| 1 | """ |
| 2 | Differential Geometry of Parametrized Surfaces. |
| 3 | |
| 4 | AUTHORS: |
| 5 | - Mikhail Malakhaltsev (2010-09-25): initial version |
| 6 | - Joris Vankerschaver (2010-10-25): implementation, doctests |
| 7 | |
| 8 | """ |
| 9 | #***************************************************************************** |
| 10 | # Copyright (C) 2010 Mikhail Malakhaltsev <mikarm@gmail.com> |
| 11 | # Copyright (C) 2010 Joris Vankerschaver <joris.vankerschaver@gmail.com> |
| 12 | # |
| 13 | # Distributed under the terms of the GNU General Public License (GPL) |
| 14 | # http://www.gnu.org/licenses/ |
| 15 | #***************************************************************************** |
| 16 | |
| 17 | |
| 18 | from sage.structure.sage_object import SageObject |
| 19 | from sage.modules.free_module_element import vector |
| 20 | from sage.matrix.constructor import matrix |
| 21 | from sage.calculus.functional import diff |
| 22 | from sage.functions.other import sqrt |
| 23 | from sage.misc.cachefunc import cached_method |
| 24 | from sage.symbolic.ring import SR |
| 25 | |
| 26 | |
| 27 | class ParametrizedSurface3D(SageObject): |
| 28 | r""" |
| 29 | |
| 30 | Class representing a parametrized two-dimensional surface in |
| 31 | Euclidian three-space. Provides methods for calculating the main |
| 32 | geometrical objects related to such a surface, such as the first |
| 33 | and the second fundamental form, the total (Gaussian) and the mean |
| 34 | curvature, the geodesic curves, parallel transport, etc. |
| 35 | |
| 36 | |
| 37 | INPUT: |
| 38 | |
| 39 | - ``surface_equation`` -- list specifying a parametric representation of the surface |
| 40 | |
| 41 | - ``variables`` -- list of coordinates on the surface |
| 42 | |
| 43 | - ``name_of_surface`` -- string with the name of the surface (optional). |
| 44 | |
| 45 | |
| 46 | .. note:: |
| 47 | |
| 48 | Throughout the documentation, we use the Einstein summation |
| 49 | convention: whenever an index appears twice, once as a |
| 50 | subscript, and once as a superscript, summation over that index |
| 51 | is implied. For instance, `g_{ij} g^{jk}` stands for `\sum_j |
| 52 | g_{ij}g^{jk}`. |
| 53 | |
| 54 | |
| 55 | EXAMPLES: |
| 56 | |
| 57 | We give several examples of standard surfaces in differential |
| 58 | geometry. First, let's specify a surface. |
| 59 | |
| 60 | The elliptic paraboloid:: |
| 61 | |
| 62 | sage: u, v = var('u,v') |
| 63 | sage: eparaboloid = ParametrizedSurface3D([u,v,u^2+v^2],[u,v],'elliptic paraboloid') |
| 64 | sage: eparaboloid |
| 65 | Parametrized surface ('elliptic paraboloid') with equation [u, v, u^2 + v^2] |
| 66 | |
| 67 | An ellipsoid with axes $a$, $b$, $c$:: |
| 68 | |
| 69 | sage: a, b,c = var('a,b,c'); |
| 70 | sage: u1, u2 = var ('u1,u2'); |
| 71 | sage: u = [u1,u2] |
| 72 | sage: ellipsoid_parametric_equation = vector([a*cos(u1)*cos(u2),b*sin(u1)*cos(u2),c*sin(u2)]) |
| 73 | sage: ellipsoid = ParametrizedSurface3D(ellipsoid_parametric_equation,u,'ellipsoid') |
| 74 | sage: print ellipsoid |
| 75 | Parametrized surface ('ellipsoid') with equation (a*cos(u1)*cos(u2), b*sin(u1)*cos(u2), c*sin(u2)) |
| 76 | |
| 77 | Latex representation of the surfaces:: |
| 78 | |
| 79 | sage: u, v = var('u, v') |
| 80 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 81 | sage: print latex(sphere) |
| 82 | \left[\cos\left(u\right) \cos\left(v\right), \sin\left(u\right) \cos\left(v\right), \sin\left(v\right)\right] |
| 83 | sage: print sphere._latex_() |
| 84 | \left[\cos\left(u\right) \cos\left(v\right), \sin\left(u\right) \cos\left(v\right), \sin\left(v\right)\right] |
| 85 | sage: print sphere |
| 86 | Parametrized surface ('sphere') with equation [cos(u)*cos(v), sin(u)*cos(v), sin(v)] |
| 87 | |
| 88 | We can plot the surface using the ``plot`` member function:: |
| 89 | |
| 90 | sage: ellipsoid_parametric_equation_abc = ellipsoid_parametric_equation.substitute(a=2,b=1.5,c=1) |
| 91 | sage: ellipsoid_abc = ParametrizedSurface3D(ellipsoid_parametric_equation_abc,[u1,u2],'ellipsoid_abc') |
| 92 | sage: ellipsoid_abc_plot = ellipsoid_abc.plot((u1,0,2*pi),(u2,-pi/2,pi/2)) |
| 93 | sage: ellipsoid_abc_plot.show(aspect_ratio=(1,1,1)) |
| 94 | |
| 95 | We find the natural frame of tangent vectors to the ellipsoid, |
| 96 | expressed in intrinsic coordinates. Note that the result is a |
| 97 | dictionary of vector fields:: |
| 98 | |
| 99 | sage: ellipsoid.natural_frame() |
| 100 | {1: (-a*sin(u1)*cos(u2), b*cos(u1)*cos(u2), 0), 2: (-a*sin(u2)*cos(u1), -b*sin(u1)*sin(u2), c*cos(u2))} |
| 101 | |
| 102 | We find the normal vector field to the surface. The normal vector |
| 103 | field is the vector product of the vectors of the natural frame, |
| 104 | and is given by:: |
| 105 | |
| 106 | sage: ellipsoid.normal_vector() |
| 107 | (b*c*cos(u1)*cos(u2)^2, a*c*sin(u1)*cos(u2)^2, a*b*sin(u2)*cos(u2)) |
| 108 | |
| 109 | By default, the normal vector field is not normalized. To obtain |
| 110 | the unit normal vector field of the elliptic paraboloid, we put:: |
| 111 | |
| 112 | sage: u, v = var('u,v') |
| 113 | sage: eparaboloid = ParametrizedSurface3D([u,v,u^2+v^2],[u,v],'elliptic paraboloid') |
| 114 | sage: eparaboloid.normal_vector(normalized=True) |
| 115 | (-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)) |
| 116 | |
| 117 | Now let us compute the coefficients of the first fundamental form of the torus:: |
| 118 | |
| 119 | sage: u, v = var('u,v') |
| 120 | sage: a, b = var('a,b') |
| 121 | sage: torus = ParametrizedSurface3D(((a + b*cos(u))*cos(v),(a + b*cos(u))*sin(v), b*sin(u)),[u,v],'torus') |
| 122 | sage: torus.first_fundamental_form_coefficients() |
| 123 | {(1, 2): 0, (1, 1): b^2, (2, 1): 0, (2, 2): b^2*cos(u)^2 + 2*a*b*cos(u) + a^2} |
| 124 | |
| 125 | The first fundamental form can be used to compute the length of a |
| 126 | curve on the surface. For example, let us find the length of the |
| 127 | curve $u^1 = t$, $u^2 = t$, $t \in [0,2\pi]$, on the ellipsoid |
| 128 | with axes $a=1$, $b=1.5$ and $c=1$. So we take the curve:: |
| 129 | |
| 130 | sage: t = var('t') |
| 131 | sage: u1 = t |
| 132 | sage: u2 = t |
| 133 | |
| 134 | Then find the tangent vector:: |
| 135 | |
| 136 | sage: du1 = diff(u1,t) |
| 137 | sage: du2 = diff(u2,t) |
| 138 | sage: du = vector([du1, du2]); du |
| 139 | (1, 1) |
| 140 | |
| 141 | Then calculate the symbolic expression for the length:: |
| 142 | |
| 143 | sage: L = sqrt(ellipsoid.first_fundamental_form(du, du).substitute(u1=u1,u2=u2)) |
| 144 | sage: integrate(L, t, 0, 2*pi) |
| 145 | integrate(sqrt(2*(a^2 - b^2)*sin(t)^2*cos(t)^2 + c^2*cos(t)^2 + (a^2*cos(t)^2 + b^2*sin(t)^2)*sin(t)^2 + (a^2*sin(t)^2 + b^2*cos(t)^2)*cos(t)^2), t, 0, 2*pi) |
| 146 | |
| 147 | Once we specify numerical values for the axes of the ellipsoid, we can |
| 148 | determine the numerical value of the length integral:: |
| 149 | |
| 150 | sage: print numerical_integral(L.substitute(a=2, b=1.5, c=1),0,1)[0] |
| 151 | 2.00127905972 |
| 152 | |
| 153 | We find the area of the sphere of radius $R$:: |
| 154 | |
| 155 | sage: R = var('R'); |
| 156 | sage: u, v = var('u,v'); |
| 157 | sage: assume(R>0) |
| 158 | sage: assume(cos(v)>0) |
| 159 | sage: sphere = ParametrizedSurface3D([R*cos(u)*cos(v),R*sin(u)*cos(v),R*sin(v)],[u,v],'sphere') |
| 160 | sage: integral(integral(sphere.area_form(),u,0,2*pi),v,-pi/2,pi/2) |
| 161 | 4*pi*R^2 |
| 162 | |
| 163 | We can find an orthonormal frame field $\{e_1, e_2\}$ of a surface |
| 164 | and calculate its structure functions. Let us first determine the |
| 165 | orthonormal frame field for the elliptic paraboloid:: |
| 166 | |
| 167 | sage: u, v = var('u,v') |
