2868:) . To represent an image in this coordinate system rather than in Cartesian coordinates, gives computational advantages when rotating or zooming in an image. Also, the photo receptors in the retina in the human eye are distributed in a way that has big similarities with the spiral coordinate system. It can also be found in the Mandelbrot fractal (see picture to the right).
2633:
2621:
2714:. The electrical network will then serve as a discrete model for the Dirichlet problem in the unit disc, where the Laplace equation takes the form of Kirchhoff's law. On the nodes on the boundary of the circle, an electrical potential (Dirichlet data) is defined, which induces an electric current (Neumann data) through the boundary nodes. The linear operator
2645:
1396:
1561:
1181:
2652:
In order to solve a PDE numerically in a domain, a discrete coordinate system must be introduced in this domain. If the domain has rotational symmetry and you want a grid consisting of rectangles, polar coordinates are a poor choice, since in the center of the circle it gives rise to triangles rather
2693:
The latter coordinate system is for instance suitable for dealing with
Dirichlet and Neumann problems. If the discrete coordinate system is interpreted as an undirected graph in the unit disc, it can be considered as a model for an electrical network. To every line segment in the graph is associated
2127:
662:
257:
1888:
1253:
1439:
766:
1898:
When one wants to solve the
Dirichlet problem in a domain with rotational symmetry, the usual thing to do is to use the method of separation of variables for partial differential equations for Laplace's equation in polar form. This means that you write
952:
1657:
545:
355:
2854:
1059:
855:
433:
i.e. the complex exponential function. From this follows that basic equations in harmonic and complex analysis will have the same simple form as in
Cartesian coordinates. This is not the case for polar coordinates.
556:
2414:
2247:
2567:
145:
1796:
1788:
2681:
is a positive integer. Use the complex exponential function to create a log-polar grid in the plane. The left half-plane is then mapped onto the unit disc, with the number of radii equal to
1756:
1391:{\displaystyle r{\frac {\partial \log R}{\partial r}}={\frac {\partial \Phi }{\partial \theta }},\ \ \ \ \ \ {\frac {\partial \log R}{\partial \theta }}=-r{\frac {\partial \Phi }{\partial r}},}
2685:. It can be even more advantageous to instead map the diagonals in these squares, which gives a discrete coordinate system in the unit disc consisting of spirals, see the figure to the right.
1967:
1959:
1724:
1692:
1556:{\displaystyle {\frac {\partial P}{\partial \rho }}={\frac {\partial \Phi }{\partial \theta }},\ \ \ \ \ \ {\frac {\partial P}{\partial \theta }}=-{\frac {\partial \Phi }{\partial \rho }}}
1245:
673:
428:
1051:
75:-axis) and the line through the origin and the point. The angular coordinate is the same as for polar coordinates, while the radial coordinate is transformed according to the rule
863:
2739:
2318:
1572:
2479:
802:
109:
456:
2772:
1431:
2712:
2653:
than rectangles. However, this can be remedied by introducing log-polar coordinates in the following way. Divide the plane into a grid of squares with side length 2
957:
has the same simple expression as in
Cartesian coordinates. This is true for all coordinate systems where the transformation to Cartesian coordinates is given by a
2671:
2150:
2443:
2276:
268:
961:. Thus, when considering Laplace's equation for a part of the plane with rotational symmetry, e.g. a circular disk, log-polar coordinates is the natural choice.
2610:
2590:
2499:
1176:{\displaystyle {\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}},\ \ \ \ \ \ {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}}
133:
2784:
807:
1401:
Just as in the case with
Laplace's equation, the simple form of Cartesian coordinates is recovered by changing polar into log-polar coordinates (let
657:{\displaystyle r{\frac {\partial }{\partial r}}\left(r{\frac {\partial u}{\partial r}}\right)+{\frac {\partial ^{2}u}{\partial \theta ^{2}}}=0}
2152:
is a constant. The first of these has constant coefficients and is easily solved. The second is a special case of Euler's equation
2973:
252:{\displaystyle {\begin{cases}\rho =\ln \left({\sqrt {x^{2}+y^{2}}}\right),\\\theta =\operatorname {atan2} (y,\,x).\end{cases}}}
1883:{\displaystyle \left({\frac {\partial }{\partial \rho }}+i{\frac {\partial }{\partial \theta }}\right)f(e^{\rho +i\theta })=0}
2612:. Thus, once again the natural choice for a domain with rotational symmetry is not polar, but rather log-polar, coordinates.
