| 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 | return (v*NN).simplify_full() |
| 593 | |
| 594 | |
| 595 | def second_fundamental_form_coefficient(self, index): |
| 596 | index = tuple(index) |
| 597 | if index not in self.index_list: |
| 598 | raise ValueError, "Index %s out of bounds." % str(index) |
| 599 | return self._compute_second_fundamental_form_coefficient(index) |
| 600 | |
| 601 | |
| 602 | @cached_method |
| 603 | def second_fundamental_form_coefficients(self): |
| 604 | """ |
| 605 | This function gives the coefficients $h_{ij}$ of the second fundamental form. |
| 606 | |
| 607 | INPUT: |
| 608 | empty argument, or one of the the following pair of indices: 1,1; 1,2; 2,1; 2,2. |
| 609 | |
| 610 | OUTPUT: |
| 611 | with empty argument the output is the dictionary of coefficients of the second fundamental form |
| 612 | |
| 613 | with given indices the output is the corresponding coefficient of the second fundamental form, |
| 614 | for example index1 = 1, index2 = 2 gives the coefficient $h_{12}$ |
| 615 | |
| 616 | EXAMPLES:: |
| 617 | |
| 618 | sage: var('u,v') |
| 619 | sage: assume(cos(v)>0) |
| 620 | sage: sphere = parametrized_surface3d([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 621 | |
| 622 | sage: sphere.second_fundamental_form_coefficients() |
| 623 | {(1, 2): 0, (1, 1): -cos(v)^2, (2, 1): 0, (2, 2): -1} |
| 624 | sage: sphere.second_fundamental_form_coefficients(1,1) |
| 625 | -cos(v)^2 |
| 626 | sage: sphere.second_fundamental_form_coefficients(2,1) |
| 627 | 0 |
| 628 | |
| 629 | sage: sphere.second_fundamental_form_coefficients(3,2) |
| 630 | 'The argument is not appropriate. Read doc' |
| 631 | """ |
| 632 | |
| 633 | coefficients = {} |
| 634 | for index in self.index_list: |
| 635 | coefficients[index] = \ |
| 636 | self._compute_second_fundamental_form_coefficient(index) |
| 637 | return coefficients |
| 638 | |
| 639 | |
| 640 | def second_fundamental_form(self,vector1,vector2): |
| 641 | """ |
| 642 | Finds the value of second fundamental form on two vectors |
| 643 | |
| 644 | INPUT: |
| 645 | Two vectors $v=(v^1,v^2)$ and $w=(w^1,w^2)$ |
| 646 | |
| 647 | OUTPUT: |
| 648 | $h_{11} v^1 w^1 + h_{12}(v^1 w^2 + v^2 w^1) + h_{22} v^2 w^2$ |
| 649 | |
| 650 | EXAMPLES:: |
| 651 | |
| 652 | sage: var('u,v') |
| 653 | sage: var ('v1,v2,w1,w2') |
| 654 | sage: assume(cos(v) > 0) |
| 655 | sage: sphere = parametrized_surface3d([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 656 | sage: sphere.second_fundamental_form(vector([v1,v2]),vector([w1,w2])) |
| 657 | -v1*w1*cos(v)^2 - v2*w2 |
| 658 | sage: vv = vector([1,2]) |
| 659 | sage: sphere.second_fundamental_form(vv,vv) |
| 660 | -cos(v)^2 - 4 |
| 661 | sage: sphere.second_fundamental_form([1,1],[2,1]) |
| 662 | -2*cos(v)^2 - 1 |
| 663 | |
| 664 | """ |
| 665 | hh = self.second_fundamental_form_coefficients() |
| 666 | return sum(hh[ind]*vector1[ind[0]-1]*vector2[ind[1]-1] for ind in self.index_list) |
| 667 | |
| 668 | |
| 669 | @cached_method |
