | if kk[1]==kk[2]: |

860 | return "This is a sphere, so any direction is principal" |

861 | pd1 = simplify_vector([hh[(1,2)]-kk[1]*gg[(1,2)],-hh[(1,1)]+kk[1]*gg[(1,1)]]) |

862 | if pd1==vector([0,0]): |

863 | pd1 = vector([1,0]) |

864 | pd2 = simplify_vector([hh[(1,2)]-kk[2]*gg[(1,2)],-hh[(1,1)]+kk[2]*gg[(1,1)]]) |

865 | if pd2==vector([0,0]): |

866 | pd2 = vector([1,0]) |

867 | return {1:[kk[1],pd1],2:[kk[2],pd2]} |

868 | |

869 | |

870 | @cached_method |

871 | def connection_coefficients(self): |

872 | """ |

873 | Finds the connection coefficients of the surface |

874 | |

875 | INPUT: |

876 | No arguments |

877 | |

878 | OUTPUT: |

879 | The dictionary of connection coefficients. |

880 | |

881 | Warning: the triple $(i,j,k)$ corresponds to $\Gamma^k_{ij}$. |

882 | |

883 | EXAMPLES:: |

884 | |

885 | sage: var('r') |

886 | sage: assume(r > 0) |

887 | sage: var('u,v') |

888 | sage: assume(cos(v)>0) |

889 | sage: sphere = parametrized_surface3d([r*cos(u)*cos(v),r*sin(u)*cos(v),r*sin(v)],[u,v],'sphere') |

890 | sage: sphere.connection_coefficients() |

891 | {(1, 2, 1): -sin(v)/cos(v), (2, 2, 2): 0, (1, 2, 2): 0, (2, 1, 1): -sin(v)/cos(v), (1, 1, 2): sin(v)*cos(v), (2, 2, 1): 0, (2, 1, 2): 0, (1, 1, 1): 0} |

892 | |

893 | """ |

894 | x = self.variables |

895 | gg = self.first_fundamental_form_coefficients() |

896 | gi = self.first_fundamental_form_inverse_coefficients() |

897 | |

898 | dg = {} |

899 | for kkk in self.index_list_3: |

900 | dg[kkk]=gg[(kkk[1],kkk[2])].differentiate(x[kkk[0]]).simplify_full() |

901 | structfun={} |

902 | |

903 | for kkk in self.index_list_3: |

904 | structfun[kkk]=sum(gi[(kkk[2],s)]*(dg[(kkk[0],kkk[1],s)]+dg[(kkk[1],kkk[0],s)]-dg[(s,kkk[0],kkk[1])])/2 for s in (1,2)).full_simplify() |

905 | return structfun |

906 | |

907 | |

908 | # jvkersch: this private method creates an ode_solver object, which can be used |

909 | # to integrate the geodesic equations numerically. |

910 | |

911 | @cached_method |

912 | def _create_geodesic_ode_system(self): |

913 | from sage.ext.fast_eval import fast_float |

914 | from sage.calculus.var import var |

915 | from sage.gsl.ode import ode_solver |

916 | |

917 | u1 = self.variables[1] |

918 | u2 = self.variables[2] |

919 | v1, v2 = var('v1, v2') |

920 | |

921 | C = self.connection_coefficients() |

922 | |

923 | dv1 = - C[(1,1,1)]*v1**2 - 2*C[(1,2,1)]*v1*v2 - C[(2,2,1)]*v2**2 |

924 | dv2 = - C[(1,1,2)]*v1**2 - 2*C[(1,2,2)]*v1*v2 - C[(2,2,2)]*v2**2 |

925 | fun1 = fast_float(dv1, str(u1), str(u2), str(v1), str(v2)) |

926 | fun2 = fast_float(dv2, str(u1), str(u2), str(v1), str(v2)) |

927 | |

928 | geodesic_ode = ode_solver() |

929 | geodesic_ode.function = \ |

930 | lambda t, (u1, u2, v1, v2) : \ |

931 | [v1, v2, fun1(u1, u2, v1, v2), fun2(u1, u2, v1, v2)] |

932 | return geodesic_ode |

933 | |

934 | |

935 | # jvkersch: integrate the geodesic equations numerically |

936 | # mikarm: Very good. I cheked it restarting each time the worksheet, it works much faster. |

