"ARMÓNICOS ESFÉRICOS"


En la tabla que sigue se muestran superficies conocidas comunmente como "armónicos esféricos" (spherical harmonics),
y su descripción se ha obtenido en la página "web" de Paul Bourke, en la que pueden encontrarse más detalles.
Su descripción general, en términos de coordonedas polares esféricas, es muy sencilla:

                r = Sin[m0 j]m1 + Cos[m2 j]m3 + Sin[m4 q]m5 + Cos[m6 q]m7

y la correspondiente parametrización inducida en coordenadas cartesianas se escribe como sigue:

    x[m0,m1,m2,m3,m4,m5,m6,m7](q,j) =
                           {(Sin[m0
j]m1 + Cos[m2 j]m3 + Sin[m4 q]m5 + Cos[m6 q]m7)Cos[q]Sin[j],
                             (Sin[m0
j]m1 + Cos[m2 j]m3 + Sin[m4 q]m5 + Cos[m6 q]m7)Cos[j],
                             (Sin[m0
j]m1 + Cos[m2 j]m3 + Sin[m4 q]m5 + Cos[m6 q]m7)Sin[q]Sin[j]}

Se ha etiquetado cada una de las superficies por sus parámetros: (m0,m1,m2,m3,m4,m5,m6,m7).

sphehar02350530_sm.gif (5116 bytes) sphehar03631410_sm.gif (6184 bytes) sphehar12341234_sm.gif (5943 bytes) sphehar13524653_sm.gif (6670 bytes) sphehar14002621_sm.gif (5814 bytes)

(0,2,3,5,0,5,3,0)

(0,3,6,3,1,4,1,0)

(1,2,3,4,1,2,3,4)

(1,3,5,2,4,6,5,3)

(1,4,0,0,2,6,2,1)

sphehar21042302_sm.gif (6593 bytes) sphehar22442244_sm.gif (6180 bytes) sphehar23126440_sm.gif (5557 bytes) sphehar23126442_sm.gif (5188 bytes) sphehar23232323_sm.gif (5304 bytes)

(2,1,0,4,2,3,0,2)

(2,2,4,4,2,2,4,4)

(2,3,1,2,6,4,4,0)

(2,3,1,2,6,4,4,2)

(2,3,2,3,2,3,2,3)

sphehar23423456_sm.gif (5449 bytes) sphehar30056542_sm.gif (7389 bytes) sphehar30056543_sm.gif (6493 bytes) sphehar34025623_sm.gif (6417 bytes) sphehar34343434_sm.gif (6019 bytes)

(2,3,4,2,3,4,5,6)

(3,0,0,5,6,5,4,2)

(3,0,0,5,6,5,4,3)

(3,4,0,2,5,6,2,3)

(3,4,3,4,3,4,3,4)

sphehar43211234_sm.gif (5304 bytes) sphehar43246263_sm.gif (6180 bytes) sphehar43246264_sm.gif (6457 bytes) sphehar44664466_sm.gif (5931 bytes) sphehar45454545_sm.gif (5995 bytes)

(4,3,2,1,1,2,3,4)

(4,3,2,4,6,2,6,3)

(4,3,2,4,6,2,6,4)

(4,4,6,6,4,4,6,6)

(4,5,4,5,4,5,4,5)

sphehar50513334_sm.gif (6365 bytes) sphehar50513335_sm.gif (5654 bytes) sphehar51340202_sm.gif (6300 bytes) sphehar51455250_sm.gif (6180 bytes) sphehar51455255_sm.gif (6212 bytes)

(5,0,5,1,3,3,3,4)

(5,0,5,1,3,3,3,5)

(5,1,3,4,0,2,0,2)

(5,1,4,5,5,2,5,0)

(5,1,4,5,5,2,5,5)

sphehar52552101_sm.gif (6344 bytes) sphehar54233245_sm.gif (6336 bytes) sphehar54545454_sm.gif (6650 bytes) sphehar63121210_sm.gif (6079 bytes) sphehar63121212_sm.gif (6352 bytes)

(5,2,5,5,2,1,0,1)

(5,4,2,3,3,2,4,5)

(5,4,5,4,5,4,5,4)

(6,3,1,2,1,2,1,0)

(6,3,1,2,1,2,1,2)

sphehar63121216_sm.gif (6007 bytes) sphehar65433456_sm.gif (5790 bytes) sphehar65434365_sm.gif (5838 bytes) sphehar66402401_sm.gif (6927 bytes) sphehar66446644_sm.gif (6356 bytes)

(6,3,1,2,1,2,1,6)

(6,5,4,3,3,4,5,6)

(6,5,4,3,4,3,6,5)

(6,6,4,0,2,4,0,1)

(6,6,4,4,6,6,4,4)


© L. A. Cordero