| 670 | def gauss_curvature(self): |
| 671 | """ |
| 672 | Finds the gaussian curvature $K = \\frac{h_{11}h_{22} - h_{12}^2}{g_{11}g_{22} - g_{12}^2}$. |
| 673 | |
| 674 | INPUT: |
| 675 | No arguments |
| 676 | |
| 677 | OUTPUT: |
| 678 | $K = \\frac{h_{11}h_{22} - h_{12}^2}{g_{11}g_{22} - g_{12}^2}$ |
| 679 | |
| 680 | EXAMPLES:: |
| 681 | |
| 682 | sage: var('R') |
| 683 | sage: assume(R>0) |
| 684 | sage: var('u,v') |
| 685 | sage: assume(cos(v)>0) |
| 686 | sage: sphere = parametrized_surface3d([R*cos(u)*cos(v),R*sin(u)*cos(v),R*sin(v)],[u,v],'sphere') |
| 687 | sage: sphere.gauss_curvature() |
| 688 | R^(-2) |
| 689 | |
| 690 | """ |
| 691 | hh = self.second_fundamental_form_coefficients() |
| 692 | return ((hh[(1,1)]*hh[(2,2)]-hh[(1,2)]**2)/self.area_form_squared()).simplify_full() |
| 693 | |
| 694 | |
| 695 | @cached_method |
| 696 | def mean_curvature(self): |
| 697 | """ |
| 698 | Finds the mean curvature $H = \\frac{1}{2}\\frac{g_{22}h_{11} - 2g_{12}h_{12} + g_{11}h_{22}}{g_{11}g_{22} - g_{12}^2}$. |
| 699 | |
| 700 | INPUT: |
| 701 | No arguments |
| 702 | |
| 703 | OUTPUT: |
| 704 | $H = \\frac{1}{2}\\frac{g_{22}h_{11} - 2g_{12}h_{12} + g_{11}h_{22}}{g_{11}g_{22} - g_{12}^2}$. |
| 705 | |
| 706 | EXAMPLES:: |
| 707 | |
| 708 | sage: var('R') |
| 709 | sage: assume(R>0) |
| 710 | sage: var('u,v') |
| 711 | sage: assume(cos(v)>0) |
| 712 | sage: sphere = parametrized_surface3d([R*cos(u)*cos(v),R*sin(u)*cos(v),R*sin(v)],[u,v],'sphere') |
| 713 | sage: sphere.mean_curvature() |
| 714 | -1/R |
| 715 | |
| 716 | """ |
| 717 | gg = self.first_fundamental_form_coefficients() |
| 718 | hh = self.second_fundamental_form_coefficients() |
| 719 | denom = 2*self.area_form_squared() |
| 720 | enum =(gg[(2,2)]*hh[(1,1)]-2*gg[(1,2)]*hh[(1,2)]+gg[(1,1)]*hh[(2,2)]).simplify_full() |
| 721 | return (enum/denom).simplify_full() |
| 722 | |
| 723 | |
| 724 | @cached_method |
| 725 | def principal_curvatures(self): |
| 726 | """ |
| 727 | Finds the principal curvatures of the surface |
| 728 | |
| 729 | INPUT: |
| 730 | No arguments |
| 731 | |
| 732 | OUTPUT: |
| 733 | The dictionary of principal curvatures |
| 734 | |
| 735 | EXAMPLES:: |
| 736 | |
| 737 | sage: var('R') |
| 738 | sage: assume(R>0) |
| 739 | sage: var('u,v') |
| 740 | sage: assume(cos(v)>0) |
| 741 | sage: sphere = parametrized_surface3d([R*cos(u)*cos(v),R*sin(u)*cos(v),R*sin(v)],[u,v],'sphere') |
| 742 | sage: sphere.principal_curvatures() |
| 743 | {1: -1/R, 2: -1/R} |
| 744 | |
| 745 | sage: var('u,v') |
| 746 | sage: var('R,r') |
| 747 | sage: assume(R>r,r>0) |
| 748 | sage: torus = parametrized_surface3d([(R+r*cos(v))*cos(u),(R+r*cos(v))*sin(u),r*sin(v)],[u,v],'torus') |
| 749 | sage: torus.principal_curvatures() |
| 750 | {1: -cos(v)/(r*cos(v) + R), 2: -1/r} |
| 751 | |
| 752 | """ |
| 753 | |
| 754 | from sage.symbolic.assumptions import assume |
| 755 | from sage.symbolic.relation import solve |
| 756 | from sage.calculus.var import var |
| 757 | |