| 168 | sage: eparaboloid = ParametrizedSurface3D([u,v,u^2+v^2],[u,v],'elliptic paraboloid') |
| 169 | sage: eparaboloid.orthonormal_frame() |
| 170 | {1: (1/sqrt(4*u^2 + 1), 0, 2*u/sqrt(4*u^2 + 1)), 2: (-4*u*v/(sqrt(4*u^2 + 1)*sqrt(4*u^2 + 4*v^2 + 1)), sqrt(4*u^2 + 1)/sqrt(4*u^2 + 4*v^2 + 1), 2*v/(sqrt(4*u^2 + 1)*sqrt(4*u^2 + 4*v^2 + 1)))} |
| 171 | |
| 172 | We can express the orthogonal frame field both in exterior |
| 173 | coordinates (i.e. expressed as vector field fields in the ambient |
| 174 | space $\mathbb{R}^3$, the default) or in intrinsic coordinates |
| 175 | (with respect to the natural frame). Here we use intrinsic |
| 176 | coordinates:: |
| 177 | |
| 178 | sage: eparaboloid.orthonormal_frame(coordinates='int') |
| 179 | {1: (1/sqrt(4*u^2 + 1), 0), 2: (-4*u*v/(sqrt(4*u^2 + 1)*sqrt(4*u^2 + 4*v^2 + 1)), sqrt(4*u^2 + 1)/sqrt(4*u^2 + 4*v^2 + 1))} |
| 180 | |
| 181 | Using the orthonormal frame in interior coordinates, we can calculate the structure |
| 182 | functions $c^k_{ij}$ of the surface, defined by $[e_i,e_j] = c^k_{ij} e_k$, where |
| 183 | $[e_i, e_j]$ represents the Lie bracket of two frame vector fields $e_i, e_j$. For the |
| 184 | elliptic paraboloid, we get:: |
| 185 | |
| 186 | sage: EE = eparaboloid.orthonormal_frame(coordinates='int') |
| 187 | sage: E1 = EE[1]; E2 = EE[2] |
| 188 | sage: CC = eparaboloid.frame_structure_functions(E1,E2) |
| 189 | sage: CC[1,2,1].simplify_full() |
| 190 | 4*sqrt(4*u^2 + 4*v^2 + 1)*v/(sqrt(4*u^2 + 1)*(16*u^4 + 4*(4*u^2 + 1)*v^2 + 8*u^2 + 1)) |
| 191 | |
| 192 | We compute the Gaussian and mean curvatures of the sphere:: |
| 193 | |
| 194 | sage: u, v = var('u,v') |
| 195 | sage: sphere = ParametrizedSurface3D((u,v,sqrt(1-u^2-v^2)),[u,v],'sphere') |
| 196 | sage: sphere.gauss_curvature() |
| 197 | 1 |
| 198 | sage: sphere.mean_curvature() |
| 199 | 1 |
| 200 | |
| 201 | We can easily generate a color plot of the Gaussian curvature of a surface. Here we |
| 202 | deal with the ellipsoid:: |
| 203 | |
| 204 | sage: u1, u2 = var('u1,u2'); |
| 205 | sage: u = [u1,u2] |
| 206 | sage: ellipsoid_equation(u1,u2) = [2*cos(u1)*cos(u2),1.5*cos(u1)*sin(u2),sin(u1)] |
| 207 | sage: ellipsoid = ParametrizedSurface3D(ellipsoid_equation(u1,u2), [u1, u2],'ellipsoid') |
| 208 | sage: # set intervals for variables and the number of division points |
| 209 | sage: u1min, u1max = -1.5, 1.5 |
| 210 | sage: u2min, u2max = 0, 6.28 |
| 211 | sage: u1num, u2num = 10, 20 |
| 212 | sage: # make the arguments array |
| 213 | sage: from numpy import linspace |
| 214 | sage: u1_array = linspace(u1min, u1max, u1num) |
| 215 | sage: u2_array = linspace(u2min, u2max, u2num) |
| 216 | sage: u_array = [ (uu1,uu2) for uu1 in u1_array for uu2 in u2_array] |
| 217 | sage: # Find the gaussian curvature |
| 218 | sage: K(u1,u2) = ellipsoid.gauss_curvature() |
| 219 | sage: # Make array of K values |
| 220 | sage: K_array = [K(uu[0],uu[1]) for uu in u_array] |
| 221 | sage: # Find minimum and max of the gauss curvature |
| 222 | sage: K_max = max(K_array) |
| 223 | sage: K_min = min(K_array) |
| 224 | sage: # Make the array of color coefficients |
| 225 | sage: cc_array = [ (ccc - K_min)/(K_max - K_min) for ccc in K_array ] |
| 226 | sage: points_array = [ellipsoid_equation(u_array[counter][0],u_array[counter][1]) for counter in range(0,len(u_array)) ] |
| 227 | sage: curvature_ellipsoid_plot = sum( point([xx for xx in points_array[counter]],color=hue(cc_array[counter]/2)) for counter in range(0,len(u_array)) ) |
| 228 | sage: curvature_ellipsoid_plot.show(aspect_ratio=1) |
| 229 | |
| 230 | We can find the principal curvatures and principal directions of the elliptic paraboloid:: |
| 231 | |
| 232 | sage: u, v = var('u, v') |
| 233 | sage: eparaboloid = ParametrizedSurface3D([u, v, u^2+v^2], [u, v], 'elliptic paraboloid') |
| 234 | sage: pd = eparaboloid.principal_directions(); pd |
| 235 | [(2*sqrt(4*u^2 + 4*v^2 + 1)/(16*u^4 + 8*u^2 + 16*v^4 + 8*(4*u^2 + 1)*v^2 + 1), [(1, v/u)], 1), (2/sqrt(4*u^2 + 4*v^2 + 1), [(1, -u/v)], 1)] |
| 236 | |
| 237 | We extract the principal curvatures:: |
| 238 | |
| 239 | sage: k1 = pd[0][0].simplify_full() |
| 240 | sage: k1 |
| 241 | 2*sqrt(4*u^2 + 4*v^2 + 1)/(16*u^4 + 8*u^2 + 16*v^4 + 8*(4*u^2 + 1)*v^2 + 1) |
| 242 | sage: k2 = pd[1][0].simplify_full() |
| 243 | sage: k2 |
| 244 | 2/sqrt(4*u^2 + 4*v^2 + 1) |
| 245 | |
| 246 | and check them by comparison with the Gaussian and mean curvature |
| 247 | expressed in terms of the principal curvatures:: |
| 248 | |
| 249 | sage: K = eparaboloid.gauss_curvature().simplify_full() |
| 250 | sage: K |
| 251 | 4/(16*u^4 + 8*u^2 + 16*v^4 + 8*(4*u^2 + 1)*v^2 + 1) |
| 252 | sage: H = eparaboloid.mean_curvature().simplify_full() |
| 253 | sage: H |
| 254 | 2*(2*u^2 + 2*v^2 + 1)/(4*u^2 + 4*v^2 + 1)^(3/2) |
| 255 | sage: (K - k1*k2).simplify_full() |
| 256 | 0 |
| 257 | sage: (2*H - k1 - k2).simplify_full() |
| 258 | 0 |
| 259 | |
| 260 | We can find the intrinsic (local coordinates) of the principal directions:: |
| 261 | |
| 262 | sage: pd[0][1] |
| 263 | [(1, v/u)] |
| 264 | sage: pd[1][1] |
| 265 | [(1, -u/v)] |
| 266 | |
| 267 | The ParametrizedSurface3D class also contains functionality to |
| 268 | compute the coefficients of the second fundamental form, the shape |
| 269 | operator, the rotation on the surface at a given angle, the |
| 270 | connection coefficients. One can also calculate numerically the |
| 271 | geodesics and the parallel translation along a curve (see the |
| 272 | documentation below).""" |
| 273 | |
| 274 | def __init__(self,equation,variables,*name): |
| 275 | """ |
| 276 | See ``ParametrizedSurface3D`` for full documentation. |
| 277 | |
| 278 | .. note:: |
| 279 | |
| 280 | The orientation of the surface is determined by the |
| 281 | parametrization, that is, the natural frame with positive |
| 282 | orientation is given by `\partial_1 \vec r`, `\partial_2 \vec |
| 283 | r`. |
| 284 | |
| 285 | |
| 286 | EXAMPLES:: |
| 287 | |
| 288 | sage: u, v = var('u,v') |
| 289 | sage: eq = [3*u + 3*u*v^2 - u^3, 3*v + 3*u^2*v - v^3, 3*(u^2-v^2)] |
| 290 | sage: enneper = ParametrizedSurface3D(eq,[u,v],'Enneper Surface'); enneper |
| 291 | Parametrized surface ('Enneper Surface') with equation [-u^3 + 3*u*v^2 + 3*u, 3*u^2*v - v^3 + 3*v, 3*u^2 - 3*v^2] |
| 292 | |
| 293 | """ |
| 294 | self.equation = equation |
| 295 | self.variables_list = variables |
| 296 | self.variables = {1:self.variables_list[0],2:self.variables_list[1]} |
| 297 | self.name = name |
| 298 | |
| 299 | # Callable version of the underlying equation |
| 300 | def eq_callable(u, v): |
| 301 | u1, u2 = self.variables_list |
| 302 | return [fun.subs({u1: u, u2: v}) for fun in self.equation] |
| 303 | |
| 304 | self.eq_callable = eq_callable |
| 305 | |
| 306 | # Precompute various index tuples |
| 307 | self.index_list = [(i,j) for i in (1,2) for j in (1,2)] |
| 308 | self.index_list_3 = [(i,j,k) for i in (1,2) for j in (1,2) for k in (1,2)] |
| 309 | |
| 310 | |
| 311 | def _latex_(self): |
| 312 | r""" |
| 313 | Returns the LaTeX representation of this parametrized surface. |
| 314 | |
| 315 | EXAMPLES:: |
| 316 | |
| 317 | sage: u, v = var('u, v') |
| 318 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 319 | sage: latex(sphere) |
| 320 | \left[\cos\left(u\right) \cos\left(v\right), \sin\left(u\right) \cos\left(v\right), \sin\left(v\right)\right] |
| 321 | sage: sphere._latex_() |
| 322 | \left[\cos\left(u\right) \cos\left(v\right), \sin\left(u\right) \cos\left(v\right), \sin\left(v\right)\right] |
| 323 | |
| 324 | """ |
| 325 | from sage.misc.latex import latex |
| 326 | return latex(self.equation) |
| 327 | |
| 328 | |
| 329 | def _repr_(self): |
| 330 | r""" |