67:
in the plane consist of a pair of real numbers (ρ,θ), where ρ is the logarithm of the distance between a given point and the
2326:
1761:
2775:
2742:
2122:{\displaystyle {\begin{cases}\Theta ''(\theta )+\nu ^{2}\Theta (\theta )=0\\r^{2}R''(r)+rR'(r)-\nu ^{2}R(r)=0\end{cases}}}
1729:
2864:
Already at the end of the 1970s, applications for the discrete spiral coordinate system were given in image analysis (
1902:
2968:
1697:
1665:
2507:
761:{\displaystyle \left(r{\frac {\partial }{\partial r}}\right)^{2}u+{\frac {\partial ^{2}u}{\partial \theta ^{2}}}=0}
1189:
2158:
947:{\displaystyle {\frac {\partial ^{2}u}{\partial \rho ^{2}}}+{\frac {\partial ^{2}u}{\partial \theta ^{2}}}=0}
550:
in
Cartesian coordinates. Writing the same equation in polar coordinates gives the more complicated equation
382:
2890:
976:
2871:
Log-polar coordinates can also be used to construct fast methods for the Radon transform and its inverse.
2895:
2885:
2717:
136:
2281:
447:
2320:, but through use of log-polar radius, it can be changed into an equation with constant coefficients:
1652:{\displaystyle \left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right)f(x+iy)=0}
2636:
Discrete coordinate system in a circular disc that can easily be expressed in log-polar coordinates (
68:
1976:
277:
154:
2748:
In the case with the continuous disc, it follows that if the conductance is homogeneous, let's say
970:
540:{\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0}
45:
2572:
When solving the
Dirichlet problem in Cartesian coordinates, these are exactly the equations for
2865:
2448:
774:
81:
2751:
2938:
Fast
Inversion of the Radon Transform Using Log-polar Coordinates and Partial Back-Projections
2880:
1404:
958:
49:
41:
29:
2697:
2900:
350:{\displaystyle {\begin{cases}x=e^{\rho }\cos \theta ,\\y=e^{\rho }\sin \theta .\end{cases}}}
53:
2656:
2135:
2422:
2255:
2849:{\displaystyle \Lambda _{\gamma }^{2}+{\frac {\partial ^{2}\ }{\partial \theta ^{2}}}=0}
2595:
2575:
2484:
118:
2953:
2962:
850:{\displaystyle r{\frac {\partial }{\partial r}}={\frac {\partial }{\partial \rho }}}
1566:
The Cauchy–Riemann equations can also be written in one single equation as
1961:. Laplace's equation is then separated into two ordinary differential equations
262:
and the formulas for transformation from log-polar to
Cartesian coordinates are
17:
1053:
written in
Cartesian coordinates satisfies the Cauchy–Riemann equations:
2904:
2644:
2624:
Discrete coordinate system in a circular disc given by log-polar coordinates (
44:, which are usually used to describe domains in the plane with some sort of
33:
2632:
32:
in two dimensions, where a point is identified by two numbers, one for the
2620:
56:, the log-polar coordinates are more canonical than polar coordinates.
1247:, the Cauchy–Riemann equations take the more complicated form
135:
is the distance to the origin. The formulas for transformation from
2927:, Computer Graphics and Image Processing 11, 197–226 (1979).
2643:
2631:
2619:
37:
2745:, and depends on the topology and conductance of the network.