| 758 | KK = self.gauss_curvature() |
| 759 | HH = self.mean_curvature() |
| 760 | |
| 761 | # jvkersch: when this assumption is uncommented, Sage raises an error stating that the assumption |
| 762 | # is redundant... Can we safely omit this, based on some geometric reasoning? |
| 763 | |
| 764 | # assume(HH**2-KK>=0) |
| 765 | # mikarm: This is a problem, I had a lot of trobles here. Of course, this inequality always hold true. |
| 766 | # I insert this assumption because sage sometimes, in simplification, uses the complex numbers, though the roots |
| 767 | # are, for sure, real. |
| 768 | # This, in turn, causes problems when we substitute coordinates into the expression of principal curvatures. |
| 769 | # Unfortunately, I did not manage to tell Sage that they are real (to declare the variables as real). |
| 770 | # Moreover, in a neighborhood of an umbilic point we even cannot "smoothly" order the set of principal curvatures. |
| 771 | # So, in general, at present this method is far from the final form. |
| 772 | |
| 773 | |
| 774 | |
| 775 | x = var('x') |
| 776 | sol = solve(x**2 -2*HH*x + KK == 0,x) |
| 777 | |
| 778 | #k1=var('k1') |
| 779 | #k2=var('k2') |
| 780 | |
| 781 | # 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. |
| 782 | |
| 783 | k1 = (x.substitute(sol[0])).simplify_full() |
| 784 | if len(sol) == 1: |
| 785 | k2 = k1 |
| 786 | else: |
| 787 | k2 = (x.substitute(sol[1])).simplify_full() |
| 788 | |
| 789 | return {1:k1,2: k2} |
| 790 | |
| 791 | @cached_method |
| 792 | def principal_directions(self): |
| 793 | """ |
| 794 | Finds the principal curvatures and principal directions of the surface |
| 795 | |
| 796 | INPUT: |
| 797 | No arguments |
| 798 | |
| 799 | OUTPUT: |
| 800 | The dictionary of lists [a principal curvature, the corresponding principal direction] |
| 801 | |
| 802 | If principal curvatures coincide, gives the warning that the surface is a sphere. |
| 803 | |
| 804 | EXAMPLES:: |
| 805 | |
| 806 | sage: var('u,v') |
| 807 | sage: var('R,r') |
| 808 | sage: assume(R>r,r>0) |
| 809 | sage: torus = parametrized_surface3d([(R+r*cos(v))*cos(u),(R+r*cos(v))*sin(u),r*sin(v)],[u,v],'torus') |
| 810 | sage: pdd = torus.principal_directions() |
| 811 | sage: pdd[1] |
| 812 | [-cos(v)/(r*cos(v) + R), (1, 0)] |
| 813 | sage: pdd[2] |
| 814 | [-1/r, (0, -(R*r*cos(v) + R^2)/r)] |
| 815 | |
| 816 | sage: var('RR') |
| 817 | sage: assume(RR>0) |
| 818 | sage: var('u,v') |
| 819 | sage: assume(cos(v)>0) |
| 820 | sage: sphere = parametrized_surface3d([RR*cos(u)*cos(v),RR*sin(u)*cos(v),RR*sin(v)],[u,v],'sphere') |
| 821 | sage: sphere.principal_directions() |
| 822 | 'This is a sphere, so any direction is principal' |
| 823 | |
| 824 | sage: var('aa') |
| 825 | sage: assume(aa>0) |
| 826 | sage: catenoid = parametrized_surface3d([aa*cosh(v)*cos(u),aa*cosh(v)*sin(u),v],[u,v],'catenoid') |
| 827 | sage: pd = catenoid.principal_directions() |
| 828 | sage: pd[1][1] |
| 829 | (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)) |