| 331 | Returns the string representation of this parametrized surface. |
| 332 | |
| 333 | EXAMPLES:: |
| 334 | |
| 335 | sage: u, v = var('u, v') |
| 336 | sage: eq = [3*u + 3*u*v^2 - u^3, 3*v + 3*u^2*v - v^3, 3*(u^2-v^2)] |
| 337 | sage: enneper = ParametrizedSurface3D(eq,[u,v],'enneper_surface') |
| 338 | sage: print enneper |
| 339 | Parametrized surface ('enneper_surface') with equation [-u^3 + 3*u*v^2 + 3*u, 3*u^2*v - v^3 + 3*v, 3*u^2 - 3*v^2] |
| 340 | sage: enneper._repr_() |
| 341 | "Parametrized surface ('enneper_surface') with equation [-u^3 + 3*u*v^2 + 3*u, 3*u^2*v - v^3 + 3*v, 3*u^2 - 3*v^2]" |
| 342 | |
| 343 | """ |
| 344 | name = 'Parametrized surface' |
| 345 | if self.name is not None: |
| 346 | name += " ('%s')" % self.name |
| 347 | return '%(designation)s with equation %(eq)s' % \ |
| 348 | {'designation': name, 'eq': str(self.equation)} |
| 349 | |
| 350 | |
| 351 | def plot(self, urange, vrange, **kwds): |
| 352 | """ |
| 353 | Enable easy plotting directly from the surface class. |
| 354 | |
| 355 | INPUT: |
| 356 | |
| 357 | - ``urange`` - 3-tuple ``(u, u_min, u_max)`` |
| 358 | - ``vrange`` - 3-tuple ``(v, v_min, v_max)`` |
| 359 | |
| 360 | EXAMPLES:: |
| 361 | |
| 362 | sage: u, v = var('u, v') |
| 363 | sage: eq = [3*u + 3*u*v^2 - u^3, 3*v + 3*u^2*v - v^3, 3*(u^2-v^2)] |
| 364 | sage: enneper = ParametrizedSurface3D(eq,[u,v],'Enneper Surface') |
| 365 | sage: enneper.plot((u, -5, 5), (v, -5, 5)) |
| 366 | """ |
| 367 | from sage.plot.plot3d.parametric_plot3d import parametric_plot3d |
| 368 | P = parametric_plot3d(self.equation, urange, vrange, **kwds) |
| 369 | return P |
| 370 | |
| 371 | |
| 372 | @cached_method |
| 373 | def natural_frame(self): |
| 374 | """ |
| 375 | Returns the natural tangent frame on the parametrized surface. |
| 376 | The vectors of this frame are tangent to the coordinate lines |
| 377 | on the surface. |
| 378 | |
| 379 | OUTPUT: |
| 380 | |
| 381 | - The natural frame as a dictionary. |
| 382 | |
| 383 | EXAMPLES:: |
| 384 | |
| 385 | sage: u, v = var('u, v') |
| 386 | sage: eparaboloid = ParametrizedSurface3D([u,v,u^2+v^2],[u,v],'elliptic paraboloid') |
| 387 | sage: eparaboloid.natural_frame() |
| 388 | {1: (1, 0, 2*u), 2: (0, 1, 2*v)} |
| 389 | """ |
| 390 | |
| 391 | dr1 = vector([diff(f,self.variables[1]).simplify_full() for f in self.equation]) |
| 392 | dr2 = vector([diff(f,self.variables[2]).simplify_full() for f in self.equation]) |
| 393 | |
| 394 | return {1:dr1, 2:dr2} |
| 395 | |
| 396 | |
| 397 | @cached_method |
| 398 | def normal_vector(self, normalized=False): |
| 399 | """ |
| 400 | Returns the normal vector field of the parametrized surface. |
| 401 | |
| 402 | INPUT: |
| 403 | |
| 404 | - ``normalized`` - default ``False`` - specifies whether the normal vector should be normalized. |
| 405 | |
| 406 | OUTPUT: |
| 407 | |
| 408 | - Normal vector field. |
| 409 | |
| 410 | EXAMPLES:: |
| 411 | |
| 412 | sage: u, v = var('u, v') |
| 413 | sage: eparaboloid = ParametrizedSurface3D([u,v,u^2+v^2],[u,v],'elliptic paraboloid') |
| 414 | sage: eparaboloid.normal_vector(normalized=False) |
| 415 | (-2*u, -2*v, 1) |
| 416 | sage: eparaboloid.normal_vector(normalized=True) |
| 417 | (-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)) |
| 418 | |
| 419 | """ |
| 420 | |
| 421 | dr = self.natural_frame() |
| 422 | normal = dr[1].cross_product(dr[2]) |
| 423 | |
| 424 | if normalized: |
| 425 | normal /= normal.norm() |
| 426 | return normal.simplify_full() |
| 427 | |
| 428 | |
| 429 | @cached_method |
| 430 | def _compute_first_fundamental_form_coefficient(self, index): |
| 431 | """ |
| 432 | Helper function to compute coefficients of the first fundamental form. |
| 433 | |
| 434 | Do not call this method directly; instead use |
| 435 | ``first_fundamental_form_coefficient``. |
| 436 | This method expects its argument to be a list, and hence can be cached. |
| 437 | |
| 438 | EXAMPLES:: |
| 439 | |
| 440 | sage: u, v = var('u, v') |
| 441 | sage: eparaboloid = ParametrizedSurface3D([u,v,u^2+v^2],[u,v]) |
| 442 | sage: eparaboloid._compute_first_fundamental_form_coefficient((1,2)) |
| 443 | 4*u*v |
| 444 | |
| 445 | """ |
| 446 | dr = self.natural_frame() |
| 447 | return (dr[index[0]]*dr[index[1]]).simplify_full() |
| 448 | |
| 449 | |
| 450 | def first_fundamental_form_coefficient(self, index): |
| 451 | r""" |
| 452 | Compute a single component $g_{ij}$ of the first fundamental form. If |
| 453 | the parametric representation of the surface is given by the vector |
| 454 | function $\vec r(u^i)$, where $u^i$, $i = 1, 2$ are curvilinear |
| 455 | coordinates, then $g_{ij} = \frac{\partial \vec r}{\partial u^i} \cdot \frac{\partial \vec r}{\partial u^j}$. |
| 456 | |
| 457 | INPUT: |
| 458 | |
| 459 | - ``index`` - length-2 index ``(i, j)`` of the component |
| 460 | |
| 461 | OUTPUT: |
| 462 | |
| 463 | - Component ``g_ij`` of the first fundamental form |
| 464 | |
| 465 | EXAMPLES:: |
| 466 | |
| 467 | sage: u, v = var('u, v') |
| 468 | sage: eparaboloid = ParametrizedSurface3D([u,v,u^2+v^2],[u,v]) |
| 469 | sage: eparaboloid.first_fundamental_form_coefficient((1,2)) |
| 470 | 4*u*v |
| 471 | |
| 472 | When the index is invalid, an error is raised:: |
| 473 | |
| 474 | sage: u, v = var('u, v') |
| 475 | sage: eparaboloid = ParametrizedSurface3D([u,v,u^2+v^2],[u,v]) |
| 476 | sage: eparaboloid.first_fundamental_form_coefficient((1,5)) |
| 477 | Traceback (most recent call last): |
| 478 | ... |
| 479 | ValueError: Index (1, 5) out of bounds. |
| 480 | |
| 481 | """ |
| 482 | index = tuple(sorted(index)) |
| 483 | if index not in self.index_list: |
| 484 | raise ValueError, "Index %s out of bounds." % str(index) |
| 485 | return self._compute_first_fundamental_form_coefficient(index) |
| 486 | |
| 487 | |
| 488 | @cached_method |
| 489 | def first_fundamental_form_coefficients(self): |
| 490 | """ |
| 491 | Returns the coefficients of the first fundamental form as a dictionary. |
| 492 | The keys are tuples $(i, j)$, where $i$ and $j$ range over $1, 2$, |
| 493 | while the values are the corresponding coefficients $g_{ij}$. |
| 494 | |
| 495 | OUTPUT: |
| 496 | |
| 497 | - Dictionary of first fundamental form coefficients. |
| 498 | |
| 499 | EXAMPLES:: |
| 500 | |
| 501 | sage: u, v = var('u,v') |
| 502 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 503 | sage: sphere.first_fundamental_form_coefficients() |
| 504 | {(1, 2): 0, (1, 1): cos(v)^2, (2, 1): 0, (2, 2): 1} |
| 505 | |
| 506 | """ |
| 507 | coefficients = {} |
| 508 | for index in self.index_list: |
| 509 | sorted_index = list(sorted(index)) |
| 510 | coefficients[index] = \ |
| 511 | self._compute_first_fundamental_form_coefficient(index) |
| 512 | return coefficients |
| 513 | |
| 514 | |
| 515 | def first_fundamental_form(self, vector1, vector2): |
| 516 | r""" |
| 517 | Evaluate the first fundamental form on two vectors expressed with |
| 518 | respect to the natural coordinate frame on the surface. In other words, |
| 519 | if the vectors are $v = (v^1, v^2)$ and $w = (w^1, w^2)$, calculate |
| 520 | $g_{11} v^1 w^1 + g_{12}(v^1 w^2 + v^2 w^1) + g_{22} v^2 w^2$, with |
| 521 | $g_{ij}$ the coefficients of the first fundamental form. |
| 522 | |
| 523 | INPUT: |
| 524 | |
| 525 | - ``vector1``, ``vector2`` - vectors on the surface. |
| 526 | |
| 527 | OUTPUT: |
| 528 | |
| 529 | - First fundamental form evaluated on the input vectors. |
| 530 | |
| 531 | EXAMPLES:: |
| 532 | |
| 533 | sage: u, v = var('u, v') |
| 534 | sage: v1, v2, w1, w2 = var('v1, v2, w1, w2') |
| 535 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 536 | sage: sphere.first_fundamental_form(vector([v1,v2]),vector([w1,w2])) |
| 537 | v1*w1*cos(v)^2 + v2*w2 |