937 | |

938 | def geodesics_numerical(self, p0, v0, tinterval): |

939 | """ |

940 | This method gives the numerical solution for the geodesic equations |

941 | |

942 | INPUT: |

943 | p0 is the list of the coordinates of the initial point |

944 | |

945 | v0 is the list of the coordinates of the initial vector |

946 | |

947 | tinterval is the list [a,b,M], where (a,b) is the domain of the geodesic, M is the number of division points |

948 | |

949 | OUTPUT: |

950 | The list consisting of lists [t, [u1(t),u2(t)], [v1(t),v2(t)], [x1(t),x2(t),x3(t)]], where |

951 | |

952 | t is a subdivision point |

953 | |

954 | [u1(t),u2(t)] is the list of coordinates of the geodesic point |

955 | |

956 | [v1(t),v2(t)] is the list of coordinates of the vector tanget to the geodesic |

957 | |

958 | [x1(t),x2(t),x3(t)] is the list of coordinates of the geodesic point in the three-dimensional space |

959 | |

960 | |

961 | EXAMPLES:: |

962 | |

963 | sage: var('p,q') |

964 | sage: v = [p,q] |

965 | sage: assume(cos(q)>0) |

966 | sage: sphere = parametrized_surface3d([cos(q)*cos(p),sin(q)*cos(p),sin(p)],v,'sphere') |

967 | sage: gg_array = sphere.geodesics_numerical([0,0],[1,1],[0,2*pi,5]) |

968 | sage: gg_array[0] |

969 | [0.0, [0.0, 0.0], [1.0, 1.0], (1, 0, 0)] |

970 | sage: gg_array[1] |

971 | [1.2566370614359172, [0.76440104189216407, 1.8586223516062499], [-0.2838683571264714, 1.9194187087799912], (-0.204895333443, 0.692104654602, 0.692104796553)] |

972 | |

973 | |

974 | """ |

975 | |

976 | u1 = self.variables[1] |

977 | u2 = self.variables[2] |

978 | |

979 | solver = self._create_geodesic_ode_system() |

980 | |

981 | t_interval, n = tinterval[0:2], tinterval[2] |

982 | solver.y_0 = [p0[0], p0[1], v0[0], v0[1]] |

983 | solver.ode_solve(t_span=t_interval, num_points=n) |

984 | |

985 | parsed_solution = \ |

986 | [[vec[0], vec[1][0:2], vec[1][2:], self.eq_callable(vec[1][0], vec[1][1])] \ |

987 | for vec in solver.solution] |

988 | |

989 | return parsed_solution |

990 | |

991 | |

992 | @cached_method |

993 | def _create_pt_ode_system(self, curve, t): |

994 | from sage.ext.fast_eval import fast_float |

995 | from sage.calculus.var import var |

996 | from sage.gsl.ode import ode_solver |

997 | |

998 | u1 = self.variables[1] |

999 | u2 = self.variables[2] |

1000 | v1, v2 = var('v1, v2') |

1001 | |

1002 | du1 = diff(curve[0], t) |

1003 | du2 = diff(curve[1], t) |

1004 | |

1005 | C = self.connection_coefficients() |

1006 | for coef in C: |

1007 | C[coef] = C[coef].subs({u1: curve[0], u2: curve[1]}) |

1008 | |

1009 | dv1 = - C[(1,1,1)]*v1*du1 - C[(1,2,1)]*(du1*v2 + du2*v1) - C[(2,2,1)]*du2*v2 |