| 830 | sage: pd[2][1] |
| 831 | (1, 0) |
| 832 | sage: pd[1][1]*pd[2][1] |
| 833 | 0 |
| 834 | """ |
| 835 | gg = self.first_fundamental_form_coefficients() |
| 836 | hh = self.second_fundamental_form_coefficients() |
| 837 | kk = self.principal_curvatures() |
| 838 | if kk[1]==kk[2]: |
| 839 | return "This is a sphere, so any direction is principal" |
| 840 | pd1 = simplify_vector([hh[(1,2)]-kk[1]*gg[(1,2)],-hh[(1,1)]+kk[1]*gg[(1,1)]]) |
| 841 | if pd1==vector([0,0]): |
| 842 | pd1 = vector([1,0]) |
| 843 | pd2 = simplify_vector([hh[(1,2)]-kk[2]*gg[(1,2)],-hh[(1,1)]+kk[2]*gg[(1,1)]]) |
| 844 | if pd2==vector([0,0]): |
| 845 | pd2 = vector([1,0]) |
| 846 | return {1:[kk[1],pd1],2:[kk[2],pd2]} |
| 847 | |
| 848 | |
| 849 | @cached_method |
| 850 | def connection_coefficients(self): |
| 851 | """ |
| 852 | Finds the connection coefficients of the surface |
| 853 | |
| 854 | INPUT: |
| 855 | No arguments |
| 856 | |
| 857 | OUTPUT: |
| 858 | The dictionary of connection coefficients. |
| 859 | |
| 860 | Warning: the triple $(i,j,k)$ corresponds to $\Gamma^k_{ij}$. |
| 861 | |
| 862 | EXAMPLES:: |
| 863 | |
| 864 | sage: var('r') |
| 865 | sage: assume(r > 0) |
| 866 | sage: var('u,v') |
| 867 | sage: assume(cos(v)>0) |
| 868 | sage: sphere = parametrized_surface3d([r*cos(u)*cos(v),r*sin(u)*cos(v),r*sin(v)],[u,v],'sphere') |
| 869 | sage: sphere.connection_coefficients() |
| 870 | {(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} |
| 871 | |
| 872 | """ |
| 873 | x = self.variables |
| 874 | gg = self.first_fundamental_form_coefficients() |
| 875 | gi = self.first_fundamental_form_inverse_coefficients() |
| 876 | |
| 877 | dg = {} |
| 878 | for kkk in self.index_list_3: |
| 879 | dg[kkk]=gg[(kkk[1],kkk[2])].differentiate(x[kkk[0]]).simplify_full() |
| 880 | structfun={} |
| 881 | |
| 882 | for kkk in self.index_list_3: |
| 883 | 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() |
| 884 | return structfun |
| 885 | |
| 886 | |
| 887 | # jvkersch: this private method creates an ode_solver object, which can be used |
| 888 | # to integrate the geodesic equations numerically. |
| 889 | |
| 890 | @cached_method |
| 891 | def _create_geodesic_ode_system(self): |
| 892 | from sage.ext.fast_eval import fast_float |
| 893 | from sage.calculus.var import var |
| 894 | from sage.gsl.ode import ode_solver |
| 895 | |
| 896 | u1 = self.variables[1] |
| 897 | u2 = self.variables[2] |
| 898 | v1, v2 = var('v1, v2') |
| 899 | |
| 900 | C = self.connection_coefficients() |
| 901 | |
| 902 | dv1 = - C[(1,1,1)]*v1**2 - 2*C[(1,2,1)]*v1*v2 - C[(2,2,1)]*v2**2 |
| 903 | dv2 = - C[(1,1,2)]*v1**2 - 2*C[(1,2,2)]*v1*v2 - C[(2,2,2)]*v2**2 |
| 904 | fun1 = fast_float(dv1, str(u1), str(u2), str(v1), str(v2)) |
| 905 | fun2 = fast_float(dv2, str(u1), str(u2), str(v1), str(v2)) |
| 906 | |
| 907 | geodesic_ode = ode_solver() |
| 908 | geodesic_ode.function = \ |
| 909 | lambda t, (u1, u2, v1, v2) : \ |
| 910 | [v1, v2, fun1(u1, u2, v1, v2), fun2(u1, u2, v1, v2)] |