| 538 | |
| 539 | sage: vv = vector([1,2]) |
| 540 | sage: sphere.first_fundamental_form(vv,vv) |
| 541 | cos(v)^2 + 4 |
| 542 | |
| 543 | sage: sphere.first_fundamental_form([1,1],[2,1]) |
| 544 | 2*cos(v)^2 + 1 |
| 545 | """ |
| 546 | gamma = self.first_fundamental_form_coefficients() |
| 547 | return sum(gamma[ind]*vector1[ind[0]-1]*vector2[ind[1]-1] |
| 548 | for ind in self.index_list) |
| 549 | |
| 550 | |
| 551 | @cached_method |
| 552 | def area_form_squared(self): |
| 553 | """ |
| 554 | Returns the square of the coefficient of the area form on the surface. |
| 555 | In terms of the coefficients $g_{ij}$ (where $i, j = 1, 2$) of the |
| 556 | first fundamental form, this invariant is given by |
| 557 | $A^2 = g_{11}g_{22} - g_{12}^2$. |
| 558 | |
| 559 | See also :meth:`.area_form`. |
| 560 | |
| 561 | OUTPUT: |
| 562 | |
| 563 | - Square of the area form |
| 564 | |
| 565 | EXAMPLES:: |
| 566 | |
| 567 | sage: u, v = var('u, v') |
| 568 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 569 | sage: sphere.area_form_squared() |
| 570 | cos(v)^2 |
| 571 | |
| 572 | """ |
| 573 | gamma = self.first_fundamental_form_coefficients() |
| 574 | return (gamma[(1,1)]*gamma[(2,2)]-gamma[(1,2)]**2).simplify_full() |
| 575 | |
| 576 | |
| 577 | @cached_method |
| 578 | def area_form(self): |
| 579 | r""" |
| 580 | Returns the coefficient of the area form on the surface. In terms of |
| 581 | the coefficients $g_{ij}$ (where $i, j = 1, 2$) of the first |
| 582 | fundamental form, the coefficient of the area form is given by |
| 583 | $A = \sqrt{g_{11}g_{22} - g_{12}^2}$. |
| 584 | |
| 585 | See also ``area_form_squared``. |
| 586 | |
| 587 | OUTPUT: |
| 588 | |
| 589 | - Coefficient of the area form |
| 590 | |
| 591 | EXAMPLES:: |
| 592 | |
| 593 | sage: u, v = var('u,v') |
| 594 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 595 | sage: sphere.area_form() |
| 596 | cos(v) |
| 597 | |
| 598 | """ |
| 599 | return sqrt(self.area_form_squared()).simplify_full() |
| 600 | |
| 601 | |
| 602 | @cached_method |
| 603 | def first_fundamental_form_inverse_coefficients(self): |
| 604 | r""" |
| 605 | Returns the coefficients $g^{ij}$ of the inverse of the fundamental |
| 606 | form, as a dictionary. The inverse coefficients are defined by |
| 607 | $g^{ij} g_{jk} = \delta^i_k$ with $\delta^i_k$ the Kronecker |
| 608 | delta. |
| 609 | |
| 610 | OUTPUT: |
| 611 | |
| 612 | - Dictionary of the inverse coefficients. |
| 613 | |
| 614 | EXAMPLES:: |
| 615 | |
| 616 | sage: u, v = var('u, v') |
| 617 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 618 | sage: sphere.first_fundamental_form_inverse_coefficients() |
| 619 | {(1, 2): 0, (1, 1): cos(v)^(-2), (2, 1): 0, (2, 2): 1} |
| 620 | |
| 621 | """ |
| 622 | |
| 623 | g = self.first_fundamental_form_coefficients() |
| 624 | D = g[(1,1)]*g[(2,2)]-g[(1,2)]**2 |
| 625 | |
| 626 | gi11 = (g[(2,2)]/D).simplify_full() |
| 627 | gi12 = (-g[(1,2)]/D).simplify_full() |
| 628 | gi21 = gi12 |
| 629 | gi22 = (g[(1,1)]/D).simplify_full() |
| 630 | |
| 631 | return {(1,1): gi11, (1,2): gi12, (2,1): gi21, (2,2): gi22} |
| 632 | |
| 633 | |
| 634 | def first_fundamental_form_inverse_coefficient(self, index): |
| 635 | r""" |
| 636 | Returns a specific component $g^{ij}$ of the inverse of the fundamental |
| 637 | form. |
| 638 | |
| 639 | INPUT: |
| 640 | |
| 641 | - ``index`` - length-2 index ``(i, j)`` of the component ``g^{ij}``. |
| 642 | |
| 643 | OUTPUT: |
| 644 | |
| 645 | - Component of the inverse of the fundamental form. |
| 646 | |
| 647 | EXAMPLES:: |
| 648 | |
| 649 | sage: u, v = var('u, v') |
| 650 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 651 | sage: sphere.first_fundamental_form_inverse_coefficient((1, 2)) |
| 652 | 0 |
| 653 | sage: sphere.first_fundamental_form_inverse_coefficient((1, 1)) |
| 654 | cos(v)^(-2) |
| 655 | |
| 656 | """ |
| 657 | |
| 658 | index = tuple(sorted(index)) |
| 659 | if index not in self.index_list: |
| 660 | raise ValueError, "Index %s out of bounds." % str(index) |
| 661 | return self.first_fundamental_form_inverse_coefficients()[index] |
| 662 | |
| 663 | |
| 664 | @cached_method |
| 665 | def rotation(self,theta): |
| 666 | r""" |
| 667 | Gives the matrix of the rotation operator over a given angle $\theta$ |
| 668 | with respect to the natural frame. |
| 669 | |
| 670 | INPUT: |
| 671 | |
| 672 | - ``theta`` - rotation angle |
| 673 | |
| 674 | OUTPUT: |
| 675 | |
| 676 | - Rotation matrix with respect to the natural frame. |
| 677 | |
| 678 | ALGORITHM: |
| 679 | |
| 680 | The operator of rotation over $\pi/2$ is $J^i_j = g^{ik}\omega_{jk}$, |
| 681 | where $\omega$ is the area form. The operator of rotation over an |
| 682 | angle $\theta$ is $\cos(\theta) I + sin(\theta) J$. |
| 683 | |
| 684 | EXAMPLES:: |
| 685 | |
| 686 | sage: u, v = var('u, v') |
| 687 | sage: assume(cos(v)>0) |
| 688 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 689 | |
| 690 | We first compute the matrix of rotation over pi/3:: |
| 691 | |
| 692 | sage: rotation = sphere.rotation(pi/3); rotation |
| 693 | [ 1/2 -1/2*sqrt(3)/cos(v)] |
| 694 | [ 1/2*sqrt(3)*cos(v) 1/2] |
| 695 | |
| 696 | We verify that three succesive rotations over pi/3 yield minus the identity:: |
| 697 | |
| 698 | sage: rotation^3 |
| 699 | [-1 0] |
| 700 | [ 0 -1] |
| 701 | |
| 702 | """ |
| 703 | |
| 704 | from sage.functions.trig import sin, cos |
| 705 | |
| 706 | gi = self.first_fundamental_form_inverse_coefficients() |
| 707 | w12 = self.area_form() |
| 708 | R11 = (cos(theta) + sin(theta)*gi[1,2]*w12).simplify_full() |
| 709 | R12 = (- sin(theta)*gi[1,1]*w12).simplify_full() |
| 710 | R21 = (sin(theta)*gi[2,2]*w12).simplify_full() |
| 711 | R22 = (cos(theta) - sin(theta)*gi[2,1]*w12).simplify_full() |
| 712 | return matrix([[R11,R12],[R21,R22]]) |
| 713 | |
| 714 | |
| 715 | @cached_method |
| 716 | def orthonormal_frame(self, coordinates='ext'): |
| 717 | r""" |
| 718 | Returns the orthonormal frame field on the surface, expressed either |
| 719 | in exterior coordinates (i.e. expressed as vector fields in the |
| 720 | ambient space $\mathbb{R}^3$, the default) or interior coordinates |
| 721 | (with respect to the natural frame) |
| 722 | |
| 723 | INPUT: |
| 724 | |
| 725 | - ``coordinates`` - either ``ext`` (default) or ``int``. |
| 726 | |
| 727 | OUTPUT: |
| 728 | |
| 729 | - Orthogonal frame field as a dictionary. |
| 730 | |
| 731 | ALGORITHM: |
| 732 | |
| 733 | We normalize the first vector $\vec e_1$ of the natural frame and then |
| 734 | get the second frame vector as $\vec e_2 = [\vec n, \vec e_1]$, where |
| 735 | $\vec n$ is the unit normal to the surface. |
| 736 | |
| 737 | EXAMPLES:: |
| 738 | |
| 739 | sage: u, v = var('u,v') |
| 740 | sage: assume(cos(v)>0) |
| 741 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v), sin(u)*cos(v), sin(v)], [u, v],'sphere') |
| 742 | sage: frame = sphere.orthonormal_frame(); frame |
| 743 | {1: (-sin(u), cos(u), 0), 2: (-sin(v)*cos(u), -sin(u)*sin(v), cos(v))} |
| 744 | sage: (frame[1]*frame[1]).simplify_full() |
| 745 | 1 |
| 746 | sage: (frame[1]*frame[2]).simplify_full() |
| 747 | 0 |
| 748 | sage: frame[1] == sphere.orthonormal_frame_vector(1) |
| 749 | True |
| 750 | |
| 751 | We compute the orthonormal frame with respect to the natural frame on |
| 752 | the surface:: |
| 753 | |
| 754 | sage: frame_int = sphere.orthonormal_frame(coordinates='int'); frame_int |
| 755 | {1: (1/cos(v), 0), 2: (0, 1)} |
| 756 | sage: sphere.first_fundamental_form(frame_int[1], frame_int[1]) |
| 757 | 1 |
| 758 | sage: sphere.first_fundamental_form(frame_int[1], frame_int[2]) |
| 759 | 0 |
| 760 | sage: sphere.first_fundamental_form(frame_int[2], frame_int[2]) |