2278:
are constants. This equation is usually solved by the ansatz
2115:
343:
245:
2925:
Logarithmic Spiral Grids for Image Processing and Display
2787:
2754:
2720:
2700:
2659:
2648:
Part of a Mandelbrot fractal showing spiral behaviour
2598:
2578:
2510:
2487:
2451:
2425:
2409:{\displaystyle P''(\rho )+(c-1)P'(\rho )+dP(\rho )=0}
2329:
2284:
2258:
2161:
2138:
1970:
1905:
1799:
1783:{\displaystyle {\frac {\partial }{\partial \theta }}}
1764:
1732:
1700:
1668:
1575:
1442:
1407:
1256:
1192:
1062:
979:
866:
810:
777:
676:
559:
459:
385:
271:
148:
121:
84:
1790:
this equation can be written in the equivalent form
71:
and θ is the angle between a line of reference (the
1751:{\displaystyle {\frac {\partial }{\partial \rho }}}
1186:If the function instead is expressed in polar form
36:of the distance to a certain point, and one for an
2848:
2766:
2733:
2706:
2665:
2604:
2584:
2561:
2493:
2473:
2437:
2408:
2312:
2270:
2241:
2144:
2121:
1953:
1882:
1782:
1750:
1718:
1686:
1651:
1555:
1425:
1390:
1239:
1175:
1045:
946:
849:
796:
760:
656:
539:
422:
349:
251:
127:
103:
1954:{\displaystyle u(r,\theta )=R(r)\Theta (\theta )}
438:Some important equations in log-polar coordinates
40:. Log-polar coordinates are closely connected to
2741:from Dirichlet data to Neumann data is called a
857:so Laplace's equation in log-polar coordinates,
2940:, SIAM J. Appl. Math. 65, 818–837 (2005).
1719:{\displaystyle {\frac {\partial }{\partial y}}}
1687:{\displaystyle {\frac {\partial }{\partial x}}}
376:, the latter transformation can be written as
2562:{\displaystyle P''(\rho )-\nu ^{2}P(\rho )=0}
8:
1240:{\displaystyle f(re^{i\theta })=Re^{i\Phi }}
969:A similar situation arises when considering
2242:{\displaystyle r^{2}R''(r)+crR'(r)+dR(r)=0}
2831:
2813:
2806:
2797:
2792:
2786:
2753:
2725:
2719:
2699:
2658:
2597:
2577:
2535:
2509:
2486:
2465:
2450:
2424:
2328:
2304:
2283:
2257:
2166:
2160:
2137:
2088:
2035:
2003:
1971:
1969:
1904:
1856:
1826:
1805:
1798:
1765:
1763:
1733:
1731:
1701:
1699:
1669:
1667:
1602:
1581:
1574:
1533:
1507:
1466:
1443:
1441:
1406:
1365:
1330:
1289:
1260:
1255:
1228:
1206:
1191:
1153:
1127:
1086:
1063:
1061:
978:
929:
911:
904:
892:
874:
867:
865:
832:
814:
809:
788:
776:
743:
725:
718:
706:
686:
675:
639:
621:
614:
586:
563:
558:
522:
504:
497:
485:
467:
460:
458:
405:
384:
322:
290:
272:
270:
232:
192:
179:
173:
149:
147:
120:
95:
83:
60:Definition and coordinate transformations
2916:
423:{\displaystyle x+iy=e^{\rho +i\theta }}
139:to log-polar coordinates are given by
2419:When considering Laplace's equation,
1046:{\displaystyle f(x,y)=u(x,y)+iv(x,y)}
7:
2824:
2810:
2789:
2734:{\displaystyle \Lambda _{\gamma }}
2722:
2694:a conductance given by a function
2009:
1980:
1939:
1832:
1828:
1811:
1807:
1771:
1767:
1739:
1735:
1707:
1703:
1675:
1671:
1608:
1604:
1587:
1583:
1544:
1539:
1536:
1518:
1510:
1477:
1472:
1469:
1454:
1446:
1376:
1371:
1368:
1347:
1333:
1300:
1295:
1292:
1277:
1263:
1232:
1164:
1156:
1138:
1130:
1097:
1089:
1074:
1066:
922:
908:
885:
871:
838:
834:
820:
816:
736:
722:
692:
688:
632:
618:
597:
589:
569:
565:
515:
501:
478:
464:
14:
2778:satisfies the following equation
2313:{\displaystyle R(r)=r^{\lambda }}
2954:Non-Newtonian calculus website
2550:
2544:
2525:
2519:
2397:
2391:
2379:
2373:
2362:
2350:
2344:
2338:
2294:
2288:
2230:
2224:
2212:
2206:
2186:
2180:
2103:
2097:
2078:
2072:
2055:
2049:
2018:
2012:
1993:
1987:
1948:
1942:
1936:
1930:
1921:
1909:
1871:
1849:
1640:
1625:
1215:
1196:
1040:
1028:
1016:
1004:
995:
983:
965:Cauchy–Riemann equations
450:in two dimensions is given by
236:
223:
1:
2776:Dirichlet-to-Neumann operator
2743:Dirichlet-to-Neumann operator
2689:Dirichlet-to-Neumann operator
26:logarithmic polar coordinates
2474:{\displaystyle d=-\nu ^{2}}
797:{\displaystyle r=e^{\rho }}
771:However, from the relation
104:{\displaystyle r=e^{\rho }}
2990:
360:By using complex numbers (
2767:{\displaystyle \gamma =1}
973:. An analytical function
1893:
1426:{\displaystyle P=\log R}
442:
2891:Cylindrical coordinates
2707:{\displaystyle \gamma }
2974:Non-Newtonian calculus
2850:
2768:
2735:
2708:
2667:
2649:
2641:
2629:
2606:
2586:
2563:
2501:takes the simple form
2495:
2475:
2439:
2410:
2314:
2272:
2243:
2146:
2123:
1955:
1884:
1784:
1752:
1720:
1688:
1653:
1557:
1427:
1392:
1241:
1177:
1047:
948:
851:
798:
762:
658:
541:
424:
351:
253:
129:
105:
2896:Spherical coordinates
2886:Cartesian coordinates
2851:
2774:everywhere, then the
2769:
2736:
2709:
2668:
2647:
2635:
2623:
2607:
2587:
2564:
2496:
2476:
2440:
2411:
2315:
2273:
2244:
2147:
2124:
1956:
1885:
1785:
1753:
1721:
1689:
1654:
1558:
1428:
1393:
1242:
1178:
1048:
949:
852:
799:
763:
659:
542:
425:
352:
254:
137:Cartesian coordinates
130:
106:
65:Log-polar coordinates
22:log-polar coordinates
2936:Andersson, Fredrik,
2785:
2752:
2718:
2698:
2666:{\displaystyle \pi }
2657:
2596:
2576:
2508:
2485:
2481:so the equation for
2449:
2423:
2327:
2282:
2256:
2159:
2145:{\displaystyle \nu }
2136:
1968:
1903:
1797:
1762:
1730:
1698:
1666:
1573:
1440:
1405:
1254:
1190:
1060:
977:
971:analytical functions
864:
808:
775:
674:
557:
457:
383:
269:
146:
119:
82:
2802:
2438:{\displaystyle c=1}
2271:{\displaystyle c,d}
46:rotational symmetry
2969:Coordinate systems
2866:image registration
2846:
2788:
2764:
2731:
2704:
2663:
2650:
2642:
2630:
2602:
2582:
2559:
2491:
2471:
2435:
2406:
2310:
2268:
2239:
2142:
2119:
2114:
1951:
1880:
1780:
1748:
1716:
1684:
1649:
1553:
1423:
1388:
1237:
1173:
1043:
944:
847:
794:
758:
654:
537:
448:Laplace's equation
443:Laplace's equation
420:
347:
342:
249:
244:
125:
101:
2923:Weiman, Chaikin,