| 911 | return geodesic_ode |
| 912 | |
| 913 | |
| 914 | # jvkersch: integrate the geodesic equations numerically |
| 915 | # mikarm: Very good. I cheked it restarting each time the worksheet, it works much faster. |
| 916 | |
| 917 | def geodesics_numerical(self, p0, v0, tinterval): |
| 918 | """ |
| 919 | This method gives the numerical solution for the geodesic equations |
| 920 | |
| 921 | INPUT: |
| 922 | p0 is the list of the coordinates of the initial point |
| 923 | |
| 924 | v0 is the list of the coordinates of the initial vector |
| 925 | |
| 926 | tinterval is the list [a,b,M], where (a,b) is the domain of the geodesic, M is the number of division points |
| 927 | |
| 928 | OUTPUT: |
| 929 | The list consisting of lists [t, [u1(t),u2(t)], [v1(t),v2(t)], [x1(t),x2(t),x3(t)]], where |
| 930 | |
| 931 | t is a subdivision point |
| 932 | |
| 933 | [u1(t),u2(t)] is the list of coordinates of the geodesic point |
| 934 | |
| 935 | [v1(t),v2(t)] is the list of coordinates of the vector tanget to the geodesic |
| 936 | |
| 937 | [x1(t),x2(t),x3(t)] is the list of coordinates of the geodesic point in the three-dimensional space |
| 938 | |
| 939 | |
| 940 | EXAMPLES:: |
| 941 | |
| 942 | sage: var('p,q') |
| 943 | sage: v = [p,q] |
| 944 | sage: assume(cos(q)>0) |
| 945 | sage: sphere = parametrized_surface3d([cos(q)*cos(p),sin(q)*cos(p),sin(p)],v,'sphere') |
| 946 | sage: gg_array = sphere.geodesics_numerical([0,0],[1,1],[0,2*pi,5]) |
| 947 | sage: gg_array[0] |
| 948 | [0.0, [0.0, 0.0], [1.0, 1.0], (1, 0, 0)] |
| 949 | sage: gg_array[1] |
| 950 | [1.2566370614359172, [0.76440104189216407, 1.8586223516062499], [-0.2838683571264714, 1.9194187087799912], (-0.204895333443, 0.692104654602, 0.692104796553)] |
| 951 | |
| 952 | |
| 953 | """ |
| 954 | |
| 955 | u1 = self.variables[1] |
| 956 | u2 = self.variables[2] |
| 957 | |
| 958 | solver = self._create_geodesic_ode_system() |
| 959 | |
| 960 | t_interval, n = tinterval[0:2], tinterval[2] |
| 961 | solver.y_0 = [p0[0], p0[1], v0[0], v0[1]] |
| 962 | solver.ode_solve(t_span=t_interval, num_points=n) |
| 963 | |
| 964 | parsed_solution = \ |
| 965 | [[vec[0], vec[1][0:2], vec[1][2:], self.eq_callable(vec[1][0], vec[1][1])] \ |
| 966 | for vec in solver.solution] |
| 967 | |
| 968 | return parsed_solution |
| 969 | |
| 970 | |
| 971 | @cached_method |
| 972 | def _create_pt_ode_system(self, curve, t): |
| 973 | from sage.ext.fast_eval import fast_float |
| 974 | from sage.calculus.var import var |
| 975 | from sage.gsl.ode import ode_solver |
| 976 | |
| 977 | u1 = self.variables[1] |
| 978 | u2 = self.variables[2] |
| 979 | v1, v2 = var('v1, v2') |
| 980 | |
| 981 | du1 = diff(curve[0], t) |
| 982 | du2 = diff(curve[1], t) |
| 983 | |
| 984 | C = self.connection_coefficients() |
| 985 | for coef in C: |
| 986 | C[coef] = C[coef].subs({u1: curve[0], u2: curve[1]}) |
| 987 | |
| 988 | dv1 = - C[(1,1,1)]*v1*du1 - C[(1,2,1)]*(du1*v2 + du2*v1) - C[(2,2,1)]*du2*v2 |