| 761 | 1 |
| 762 | |
| 763 | """ |
| 764 | |
| 765 | |
| 766 | from sage.symbolic.constants import pi |
| 767 | |
| 768 | if coordinates not in ['ext', 'int']: |
| 769 | raise ValueError, \ |
| 770 | r"Coordinate system must be exterior ('ext') or interior ('int')." |
| 771 | |
| 772 | c = self.first_fundamental_form_coefficient([1,1]) |
| 773 | if coordinates == 'ext': |
| 774 | f1 = self.natural_frame()[1] |
| 775 | |
| 776 | E1 = (f1/sqrt(c)).simplify_full() |
| 777 | E2 = self.normal_vector(normalized=True).cross_product(E1).simplify_full() |
| 778 | else: |
| 779 | E1 = vector([(1/sqrt(c)).simplify_full(), 0]) |
| 780 | E2 = (self.rotation(pi/2)*E1).simplify_full() |
| 781 | return {1:E1, 2:E2} |
| 782 | |
| 783 | |
| 784 | @cached_method |
| 785 | def orthonormal_frame_vector(self, index, coordinates='ext'): |
| 786 | r""" |
| 787 | Returns a specific basis vector field of the orthonormal frame field on |
| 788 | the surface, expressed in exterior or interior coordinates. See |
| 789 | ``orthogonal_frame`` for more details. |
| 790 | |
| 791 | INPUT: |
| 792 | |
| 793 | - ``index`` - index ``(i, j)`` of the basis vector; |
| 794 | - ``coordinates`` - either ``ext`` (default) or ``int``. |
| 795 | |
| 796 | OUTPUT: |
| 797 | |
| 798 | - Orthonormal frame vector field. |
| 799 | |
| 800 | EXAMPLES:: |
| 801 | |
| 802 | sage: u, v = var('u, v') |
| 803 | sage: assume(cos(v)>0) |
| 804 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 805 | sage: V1 = sphere.orthonormal_frame_vector(1); V1 |
| 806 | (-sin(u), cos(u), 0) |
| 807 | sage: V2 = sphere.orthonormal_frame_vector(2); V2 |
| 808 | (-sin(v)*cos(u), -sin(u)*sin(v), cos(v)) |
| 809 | sage: (V1*V1).simplify_full() |
| 810 | 1 |
| 811 | sage: (V1*V2).simplify_full() |
| 812 | 0 |
| 813 | |
| 814 | sage: n = sphere.normal_vector(normalized=True) |
| 815 | sage: (V1.cross_product(V2) - n).simplify_full() |
| 816 | (0, 0, 0) |
| 817 | """ |
| 818 | |
| 819 | return self.orthonormal_frame(coordinates)[index] |
| 820 | |
| 821 | |
| 822 | def lie_bracket(self, v, w): |
| 823 | """ |
| 824 | Returns the Lie bracket of two vector fields that are tangent |
| 825 | to the surface. The vector fields should be given in intrinsic |
| 826 | coordinates, i.e. with respect to the natural frame. |
| 827 | |
| 828 | INPUT: |
| 829 | |
| 830 | - ``v`` and ``w`` - vector fields on the surface, expressed |
| 831 | as pairs of functions or as vectors of length 2. |
| 832 | |
| 833 | OUTPUT: |
| 834 | |
| 835 | - The Lie bracket ``[v, w]`` |
| 836 | |
| 837 | |
| 838 | EXAMPLES:: |
| 839 | |
| 840 | sage: u, v = var('u, v') |
| 841 | sage: assume(cos(v)>0) |
| 842 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 843 | sage: sphere.lie_bracket([u,v],[-v,u]) |
| 844 | (0, 0) |
| 845 | |
| 846 | sage: EE_int = sphere.orthonormal_frame(coordinates='int') |
| 847 | sage: sphere.lie_bracket(EE_int[1],EE_int[2]) |
| 848 | (sin(v)/cos(v)^2, 0) |
| 849 | """ |
| 850 | v = vector(SR, v) |
| 851 | w = vector(SR, w) |
| 852 | |
| 853 | variables = self.variables_list |
| 854 | Dv = matrix([[diff(component, u).simplify_full() for u in variables] |
| 855 | for component in v]) |
| 856 | Dw = matrix([[diff(component, u).simplify_full() for u in variables] |
| 857 | for component in w]) |
| 858 | return vector(Dv*w - Dw*v).simplify_full() |
| 859 | |
| 860 | |
| 861 | def frame_structure_functions(self, e1, e2): |
| 862 | r""" |
| 863 | Returns the structure functions $c^k_{ij}$ for a frame field |
| 864 | $e_1, e_2$, i.e. a pair of vector fields on the surface which are |
| 865 | linearly independent at each point. The structure functions are |
| 866 | defined using the Lie bracket by $[e_i,e_j] = c^k_{ij}e_k$. |
| 867 | |
| 868 | INPUT: |
| 869 | |
| 870 | - ``e1``, ``e2`` - vector fields in intrinsic coordinates on |
| 871 | the surface, expressed as pairs of functions, or as vectors of |
| 872 | length 2. |
| 873 | |
| 874 | OUTPUT: |
| 875 | |
| 876 | - Dictionary of structure functions, where the key ``(i, j, k)`` refers to |
| 877 | the structure function ``c_{i,j}^k``. |
| 878 | |
| 879 | |
| 880 | EXAMPLES:: |
| 881 | |
| 882 | sage: u, v = var('u, v') |
| 883 | sage: assume(cos(v) > 0) |
| 884 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v), sin(u)*cos(v), sin(v)], [u, v], 'sphere') |
| 885 | sage: sphere.frame_structure_functions([u, v], [-v, u]) |
| 886 | {(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} |
| 887 | |
| 888 | We construct the structure functions of the orthonormal frame on the |
| 889 | surface:: |
| 890 | |
| 891 | sage: EE_int = sphere.orthonormal_frame(coordinates='int') |
| 892 | sage: CC = sphere.frame_structure_functions(EE_int[1],EE_int[2]); CC |
| 893 | {(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} |
| 894 | sage: sphere.lie_bracket(EE_int[1],EE_int[2]) - CC[(1,2,1)]*EE_int[1] - CC[(1,2,2)]*EE_int[2] |
| 895 | (0, 0) |
| 896 | """ |
| 897 | e1 = vector(SR, e1) |
| 898 | e2 = vector(SR, e2) |
| 899 | |
| 900 | lie_bracket = self.lie_bracket(e1, e2).simplify_full() |
| 901 | transformation = matrix(SR, [e1, e2]).transpose() |
| 902 | |
| 903 | w = (transformation.inverse()*lie_bracket).simplify_full() |
| 904 | |
| 905 | return {(1,1,1): 0, (1,1,2): 0, (1,2,1): w[0], (1,2,2): w[1], |
| 906 | (2,1,1): -w[0], (2,1,2): -w[1], (2,2,1): 0, (2,2,2): 0} |
| 907 | |
| 908 | |
| 909 | @cached_method |
| 910 | def _compute_second_order_frame_element(self, index): |
| 911 | """ |
| 912 | Compute an element of the second order frame of the surface. See |
| 913 | ``second_order_natural_frame`` for more details. |
| 914 | |
| 915 | This method expects its arguments in tuple form for caching. |
| 916 | As it does no input checking, it should not be called directly. |
| 917 | |
| 918 | EXAMPLES:: |
| 919 | |
| 920 | sage: u, v = var('u, v') |
| 921 | sage: paraboloid = ParametrizedSurface3D([u, v, u^2 + v^2], [u,v], 'paraboloid') |
| 922 | sage: paraboloid._compute_second_order_frame_element((1, 2)) |
| 923 | (0, 0, 0) |
| 924 | sage: paraboloid._compute_second_order_frame_element((2, 2)) |
| 925 | (0, 0, 2) |
| 926 | |
| 927 | """ |
| 928 | variables = [self.variables[i] for i in index] |
| 929 | ddr_element = vector([diff(f, variables).simplify_full() for f in self.equation]) |
| 930 | |
| 931 | return ddr_element |
| 932 | |
| 933 | |
| 934 | @cached_method |
| 935 | def second_order_natural_frame(self): |
| 936 | r""" |
| 937 | Returns the second-order frame of the surface, i.e. computes the |
| 938 | second-order derivatives (with respect to the parameters on the |
| 939 | surface) of the parametric expression $\vec r = \vec r(u^1,u^2)$ |
| 940 | of the surface. |
| 941 | |
| 942 | OUTPUT: |
| 943 | |
| 944 | - Dictionary where the keys are 2-tuples ``(i, j)`` and the values are the corresponding derivatives ``r_ij``. |
| 945 | |
| 946 | EXAMPLES: |
| 947 | |
| 948 | We compute the second-order natural frame of the sphere:: |
| 949 | |
| 950 | sage: u, v = var('u, v') |
| 951 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 952 | sage: sphere.second_order_natural_frame() |
| 953 | {(1, 2): (sin(u)*sin(v), -sin(v)*cos(u), 0), (1, 1): (-cos(u)*cos(v), |
| 954 | -sin(u)*cos(v), 0), (2, 1): (sin(u)*sin(v), -sin(v)*cos(u), 0), (2, 2): |
| 955 | (-cos(u)*cos(v), -sin(u)*cos(v), -sin(v))} |
| 956 | |
| 957 | """ |
| 958 | |
| 959 | vectors = {} |
| 960 | for index in self.index_list: |
| 961 | sorted_index = tuple(sorted(index)) |
| 962 | vectors[index] = \ |
| 963 | self._compute_second_order_frame_element(sorted_index) |
| 964 | return vectors |
| 965 | |
| 966 | |
| 967 | def second_order_natural_frame_element(self, index): |