1010 | dv2 = - C[(1,1,2)]*v1*du1 - C[(1,2,2)]*(du1*v2 + du2*v1) - C[(2,2,2)]*du2*v2 |

1011 | fun1 = fast_float(dv1, str(t), str(v1), str(v2)) |

1012 | fun2 = fast_float(dv2, str(t), str(v1), str(v2)) |

1013 | |

1014 | pt_ode = ode_solver() |

1015 | pt_ode.function = lambda t, (v1, v2): [fun1(t, v1, v2), fun2(t, v1, v2)] |

1016 | return pt_ode |

1017 | |

1018 | |

1019 | # mikarm: We should rewrite it like you did for geodesics. |

1020 | # jvkersch: OK, see this and the above |

1021 | def parallel_translation_numerical_new(self,curve,t,v0,tinterval): |

1022 | """ |

1023 | This method gives the numerical solution to the equation of parallel translation of a vector |

1024 | |

1025 | INPUT: |

1026 | curve equation = list of functions which determine the curve wrt the local coordinate |

1027 | |

1028 | t - curve parameter |

1029 | |

1030 | v0 - initial vector |

1031 | |

1032 | tinterval = [a,b,N], (a,b) is the domain of the curve, N is the number of subdivision points |

1033 | |

1034 | OUTPUT: |

1035 | The list consisting of lists [t, [v1(t),v2(t)]], where |

1036 | |

1037 | t is a subdivision point |

1038 | |

1039 | [v1(t),v2(t)] is the list of coordinates of the vector translated parallely along the curve |

1040 | |

1041 | EXAMPLES:: |

1042 | |

1043 | sage: var('p,q') |

1044 | sage: v = [p,q] |

1045 | sage: assume(cos(q)>0) |

1046 | sage: sphere = parametrized_surface3d([cos(q)*cos(p),sin(q)*cos(p),sin(p)],v,'sphere') |

1047 | sage: var('ss') |

1048 | sage: vv_array = sphere.parallel_translation_numerical([ss,ss],ss,[1,1],[0,pi/4,20]) |

1049 | sage: vv_array[0] |

1050 | [0.0, [1.0, 1.0]] |

1051 | sage: vv_array[5] |

1052 | [0.19634954084936207, [0.98060182522638917, 1.0389930474425113]] |

1053 | |

1054 | """ |

1055 | |

1056 | u1 = self.variables[1] |

1057 | u2 = self.variables[2] |

1058 | |

1059 | solver = self._create_pt_ode_system(tuple(curve), t) |

1060 | |

1061 | t_interval, n = tinterval[0:2], tinterval[2] |

1062 | solver.y_0 = v0 |

1063 | solver.ode_solve(t_span=t_interval, num_points=n) |

1064 | |

1065 | return solver.solution |

1066 | |

1067 | |

1068 | |

1069 | |

1070 | |

1071 | def parallel_translation_numerical(self,curve,t,v0,tinterval): |

1072 | """ |

1073 | This method gives the numerical solution to the equation of parallel translation of a vector |

1074 | |

1075 | INPUT: |

1076 | curve equation = list of functions which determine the curve wrt the local coordinate |

1077 | |

1078 | t - curve parameter |

1079 | |

1080 | v0 - initial vector |

1081 | |

1082 | tinterval = [a,b,N], (a,b) is the domain of the curve, N is the number of subdivision points |

1083 | |

1084 | OUTPUT: |

1085 | The list consisting of lists [t, [v1(t),v2(t)]], where |

1086 | |

1087 | t is a subdivision point |

1088 | |

1089 | [v1(t),v2(t)] is the list of coordinates of the vector translated parallely along the curve |

1090 | |

1091 | EXAMPLES:: |

1092 | |

1093 | sage: var('p,q') |

1094 | sage: v = [p,q] |

1095 | sage: assume(cos(q)>0) |

1096 | sage: sphere = parametrized_surface3d([cos(q)*cos(p),sin(q)*cos(p),sin(p)],v,'sphere') |