| 989 | dv2 = - C[(1,1,2)]*v1*du1 - C[(1,2,2)]*(du1*v2 + du2*v1) - C[(2,2,2)]*du2*v2 |
| 990 | fun1 = fast_float(dv1, str(t), str(v1), str(v2)) |
| 991 | fun2 = fast_float(dv2, str(t), str(v1), str(v2)) |
| 992 | |
| 993 | pt_ode = ode_solver() |
| 994 | pt_ode.function = lambda t, (v1, v2): [fun1(t, v1, v2), fun2(t, v1, v2)] |
| 995 | return pt_ode |
| 996 | |
| 997 | |
| 998 | # mikarm: We should rewrite it like you did for geodesics. |
| 999 | # jvkersch: OK, see this and the above |
| 1000 | def parallel_translation_numerical_new(self,curve,t,v0,tinterval): |
| 1001 | """ |
| 1002 | This method gives the numerical solution to the equation of parallel translation of a vector |
| 1003 | |
| 1004 | INPUT: |
| 1005 | curve equation = list of functions which determine the curve wrt the local coordinate |
| 1006 | |
| 1007 | t - curve parameter |
| 1008 | |
| 1009 | v0 - initial vector |
| 1010 | |
| 1011 | tinterval = [a,b,N], (a,b) is the domain of the curve, N is the number of subdivision points |
| 1012 | |
| 1013 | OUTPUT: |
| 1014 | The list consisting of lists [t, [v1(t),v2(t)]], where |
| 1015 | |
| 1016 | t is a subdivision point |
| 1017 | |
| 1018 | [v1(t),v2(t)] is the list of coordinates of the vector translated parallely along the curve |
| 1019 | |
| 1020 | EXAMPLES:: |
| 1021 | |
| 1022 | sage: var('p,q') |
| 1023 | sage: v = [p,q] |
| 1024 | sage: assume(cos(q)>0) |
| 1025 | sage: sphere = parametrized_surface3d([cos(q)*cos(p),sin(q)*cos(p),sin(p)],v,'sphere') |
| 1026 | sage: var('ss') |
| 1027 | sage: vv_array = sphere.parallel_translation_numerical([ss,ss],ss,[1,1],[0,pi/4,20]) |
| 1028 | sage: vv_array[0] |
| 1029 | [0.0, [1.0, 1.0]] |
| 1030 | sage: vv_array[5] |
| 1031 | [0.19634954084936207, [0.98060182522638917, 1.0389930474425113]] |
| 1032 | |
| 1033 | """ |
| 1034 | |
| 1035 | u1 = self.variables[1] |
| 1036 | u2 = self.variables[2] |
| 1037 | |
| 1038 | solver = self._create_pt_ode_system(tuple(curve), t) |
| 1039 | |
| 1040 | t_interval, n = tinterval[0:2], tinterval[2] |
| 1041 | solver.y_0 = v0 |
| 1042 | solver.ode_solve(t_span=t_interval, num_points=n) |
| 1043 | |
| 1044 | return solver.solution |
| 1045 | |
| 1046 | |
| 1047 | |
| 1048 | |
| 1049 | |
| 1050 | def parallel_translation_numerical(self,curve,t,v0,tinterval): |
| 1051 | """ |
| 1052 | This method gives the numerical solution to the equation of parallel translation of a vector |
| 1053 | |
| 1054 | INPUT: |
| 1055 | curve equation = list of functions which determine the curve wrt the local coordinate |
| 1056 | |
| 1057 | t - curve parameter |
| 1058 | |
| 1059 | v0 - initial vector |
| 1060 | |
| 1061 | tinterval = [a,b,N], (a,b) is the domain of the curve, N is the number of subdivision points |
| 1062 | |
| 1063 | OUTPUT: |
| 1064 | The list consisting of lists [t, [v1(t),v2(t)]], where |
| 1065 | |
| 1066 | t is a subdivision point |
| 1067 | |
| 1068 | [v1(t),v2(t)] is the list of coordinates of the vector translated parallely along the curve |
| 1069 | |