| 968 | r""" |
| 969 | Returns a vector in the second-order frame of the surface, i.e. |
| 970 | computes the second-order derivatives of the parametric expression |
| 971 | $\vec{r}$ of the surface with respect to the parameters listed in the |
| 972 | argument. |
| 973 | |
| 974 | INPUT: |
| 975 | |
| 976 | - ``index`` - a 2-tuple ``(i, j)`` specifying the element of the second-order frame. |
| 977 | |
| 978 | OUTPUT: |
| 979 | |
| 980 | - The second-order derivative ``r_ij``. |
| 981 | |
| 982 | EXAMPLES:: |
| 983 | |
| 984 | sage: u, v = var('u, v') |
| 985 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 986 | sage: sphere.second_order_natural_frame_element((1, 2)) |
| 987 | (sin(u)*sin(v), -sin(v)*cos(u), 0) |
| 988 | |
| 989 | """ |
| 990 | |
| 991 | index = tuple(sorted(index)) |
| 992 | if index not in self.index_list: |
| 993 | raise ValueError, "Index %s out of bounds." % str(index) |
| 994 | return self._compute_second_order_frame_element(index) |
| 995 | |
| 996 | |
| 997 | @cached_method |
| 998 | def _compute_second_fundamental_form_coefficient(self, index): |
| 999 | """ |
| 1000 | Compute a coefficient of the second fundamental form of the surface. |
| 1001 | See ``second_fundamental_form_coefficient`` for more details. |
| 1002 | |
| 1003 | This method expects its arguments in tuple form for caching. As it |
| 1004 | does no input checking, it should not be called directly. |
| 1005 | |
| 1006 | EXAMPLES:: |
| 1007 | |
| 1008 | sage: u, v = var('u,v') |
| 1009 | sage: paraboloid = ParametrizedSurface3D([u, v, u^2+v^2], [u, v], 'paraboloid') |
| 1010 | sage: paraboloid._compute_second_fundamental_form_coefficient((1,1)) |
| 1011 | 2/sqrt(4*u^2 + 4*v^2 + 1) |
| 1012 | |
| 1013 | """ |
| 1014 | N = self.normal_vector(normalized=True) |
| 1015 | v = self.second_order_natural_frame_element(index) |
| 1016 | return (v*N).simplify_full() |
| 1017 | |
| 1018 | |
| 1019 | def second_fundamental_form_coefficient(self, index): |
| 1020 | r""" |
| 1021 | Returns the coefficient $h_{ij}$ of the second fundamental form |
| 1022 | corresponding to the index $(i, j)$. If the equation of the surface |
| 1023 | is $\vec{r}(u^1, u^2)$, then $h_{ij} = \vec{r}_{u^i u^j} \cdot \vec{n}$, |
| 1024 | where $\vec{n}$ is the unit normal. |
| 1025 | |
| 1026 | INPUT: |
| 1027 | |
| 1028 | - ``index`` - a 2-tuple ``(i, j)`` |
| 1029 | |
| 1030 | OUTPUT: |
| 1031 | |
| 1032 | - Component ``h_{ij}`` of the second fundamental form. |
| 1033 | |
| 1034 | EXAMPLES:: |
| 1035 | |
| 1036 | sage: u, v = var('u,v') |
| 1037 | sage: assume(cos(v)>0) |
| 1038 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 1039 | sage: sphere.second_fundamental_form_coefficient((1, 1)) |
| 1040 | -cos(v)^2 |
| 1041 | sage: sphere.second_fundamental_form_coefficient((2, 1)) |
| 1042 | 0 |
| 1043 | |
| 1044 | """ |
| 1045 | index = tuple(index) |
| 1046 | if index not in self.index_list: |
| 1047 | raise ValueError, "Index %s out of bounds." % str(index) |
| 1048 | return self._compute_second_fundamental_form_coefficient(index) |
| 1049 | |
| 1050 | |
| 1051 | @cached_method |
| 1052 | def second_fundamental_form_coefficients(self): |
| 1053 | """ |
| 1054 | Returns the coefficients $h_{ij}$ of the second fundamental form as |
| 1055 | a dictionary, where the keys are the indices $(i, j)$ and the values |
| 1056 | are the corresponding components $h_{ij}$. |
| 1057 | |
| 1058 | When only one component is needed, consider the function |
| 1059 | ``second_fundamental_form_coefficient``. |
| 1060 | |
| 1061 | OUTPUT: |
| 1062 | |
| 1063 | Dictionary of second fundamental form coefficients. |
| 1064 | |
| 1065 | EXAMPLES:: |
| 1066 | |
| 1067 | sage: u, v = var('u, v') |
| 1068 | sage: assume(cos(v)>0) |
| 1069 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 1070 | sage: sphere.second_fundamental_form_coefficients() |
| 1071 | {(1, 2): 0, (1, 1): -cos(v)^2, (2, 1): 0, (2, 2): -1} |
| 1072 | |
| 1073 | """ |
| 1074 | |
| 1075 | coefficients = {} |
| 1076 | for index in self.index_list: |
| 1077 | coefficients[index] = \ |
| 1078 | self._compute_second_fundamental_form_coefficient(index) |
| 1079 | return coefficients |
| 1080 | |
| 1081 | |
| 1082 | def second_fundamental_form(self,vector1,vector2): |
| 1083 | r""" |
| 1084 | Evaluates the second fundamental form on two vectors on the surface. |
| 1085 | If the vectors are given by $v=(v^1,v^2)$ and $w=(w^1,w^2)$, the |
| 1086 | result of this function is $h_{11} v^1 w^1 + h_{12}(v^1 w^2 + v^2 w^1) + h_{22} v^2 w^2$. |
| 1087 | |
| 1088 | INPUT: |
| 1089 | |
| 1090 | - ``vector1``, ``vector2`` - 2-tuples or vectors of length 2 representing the input vectors. |
| 1091 | |
| 1092 | OUTPUT: |
| 1093 | |
| 1094 | - Value of the second fundamental form evaluated on the given vectors. |
| 1095 | |
| 1096 | EXAMPLES: |
| 1097 | |
| 1098 | We evaluate the second fundamental form on two symbolic vectors:: |
| 1099 | |
| 1100 | sage: u, v = var('u, v') |
| 1101 | sage: v1, v2, w1, w2 = var('v1, v2, w1, w2') |
| 1102 | sage: assume(cos(v) > 0) |
| 1103 | sage: sphere = ParametrizedSurface3D([cos(u)*cos(v),sin(u)*cos(v),sin(v)],[u,v],'sphere') |
| 1104 | sage: sphere.second_fundamental_form(vector([v1, v2]), vector([w1, w2])) |
| 1105 | -v1*w1*cos(v)^2 - v2*w2 |
| 1106 | |
| 1107 | We evaluate the second fundamental form on vectors with numerical |
| 1108 | components:: |
| 1109 | |
| 1110 | sage: vect = vector([1,2]) |
| 1111 | sage: sphere.second_fundamental_form(vect, vect) |
| 1112 | -cos(v)^2 - 4 |
| 1113 | sage: sphere.second_fundamental_form([1,1], [2,1]) |
| 1114 | -2*cos(v)^2 - 1 |
| 1115 | |
| 1116 | """ |
| 1117 | hh = self.second_fundamental_form_coefficients() |
| 1118 | return sum(hh[ind]*vector1[ind[0]-1]*vector2[ind[1]-1] |
| 1119 | for ind in self.index_list) |
| 1120 | |
| 1121 | |
| 1122 | @cached_method |
| 1123 | def gauss_curvature(self): |
| 1124 | r""" |
| 1125 | Finds the gaussian curvature of the surface, given by |
| 1126 | $K = \frac{h_{11}h_{22} - h_{12}^2}{g_{11}g_{22} - g_{12}^2}$, |
| 1127 | where $g_{ij}$ and $h_{ij}$ are the coefficients of the first |
| 1128 | and second fundamental form, respectively. |
| 1129 | |
| 1130 | OUTPUT: |
| 1131 | |
| 1132 | - Gaussian curvature of the surface. |
| 1133 | |
| 1134 | EXAMPLES:: |
| 1135 | |
| 1136 | sage: R = var('R') |
| 1137 | sage: assume(R>0) |
| 1138 | sage: u, v = var('u,v') |
| 1139 | sage: assume(cos(v)>0) |
| 1140 | sage: sphere = ParametrizedSurface3D([R*cos(u)*cos(v),R*sin(u)*cos(v),R*sin(v)],[u,v],'sphere') |
| 1141 | sage: sphere.gauss_curvature() |
| 1142 | R^(-2) |
| 1143 | |
| 1144 | """ |
| 1145 | hh = self.second_fundamental_form_coefficients() |
| 1146 | return ((hh[(1,1)]*hh[(2,2)]-hh[(1,2)]**2)/self.area_form_squared()).simplify_full() |
| 1147 | |
| 1148 | |
| 1149 | @cached_method |
| 1150 | def mean_curvature(self): |
| 1151 | r""" |
| 1152 | Finds the mean curvature of the surface, given by |
| 1153 | $H = \frac{1}{2}\frac{g_{22}h_{11} - 2g_{12}h_{12} + g_{11}h_{22}}{g_{11}g_{22} - g_{12}^2}$, |
| 1154 | where $g_{ij}$ and $h_{ij}$ are the components of the first and second |
| 1155 | fundamental forms, respectively. |
| 1156 | |
| 1157 | OUTPUT: |
| 1158 | |
| 1159 | - Mean curvature of the surface |
| 1160 | |
| 1161 | EXAMPLES:: |
| 1162 | |
| 1163 | sage: R = var('R') |
| 1164 | sage: assume(R>0) |
| 1165 | sage: u, v = var('u,v') |
| 1166 | sage: assume(cos(v)>0) |
| 1167 | sage: sphere = ParametrizedSurface3D([R*cos(u)*cos(v),R*sin(u)*cos(v),R*sin(v)],[u,v],'sphere') |
| 1168 | sage: sphere.mean_curvature() |
| 1169 | -1/R |
| 1170 | |
| 1171 | """ |
| 1172 | gg = self.first_fundamental_form_coefficients() |