1097 | sage: var('ss') |

1098 | sage: vv_array = sphere.parallel_translation_numerical([ss,ss],ss,[1,1],[0,pi/4,20]) |

1099 | sage: vv_array[0] |

1100 | [0.0, [1.0, 1.0]] |

1101 | sage: vv_array[5] |

1102 | [0.19634954084936207, [0.98060182522638917, 1.0389930474425113]] |

1103 | |

1104 | """ |

1105 | u1 = self.variables[1] |

1106 | u2 = self.variables[2] |

1107 | #var('uu1,uu2') |

1108 | |

1109 | #f(u1,u2) = self.equation |

1110 | |

1111 | def f(ux, vx): |

1112 | return [self.equation[0].subs({u1: ux, u2: vx}), |

1113 | self.equation[1].subs({u1: ux, u2: vx}), |

1114 | self.equation[2].subs({u1: ux, u2: vx})] |

1115 | |

1116 | #from sage.calculus.var import var |

1117 | #t = var('t') |

1118 | tt = t |

1119 | |

1120 | C111 = self.connection_coefficients()[(1,1,1)] |

1121 | C121 = self.connection_coefficients()[(1,2,1)] |

1122 | C221 = self.connection_coefficients()[(2,2,1)] |

1123 | C112 = self.connection_coefficients()[(1,1,2)] |

1124 | C122 = self.connection_coefficients()[(1,2,2)] |

1125 | C222 = self.connection_coefficients()[(2,2,2)] |

1126 | |

1127 | du1=diff(curve[0],tt) |

1128 | du2=diff(curve[1],tt) |

1129 | uu1=curve[0] |

1130 | uu2=curve[1] |

1131 | |

1132 | |

1133 | def ode_system(unknown_functions,t1): |

1134 | du1p = du1.subs(tt==t1) |

1135 | du2p = du2.subs(tt==t1) |

1136 | uu1p = uu1.subs(tt==t1) |

1137 | uu2p = uu2.subs(tt==t1) |

1138 | |

1139 | c111 = C111.subs({u1: uu1p, u2: uu2p}) |

1140 | c121 = C121.subs({u1: uu1p, u2: uu2p}) |

1141 | c221 = C221.subs({u1: uu1p, u2: uu2p}) |

1142 | c112 = C112.subs({u1: uu1p, u2: uu2p}) |

1143 | c122 = C122.subs({u1: uu1p, u2: uu2p}) |

1144 | c222 = C222.subs({u1: uu1p, u2: uu2p}) |

1145 | |

1146 | #print c111, c121, c221, c112, c122, c222 |

1147 | |

1148 | v1,v2 = unknown_functions |

1149 | f1 = - c111*du1p*v1 - c121*(du1p*v2 + du2p*v1) - c221*du2p*v2 |

1150 | f2 = - c112*du1p*v1 - c122*(du1p*v2 + du2p*v1) - c222*du2p*v2 |

1151 | dv1 = f1.subs(tt==t1) |

1152 | dv2 = f2.subs(tt==t1) |

1153 | |

1154 | #print dv1, dv2 |

1155 | |

1156 | return float(dv1), float(dv2) |

1157 | |

1158 | step = float((tinterval[1]-tinterval[0])/tinterval[2]) |

1159 | ttt = float(tinterval[0]) |

1160 | tarray = [ ttt ] |

1161 | for counter in range(1,tinterval[2]+1): |

1162 | ttt = ttt + step |

1163 | tarray = tarray + [ ttt ] |

1164 | |

1165 | |

1166 | # import ode solving routine |

1167 | import scipy.integrate |

1168 | |

1169 | initial_data = (v0[0],v0[1]) |

1170 | solution = scipy.integrate.odeint(ode_system,initial_data,tarray) |

1171 | return [[ tarray[counter],[solution[counter,0], solution[counter,1]]] for counter in range(0,len(tarray))] |

1172 |