| 1070 | EXAMPLES:: |
| 1071 | |
| 1072 | sage: var('p,q') |
| 1073 | sage: v = [p,q] |
| 1074 | sage: assume(cos(q)>0) |
| 1075 | sage: sphere = parametrized_surface3d([cos(q)*cos(p),sin(q)*cos(p),sin(p)],v,'sphere') |
| 1076 | sage: var('ss') |
| 1077 | sage: vv_array = sphere.parallel_translation_numerical([ss,ss],ss,[1,1],[0,pi/4,20]) |
| 1078 | sage: vv_array[0] |
| 1079 | [0.0, [1.0, 1.0]] |
| 1080 | sage: vv_array[5] |
| 1081 | [0.19634954084936207, [0.98060182522638917, 1.0389930474425113]] |
| 1082 | |
| 1083 | """ |
| 1084 | u1 = self.variables[1] |
| 1085 | u2 = self.variables[2] |
| 1086 | #var('uu1,uu2') |
| 1087 | |
| 1088 | #f(u1,u2) = self.equation |
| 1089 | |
| 1090 | def f(ux, vx): |
| 1091 | return [self.equation[0].subs({u1: ux, u2: vx}), |
| 1092 | self.equation[1].subs({u1: ux, u2: vx}), |
| 1093 | self.equation[2].subs({u1: ux, u2: vx})] |
| 1094 | |
| 1095 | #from sage.calculus.var import var |
| 1096 | #t = var('t') |
| 1097 | tt = t |
| 1098 | |
| 1099 | C111 = self.connection_coefficients()[(1,1,1)] |
| 1100 | C121 = self.connection_coefficients()[(1,2,1)] |
| 1101 | C221 = self.connection_coefficients()[(2,2,1)] |
| 1102 | C112 = self.connection_coefficients()[(1,1,2)] |
| 1103 | C122 = self.connection_coefficients()[(1,2,2)] |
| 1104 | C222 = self.connection_coefficients()[(2,2,2)] |
| 1105 | |
| 1106 | du1=diff(curve[0],tt) |
| 1107 | du2=diff(curve[1],tt) |
| 1108 | uu1=curve[0] |
| 1109 | uu2=curve[1] |
| 1110 | |
| 1111 | |
| 1112 | def ode_system(unknown_functions,t1): |
| 1113 | du1p = du1.subs(tt==t1) |
| 1114 | du2p = du2.subs(tt==t1) |
| 1115 | uu1p = uu1.subs(tt==t1) |
| 1116 | uu2p = uu2.subs(tt==t1) |
| 1117 | |
| 1118 | c111 = C111.subs({u1: uu1p, u2: uu2p}) |
| 1119 | c121 = C121.subs({u1: uu1p, u2: uu2p}) |
| 1120 | c221 = C221.subs({u1: uu1p, u2: uu2p}) |
| 1121 | c112 = C112.subs({u1: uu1p, u2: uu2p}) |
| 1122 | c122 = C122.subs({u1: uu1p, u2: uu2p}) |
| 1123 | c222 = C222.subs({u1: uu1p, u2: uu2p}) |
| 1124 | |
| 1125 | #print c111, c121, c221, c112, c122, c222 |
| 1126 | |
| 1127 | v1,v2 = unknown_functions |
| 1128 | f1 = - c111*du1p*v1 - c121*(du1p*v2 + du2p*v1) - c221*du2p*v2 |
| 1129 | f2 = - c112*du1p*v1 - c122*(du1p*v2 + du2p*v1) - c222*du2p*v2 |
| 1130 | dv1 = f1.subs(tt==t1) |
| 1131 | dv2 = f2.subs(tt==t1) |
| 1132 | |
| 1133 | #print dv1, dv2 |
| 1134 | |
| 1135 | return float(dv1), float(dv2) |
| 1136 | |
| 1137 | step = float((tinterval[1]-tinterval[0])/tinterval[2]) |
| 1138 | ttt = float(tinterval[0]) |
| 1139 | tarray = [ ttt ] |
| 1140 | for counter in range(1,tinterval[2]+1): |
| 1141 | ttt = ttt + step |
| 1142 | tarray = tarray + [ ttt ] |
| 1143 | |
| 1144 | |
| 1145 | # import ode solving routine |
| 1146 | import scipy.integrate |
| 1147 | |
| 1148 | initial_data = (v0[0],v0[1]) |
| 1149 | solution = scipy.integrate.odeint(ode_system,initial_data,tarray) |
| 1150 | return [[ tarray[counter],[solution[counter,0], solution[counter,1]]] for counter in range(0,len(tarray))] |
| 1151 | |