| 1173 | hh = self.second_fundamental_form_coefficients() |
| 1174 | denom = 2*self.area_form_squared() |
| 1175 | numer = (gg[(2,2)]*hh[(1,1)] - 2*gg[(1,2)]*hh[(1,2)] + |
| 1176 | gg[(1,1)]*hh[(2,2)]).simplify_full() |
| 1177 | return (numer/denom).simplify_full() |
| 1178 | |
| 1179 | |
| 1180 | @cached_method |
| 1181 | def shape_operator_coefficients(self): |
| 1182 | r""" |
| 1183 | Returns the components of the shape operator of the surface as a |
| 1184 | dictionary. See ``shape_operator`` for more information. |
| 1185 | |
| 1186 | OUTPUT: |
| 1187 | |
| 1188 | - Dictionary where the keys are two-tuples ``(i, j)``, with values the |
| 1189 | corresponding component of the shape operator. |
| 1190 | |
| 1191 | EXAMPLES:: |
| 1192 | |
| 1193 | sage: R = var('R') |
| 1194 | sage: u, v = var('u,v') |
| 1195 | sage: assume(cos(v)>0) |
| 1196 | sage: sphere = ParametrizedSurface3D([R*cos(u)*cos(v),R*sin(u)*cos(v),R*sin(v)],[u,v],'sphere') |
| 1197 | sage: sphere.shape_operator_coefficients() |
| 1198 | {(1, 2): 0, (1, 1): -1/R, (2, 1): 0, (2, 2): -1/R} |
| 1199 | |
| 1200 | """ |
| 1201 | |
| 1202 | gi = self.first_fundamental_form_inverse_coefficients() |
| 1203 | hh = self.second_fundamental_form_coefficients() |
| 1204 | |
| 1205 | sh_op11 = (gi[(1,1)]*hh[(1,1)] + gi[(1,2)]*hh[(1,2)]).simplify_full() |
| 1206 | sh_op12 = (gi[(1,1)]*hh[(2,1)] + gi[(1,2)]*hh[(2,2)]).simplify_full() |
| 1207 | sh_op21 = (gi[(2,1)]*hh[(1,1)] + gi[(2,2)]*hh[(1,2)]).simplify_full() |
| 1208 | sh_op22 = (gi[(2,1)]*hh[(2,1)] + gi[(2,2)]*hh[(2,2)]).simplify_full() |
| 1209 | |
| 1210 | return {(1,1): sh_op11, (1,2): sh_op12, (2,1): sh_op21, (2,2): sh_op22} |
| 1211 | |
| 1212 | |
| 1213 | def shape_operator(self): |
| 1214 | r""" |
| 1215 | Returns the shape operator of the surface as a matrix. The shape |
| 1216 | operator is defined as the derivative of the Gauss map, and is |
| 1217 | computed here in terms of the first and second fundamental form by |
| 1218 | means of the Weingarten equations. |
| 1219 | |
| 1220 | OUTPUT: |
| 1221 | |
| 1222 | - Matrix of the shape operator |
| 1223 | |
| 1224 | EXAMPLES:: |
| 1225 | |
| 1226 | sage: R = var('R') |
| 1227 | sage: assume(R>0) |
| 1228 | sage: u, v = var('u,v') |
| 1229 | sage: assume(cos(v)>0) |
| 1230 | sage: sphere = ParametrizedSurface3D([R*cos(u)*cos(v),R*sin(u)*cos(v),R*sin(v)],[u,v],'sphere') |
| 1231 | sage: S = sphere.shape_operator(); S |
| 1232 | [-1/R 0] |
| 1233 | [ 0 -1/R] |
| 1234 | |
| 1235 | The eigenvalues of the shape operator are the principal curvatures of |
| 1236 | the surface:: |
| 1237 | |
| 1238 | sage: u, v = var('u,v') |
| 1239 | sage: paraboloid = ParametrizedSurface3D([u, v, u^2+v^2], [u, v], 'paraboloid') |
| 1240 | sage: S = paraboloid.shape_operator(); S |
| 1241 | [2*(4*v^2 + 1)/(4*u^2 + 4*v^2 + 1)^(3/2) -8*u*v/(4*u^2 + 4*v^2 + 1)^(3/2)] |
| 1242 | [ -8*u*v/(4*u^2 + 4*v^2 + 1)^(3/2) 2*(4*u^2 + 1)/(4*u^2 + 4*v^2 + 1)^(3/2)] |
| 1243 | sage: S.eigenvalues() |
| 1244 | [2*sqrt(4*u^2 + 4*v^2 + 1)/(16*u^4 + 8*u^2 + 16*v^4 + 8*(4*u^2 + 1)*v^2 + 1), 2/sqrt(4*u^2 + 4*v^2 + 1)] |
| 1245 | |
| 1246 | """ |
| 1247 | |
| 1248 | shop = self.shape_operator_coefficients() |
| 1249 | shop_matrix=matrix([[shop[(1,1)],shop[(1,2)]], |
| 1250 | [shop[(2,1)],shop[(2,2)]]]) |
| 1251 | return shop_matrix |
| 1252 | |
| 1253 | |
| 1254 | def principal_directions(self): |
| 1255 | r""" |
| 1256 | Finds the principal curvatures and principal directions of the surface |
| 1257 | |
| 1258 | OUTPUT: |
| 1259 | |
| 1260 | For each principal curvature, returns a list of the form |
| 1261 | ``(rho, V, n)``, where ``rho`` is the principal curvature, |
| 1262 | ``V`` is the corresponding principal direction, and ``n`` is |
| 1263 | the multiplicity. |
| 1264 | |
| 1265 | EXAMPLES:: |
| 1266 | |
| 1267 | sage: u, v = var('u, v') |
| 1268 | sage: R, r = var('R,r') |
| 1269 | sage: assume(R>r,r>0) |
| 1270 | sage: torus = ParametrizedSurface3D([(R+r*cos(v))*cos(u),(R+r*cos(v))*sin(u),r*sin(v)],[u,v],'torus') |
| 1271 | sage: torus.principal_directions() |
| 1272 | [(-cos(v)/(r*cos(v) + R), [(1, 0)], 1), (-1/r, [(0, 1)], 1)] |
| 1273 | |
| 1274 | """ |
| 1275 | return self.shape_operator().eigenvectors_left() |
| 1276 | |
| 1277 | |
| 1278 | @cached_method |
| 1279 | def connection_coefficients(self): |
| 1280 | r""" |
| 1281 | Computes the connection coefficients or Christoffel symbols |
| 1282 | $\Gamma^k_{ij}$ of the surface. If the coefficients of the first |
| 1283 | fundamental form are given by $g_{ij}$ (where $i, j = 1, 2$), then |
| 1284 | $\Gamma^k_{ij} = \frac{1}{2} g^{kl} \left( \frac{\partial g_{li}}{\partial x^j} |
| 1285 | - \frac{\partial g_{ij}}{\partial x^l} |
| 1286 | + \frac{\partial g_{lj}}{\partial x^i} \right)$. |
| 1287 | Here, $(g^{kl})$ is the inverse of the matrix $(g_{ij})$, with |
| 1288 | $i, j = 1, 2$. |
| 1289 | |
| 1290 | OUTPUT: |
| 1291 | |
| 1292 | Dictionary of connection coefficients, where the keys are 3-tuples |
| 1293 | $(i,j,k)$ and the values are the corresponding coefficients |
| 1294 | $\Gamma^k_{ij}$. |
| 1295 | |
| 1296 | EXAMPLES:: |
| 1297 | |
| 1298 | sage: r = var('r') |
| 1299 | sage: assume(r > 0) |
| 1300 | sage: u, v = var('u,v') |
| 1301 | sage: assume(cos(v)>0) |
| 1302 | sage: sphere = ParametrizedSurface3D([r*cos(u)*cos(v),r*sin(u)*cos(v),r*sin(v)],[u,v],'sphere') |
| 1303 | sage: sphere.connection_coefficients() |
| 1304 | {(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} |
| 1305 | |
| 1306 | """ |
| 1307 | x = self.variables |
| 1308 | gg = self.first_fundamental_form_coefficients() |
| 1309 | gi = self.first_fundamental_form_inverse_coefficients() |
| 1310 | |
| 1311 | dg = {} |
| 1312 | for i,j,k in self.index_list_3: |
| 1313 | dg[(i,j,k)] = gg[(j,k)].differentiate(x[i]).simplify_full() |
| 1314 | |
| 1315 | structfun={} |
| 1316 | for i,j,k in self.index_list_3: |
| 1317 | structfun[(i,j,k)] = sum(gi[(k,s)]*(dg[(i,j,s)] + dg[(j,i,s)] |
| 1318 | -dg[(s,i,j)])/2 |
| 1319 | for s in (1,2)).full_simplify() |
| 1320 | return structfun |
| 1321 | |
| 1322 | |
| 1323 | @cached_method |
| 1324 | def _create_geodesic_ode_system(self): |
| 1325 | r""" |
| 1326 | Helper method to create a fast floating-point version of the |
| 1327 | geodesic equations, used by ``geodesics_numerical``. |
| 1328 | |
| 1329 | EXAMPLES:: |
| 1330 | sage: p, q = var('p,q') |
| 1331 | sage: sphere = ParametrizedSurface3D([cos(q)*cos(p),sin(q)*cos(p),sin(p)],[p,q],'sphere') |
| 1332 | sage: ode = sphere._create_geodesic_ode_system() |
| 1333 | sage: ode.function(0.0, (1.0, 0.0, 1.0, 1.0)) |
| 1334 | [1.00000000000000, 1.00000000000000, -0.45464871341284091, 3.1148154493098041] |
| 1335 | |
| 1336 | """ |
| 1337 | from sage.ext.fast_eval import fast_float |
| 1338 | from sage.calculus.var import var |
| 1339 | from sage.gsl.ode import ode_solver |
| 1340 | |
| 1341 | u1 = self.variables[1] |
| 1342 | u2 = self.variables[2] |
| 1343 | v1, v2 = var('v1, v2') |
| 1344 | |
| 1345 | C = self.connection_coefficients() |
| 1346 | |
| 1347 | dv1 = - C[(1,1,1)]*v1**2 - 2*C[(1,2,1)]*v1*v2 - C[(2,2,1)]*v2**2 |
| 1348 | dv2 = - C[(1,1,2)]*v1**2 - 2*C[(1,2,2)]*v1*v2 - C[(2,2,2)]*v2**2 |
| 1349 | fun1 = fast_float(dv1, str(u1), str(u2), str(v1), str(v2)) |
| 1350 | fun2 = fast_float(dv2, str(u1), str(u2), str(v1), str(v2)) |
| 1351 | |
| 1352 | geodesic_ode = ode_solver() |
| 1353 | geodesic_ode.function = \ |
| 1354 | lambda t, (u1, u2, v1, v2) : \ |
| 1355 | [v1, v2, fun1(u1, u2, v1, v2), fun2(u1, u2, v1, v2)] |
| 1356 | return geodesic_ode |
| 1357 | |
| 1358 | |
| 1359 | def geodesics_numerical(self, p0, v0, tinterval): |
| 1360 | r""" |
| 1361 | Numerical integration of the geodesic equations. Explicitly, the |
| 1362 | geodesic equations are given by |
| 1363 | $\frac{d^2 u^i}{dt^2} + \Gamma^i_{jk} \frac{d u^j}{dt} \frac{d u^k}{dt} = 0$. |
| 1364 | |
| 1365 | Solving these equations gives the coordinates $(u^1, u^2)$ of |
| 1366 | the geodesic on the surface. The coordinates in space can |
| 1367 | then be found by substituting $(u^1, u^2)$ into the vector |
| 1368 | $\vec{r}(u^1, u^2)$ representing the surface. |
| 1369 | |
| 1370 | ALGORITHM: |
| 1371 | |
| 1372 | The geodesic equations are integrated forward in time using |
| 1373 | the ode solvers from ``sage.gsl.ode``. See the member |
| 1374 | function ``_create_geodesic_ode_system`` for more details. |
| 1375 | |
| 1376 | INPUT: |
| 1377 | |
| 1378 | - ``p0`` - 2-tuple with coordinates of the initial point. |
| 1379 | |
| 1380 | - ``v0`` - 2-tuple with components of the initial tangent vector to the geodesic. |
| 1381 | |
| 1382 | - ``tinterval`` - List ``[a,b,M]``, where ``(a,b)`` is the domain of the geodesic and ``M`` is the number of subdivision points used when returning the solution. |
| 1383 | |
| 1384 | OUTPUT: |
| 1385 | |
| 1386 | List of lists ``[t, [u1(t),u2(t)], [v1(t),v2(t)], [x1(t),x2(t),x3(t)]]``, where |
| 1387 | |
| 1388 | - ``t`` is a subdivision point; |
| 1389 | |
| 1390 | - ``[u1(t),u2(t)]`` are the intrinsic coordinates of the geodesic point; |
| 1391 | |
| 1392 | - ``[v1(t),v2(t)]`` are the intrinsic coordinates of the tangent vector to the geodesic; |
| 1393 | |
| 1394 | - ``[x1(t),x2(t),x3(t)]`` are the coordinates of the geodesic point in the three-dimensional space. |
| 1395 | |
| 1396 | EXAMPLES:: |
| 1397 | |
| 1398 | sage: p, q = var('p,q') |
| 1399 | sage: assume(cos(q)>0) |
| 1400 | sage: sphere = ParametrizedSurface3D([cos(q)*cos(p),sin(q)*cos(p),sin(p)],[p,q],'sphere') |
| 1401 | sage: geodesic = sphere.geodesics_numerical([0.0,0.0],[1.0,1.0],[0,2*pi,5]) |
| 1402 | sage: times, points, tangent_vectors, ext_points = zip(*geodesic) |
| 1403 | |
| 1404 | sage: round4 = lambda vec: [N(x, digits=4) for x in vec] # helper function to round to 4 digits |
| 1405 | sage: round4(times) |
| 1406 | [0.0000, 1.257, 2.513, 3.770, 5.027, 6.283] |
| 1407 | sage: [round4(p) for p in points] |
| 1408 | [[0.0000, 0.0000], [0.7644, 1.859], [-0.2876, 3.442], [-0.6137, 5.502], [0.5464, 6.937], [0.3714, 9.025]] |
| 1409 | sage: [round4(p) for p in ext_points] |
| 1410 | [[1.000, 0.0000, 0.0000], [-0.2049, 0.6921, 0.6921], [-0.9160, -0.2836, -0.2836], [0.5803, -0.5759, -0.5759], [0.6782, 0.5196, 0.5196], [-0.8582, 0.3629, 0.3629]] |
| 1411 | """ |
| 1412 | |
| 1413 | u1 = self.variables[1] |
| 1414 | u2 = self.variables[2] |
| 1415 | |
| 1416 | solver = self._create_geodesic_ode_system() |
| 1417 | |
| 1418 | t_interval, n = tinterval[0:2], tinterval[2] |
| 1419 | solver.y_0 = [p0[0], p0[1], v0[0], v0[1]] |
| 1420 | solver.ode_solve(t_span=t_interval, num_points=n) |
| 1421 | |
| 1422 | parsed_solution = \ |
| 1423 | [[vec[0], vec[1][0:2], vec[1][2:], |
| 1424 | self.eq_callable(vec[1][0], vec[1][1])] |
| 1425 | for vec in solver.solution] |
| 1426 | |
| 1427 | return parsed_solution |
| 1428 | |
| 1429 | |
| 1430 | @cached_method |
| 1431 | def _create_pt_ode_system(self, curve, t): |
| 1432 | """ |
| 1433 | Helper method to create a fast floating-point version of the parallel |
| 1434 | transport equations, used by ``parallel_translation_numerical``. |
| 1435 | |
| 1436 | INPUT: |
| 1437 | |
| 1438 | - ``curve`` - curve in intrinsic coordinates along which to do parallel transport. |
| 1439 | - ``t`` - curve parameter |
| 1440 | |
| 1441 | EXAMPLES:: |
| 1442 | sage: p, q = var('p,q') |
| 1443 | sage: sphere = ParametrizedSurface3D([cos(q)*cos(p),sin(q)*cos(p),sin(p)],[p,q],'sphere') |
| 1444 | sage: s = var('s') |
| 1445 | sage: ode = sphere._create_pt_ode_system((s, s), s) |
| 1446 | sage: ode.function(0.0, (1.0, 1.0)) |
| 1447 | [-0.0, 0.0] |
| 1448 | |
| 1449 | """ |
| 1450 | |
| 1451 | from sage.ext.fast_eval import fast_float |
| 1452 | from sage.calculus.var import var |
| 1453 | from sage.gsl.ode import ode_solver |
| 1454 | |
| 1455 | u1 = self.variables[1] |
| 1456 | u2 = self.variables[2] |
| 1457 | v1, v2 = var('v1, v2') |
| 1458 | |
| 1459 | du1 = diff(curve[0], t) |
| 1460 | du2 = diff(curve[1], t) |
| 1461 | |
| 1462 | C = self.connection_coefficients() |
| 1463 | for coef in C: |
| 1464 | C[coef] = C[coef].subs({u1: curve[0], u2: curve[1]}) |
| 1465 | |
| 1466 | dv1 = - C[(1,1,1)]*v1*du1 - C[(1,2,1)]*(du1*v2 + du2*v1) - \ |
| 1467 | C[(2,2,1)]*du2*v2 |
| 1468 | dv2 = - C[(1,1,2)]*v1*du1 - C[(1,2,2)]*(du1*v2 + du2*v1) - \ |
| 1469 | C[(2,2,2)]*du2*v2 |
| 1470 | fun1 = fast_float(dv1, str(t), str(v1), str(v2)) |
| 1471 | fun2 = fast_float(dv2, str(t), str(v1), str(v2)) |
| 1472 | |
| 1473 | pt_ode = ode_solver() |
| 1474 | pt_ode.function = lambda t, (v1, v2): [fun1(t, v1, v2), fun2(t, v1, v2)] |
| 1475 | return pt_ode |
| 1476 | |
| 1477 | |
| 1478 | def parallel_translation_numerical(self,curve,t,v0,tinterval): |
| 1479 | r""" |
| 1480 | Numerically solves the equations for parallel translation of a vector |
| 1481 | along a curve on the surface. Explicitly, the equations for parallel |
| 1482 | translation are given by |
| 1483 | $\frac{d u^i}{dt} + u^j \frac{d c^k}{dt} \Gamma^i_{jk} = 0$, |
| 1484 | where $\Gamma^i_{jk}$ are the connection coefficients of the surface, |
| 1485 | the vector to be transported has components $u^j$ and the curve along |
| 1486 | which to transport has components $c^k$. |
| 1487 | |
| 1488 | ALGORITHM: |
| 1489 | |
| 1490 | The parallel transport equations are integrated forward in time using |
| 1491 | the ode solvers from ``sage.gsl.ode``. See ``_create_pt_ode_system`` |
| 1492 | for more details. |
| 1493 | |
| 1494 | INPUT: |
| 1495 | |
| 1496 | - ``curve`` - 2-tuple of functions which determine the curve with respect to |
| 1497 | the local coordinate system; |
| 1498 | |
| 1499 | - ``t`` - symbolic variable denoting the curve parameter; |
| 1500 | |
| 1501 | - ``v0`` - 2-tuple representing the initial vector; |
| 1502 | |
| 1503 | - ``tinterval`` - list ``[a,b,N]``, where ``(a,b)`` is the domain of the curve |
| 1504 | and ``N`` is the number of subdivision points. |
| 1505 | |
| 1506 | OUTPUT: |
| 1507 | |
| 1508 | The list consisting of lists ``[t, [v1(t),v2(t)]]``, where |
| 1509 | |
| 1510 | - ``t`` is a subdivision point; |
| 1511 | |
| 1512 | - ``[v1(t),v2(t)]`` is the list of coordinates of the vector parallel translated |
| 1513 | along the curve. |
| 1514 | |
| 1515 | EXAMPLES:: |
| 1516 | |
| 1517 | sage: p, q = var('p,q') |
| 1518 | sage: v = [p,q] |
| 1519 | sage: assume(cos(q)>0) |
| 1520 | sage: sphere = ParametrizedSurface3D([cos(q)*cos(p),sin(q)*cos(p),sin(p)],v,'sphere') |
| 1521 | sage: s = var('s') |
| 1522 | sage: vector_field = sphere.parallel_translation_numerical([s,s],s,[1.0,1.0],[0.0, pi/4, 5]) |
| 1523 | sage: times, components = zip(*vector_field) |
| 1524 | |
| 1525 | sage: round4 = lambda vec: [N(x, digits=4) for x in vec] # helper function to round to 4 digits |
| 1526 | sage: round4(times) |
| 1527 | [0.0000, 0.1571, 0.3142, 0.4712, 0.6283, 0.7854] |
| 1528 | sage: [round4(v) for v in components] |
| 1529 | [[1.000, 1.000], [0.9876, 1.025], [0.9499, 1.102], [0.8853, 1.238], [0.7920, 1.448], [0.6687, 1.762]] |
| 1530 | |
| 1531 | """ |
| 1532 | |
| 1533 | u1 = self.variables[1] |
| 1534 | u2 = self.variables[2] |
| 1535 | |
| 1536 | solver = self._create_pt_ode_system(tuple(curve), t) |
| 1537 | |
| 1538 | t_interval, n = tinterval[0:2], tinterval[2] |
| 1539 | solver.y_0 = v0 |
| 1540 | solver.ode_solve(t_span=t_interval, num_points=n) |
| 1541 | |
| 1542 | return solver.solution |