2901:log-polar mapping
2881:Polar coordinates
2838:
2821:
2616:Discrete geometry
2605:{\displaystyle y}
2585:{\displaystyle x}
2494:{\displaystyle r}
1839:
1818:
1778:
1746:
1714:
1682:
1615:
1594:
1551:
1525:
1506:
1503:
1500:
1497:
1494:
1491:
1484:
1461:
1383:
1354:
1329:
1326:
1323:
1320:
1317:
1314:
1307:
1284:
1171:
1145:
1126:
1123:
1120:
1117:
1114:
1111:
1104:
1081:
959:conformal mapping
936:
899:
845:
827:
750:
699:
646:
604:
576:
529:
492:
198:
128:{\displaystyle r}
42:polar coordinates
30:coordinate system
2981:
2941:
2934:
2928:
2921:
2855:
2853:
2852:
2847:
2839:
2837:
2836:
2835:
2822:
2819:
2818:
2817:
2807:
2801:
2796:
2773:
2771:
2770:
2765:
2740:
2738:
2737:
2732:
2730:
2729:
2713:
2711:
2710:
2705:
2672:
2670:
2669:
2664:
2640: = 25)
2628: = 25)
2611:
2609:
2608:
2603:
2591:
2589:
2588:
2583:
2568:
2566:
2565:
2560:
2540:
2539:
2518:
2500:
2498:
2497:
2492:
2480:
2478:
2477:
2472:
2470:
2469:
2444:
2442:
2441:
2436:
2415:
2413:
2412:
2407:
2372:
2337:
2319:
2317:
2316:
2311:
2309:
2308:
2277:
2275:
2274:
2269:
2248:
2246:
2245:
2240:
2205:
2179:
2171:
2170:
2151:
2149:
2148:
2143:
2128:
2126:
2125:
2120:
2118:
2117:
2093:
2092:
2071:
2048:
2040:
2039:
2008:
2007:
1986:
1960:
1958:
1957:
1952:
1894:Euler's equation
1889:
1887:
1886:
1881:
1870:
1869:
1845:
1841:
1840:
1838:
1827:
1819:
1817:
1806:
1789:
1787:
1786:
1781:
1779:
1777:
1766:
1757:
1755:
1754:
1749:
1747:
1745:
1734:
1725:
1723:
1722:
1717:
1715:
1713:
1702:
1693:
1691:
1690:
1685:
1683:
1681:
1670:
1658:
1656:
1655:
1650:
1621:
1617:
1616:
1614:
1603:
1595:
1593:
1582:
1562:
1560:
1559:
1554:
1552:
1550:
1542:
1534:
1526:
1524:
1516:
1508:
1504:
1501:
1498:
1495:
1492:
1489:
1485:
1483:
1475:
1467:
1462:
1460:
1452:
1444:
1432:
1430:
1429:
1424:
1397:
1395:
1394:
1389:
1384:
1382:
1374:
1366:
1355:
1353:
1345:
1331:
1327:
1324:
1321:
1318:
1315:
1312:
1308:
1306:
1298:
1290:
1285:
1283:
1275:
1261:
1246:
1244:
1243:
1238:
1236:
1235:
1214:
1213:
1182:
1180:
1179:
1174:
1172:
1170:
1162:
1154:
1146:
1144:
1136:
1128:
1124:
1121:
1118:
1115:
1112:
1109:
1105:
1103:
1095:
1087:
1082:
1080:
1072:
1064:
1052:
1050:
1049:
1044:
953:
951:
950:
945:
937:
935:
934:
933:
920:
916:
915:
905:
900:
898:
897:
896:
883:
879:
878:
868:
856:
854:
853:
848:
846:
844:
833:
828:
826:
815:
804:it follows that
803:
801:
800:
795:
793:
792:
767:
765:
764:
759:
751:
749:
748:
747:
734:
730:
729:
719:
711:
710:
705:
701:
700:
698:
687:
667:or equivalently
663:
661:
660:
655:
647:
645:
644:
643:
630:
626:
625:
615:
610:
606:
605:
603:
595:
587:
577:
575:
564:
546:
544:
543:
538:
530:
528:
527:
526:
513:
509:
508:
498:
493:
491:
490:
489:
476:
472:
471:
461:
429:
427:
426:
421:
419:
418:
356:
354:
353:
348:
346:
345:
327:
326:
295:
294:
258:
256:
255:
250:
248:
247:
203:
199:
197:
196:
184:
183:
174:
134:
132:
131:
126:
110:
108:
107:
102:
100:
99:
54:complex analysis
48:. In areas like
2989:
2988:
2984:
2983:
2982:
2980:
2979:
2978:
2959:
2958:
2950:
2945:
2944:
2935:
2931:
2922:
2918:
2913:
2877:
2862:
2827:
2823:
2809:
2808:
2783:
2782:
2750:
2749:
2721:
2716:
2715:
2696:
2695:
2691:
2655:
2654:
2618:
2594:
2593:
2574:
2573:
2531:
2511:
2506:
2505:
2483:
2482:
2461:
2447:
2446:
2421:
2420:
2365:
2330:
2325:
2324:
2300:
2280:
2279:
2254:
2253:
2198:
2172:
2162:
2157:
2156:
2134:
2133:
2113:
2112:
2084:
2064:
2041:
2031:
2028:
2027:
1999:
1979:
1972:
1966:
1965:
1901:
1900:
1896:
1852:
1831:
1810:
1804:
1800:
1795:
1794:
1770:
1760:
1759:
1738:
1728:
1727:
1706:
1696:
1695:
1674:
1664:
1663:
1607:
1586:
1580:
1576:
1571:
1570:
1543:
1535:
1517:
1509:
1476:
1468:
1453:
1445:
1438:
1437:
1403:
1402:
1375:
1367:
1346:
1332:
1299:
1291:
1276:
1262:
1252:
1251:
1224:
1202:
1188:
1187:
1163:
1155:
1137:
1129:
1096:
1088:
1073:
1065:
1058:
1057:
975:
974:
967:
925:
921:
907:
906:
888:
884:
870:
869:
862:
861:
837:
819:
806:
805:
784:
773:
772:
739:
735:
721:
720:
691:
682:
678:
677:
672:
671:
635:
631:
617:
616:
596:
588:
582:
578:
568:
555:
554:
518:
514:
500:
499:
481:
477:
463:
462:
455:
454:
445:
440:
401:
381:
380:
341:
340:
318:
309:
308:
286:
273:
267:
266:
243:
242:
208:
207:
188:
175:
169:
150:
144:
143:
117:
116:
91:
80:
79:
62:
12:
11:
5:
2987:
2985:
2977:
2976:
2971:
2961:
2960:
2957:
2956:
2949:
2948:External links
2946:
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2942:
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2915:
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2912:
2909:
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2907:
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2893:
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2860:Image analysis
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2077:
2074:
2070:
2067:
2063:
2060:
2057:
2054:
2051:
2047:
2044:
2038:
2034:
2030:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2006:
2002:
1998:
1995:
1992:
1989:
1985:
1982:
1978:
1977:
1975:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1895:
1892:
1891:
1890:
1879:
1876:
1873:
1868:
1865:
1862:
1859:
1855:
1851:
1848:
1844:
1837:
1834:
1830:
1825:
1822:
1816:
1813:
1809:
1803:
1776:
1773:
1769:
1744:
1741:
1737:
1712:
1709:
1705:
1680:
1677:
1673:
1662:By expressing
1660:
1659:
1648:
1645:
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1613:
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1538:
1532:
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1515:
1512:
1488:
1482:
1479:
1474:
1471:
1465:
1459:
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1422:
1419:
1416:
1413:
1410:
1399:
1398:
1387:
1381:
1378:
1373:
1370:
1364:
1361:
1358:
1352:
1349:
1344:
1341:
1338:
1335:
1311:
1305:
1302:
1297:
1294:
1288:
1282:
1279:
1274:
1271:
1268:
1265:
1259:
1234:
1231:
1227:
1223:
1220:
1217:
1212:
1209:
1205:
1201:
1198:
1195:
1184:
1183:
1169:
1166:
1161:
1158:
1152:
1149:
1143:
1140:
1135:
1132:
1108:
1102:
1099:
1094:
1091:
1085:
1079:
1076:
1071:
1068:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
966:
963:
955:
954:
943:
940:
932:
928:
924:
919:
914:
910:
903:
895:
891:
887:
882:
877:
873:
843:
840:
836:
831:
825:
822:
818:
813:
791:
787:
783:
780:
769:
768:
757:
754:
746:
742:
738:
733:
728:
724:
717:
714:
709:
704:
697:
694:
690:
685:
681:
665:
664:
653:
650:
642:
638:
634:
629:
624:
620:
613:
609:
602:
599:
594:
591:
585:
581:
574:
571:
567:
562:
548:
547:
536:
533:
525:
521:
517:
512:
507:
503:
496:
488:
484:
480:
475:
470:
466:
444:
441:
439:
436:
431:
430:
417:
414:
411:
408:
404:
400:
397:
394:
391:
388:
368:) =
358:
357:
344:
339:
336:
333:
330:
325:
321:
317:
314:
311:
310:
307:
304:
301:
298:
293:
289:
285:
282:
279:
278:
276:
260:
259:
246:
241:
238:
235:
231:
228:
225:
222:
219:
216:
213:
210:
209:
206:
202:
195:
191:
187:
182:
178:
172:
168:
165:
162:
159:
156:
155:
153:
124:
113:
112:
98:
94:
90:
87:
61:
58:
13:
10:
9:
6:
4:
3:
2:
2986:
2975:
2972:
2970:
2967:
2966:
2964:
2955:
2952:
2951:
2947:
2939:
2933:
2930:
2926:
2920:
2917:
2910:
2906:
2902:
2899:
2897:
2894:
2892:
2889:
2887:
2884:
2882:
2879:
2878:
2874:
2872:
2869:
2867:
2859:
2843:
2840:
2832:
2828:
2814:
2803:
2798:
2793:
2781:
2780:
2779:
2777:
2761:
2758:
2755:
2746:
2744:
2726:
2701:
2688:
2686:
2684:
2680:
2676:
2660:
2646:
2639:
2634:
2627:
2622:
2615:
2613:
2599:
2579:
2556:
2553:
2547:
2541:
2536:
2532:
2528:
2522:
2515:
2512:
2504:
2503:
2502:
2488:
2466:
2462:
2458:
2455:
2452:
2432:
2429:
2426:
2403:
2400:
2394:
2388:
2385:
2382:
2376:
2369:
2366:
2359:
2356:
2353:
2347:
2341:
2334:
2331:
2323:
2322:
2321:
2305:
2301:
2297:
2291:
2285:
2265:
2262:
2259:
2236:
2233:
2227:
2221:
2218:
2215:
2209:
2202:
2199:
2195:
2192:
2189:
2183:
2176:
2173:
2167:
2163:
2155:
2154:
2153:
2139:
2109:
2106:
2100:
2094:
2089:
2085:
2081:
2075:
2068:
2065:
2061:
2058:
2052:
2045:
2042:
2036:
2032:
2024:
2021:
2015:
2004:
2000:
1996:
1990:
1983:
1973:
1964:
1963:
1962:
1945:
1933:
1927:
1924:
1918:
1915:
1912:
1906:
1877:
1874:
1866:
1863:
1860:
1857:
1853:
1846:
1842:
1835:
1823:
1820:
1814:
1801:
1793:
1792:
1791:
1774:
1742:
1710:
1678:
1646:
1643:
1637:
1634:
1631:
1628:
1622:
1618:
1611:
1599:
1596:
1590:
1577:
1569:
1568:
1567:
1547:
1530:
1527:
1521:
1513:
1486:
1480:
1463:
1457:
1449:
1436:
1435:
1434:
1420:
1417:
1414:
1411:
1408:
1385:
1379:
1362:
1359:
1356:
1350:
1342:
1339:
1336:
1309:
1303:
1286:
1280:
1272:
1269:
1266:
1257:
1250:
1249:
1248:
1229:
1225:
1221:
1218:
1210:
1207:
1203:
1199:
1193:
1167:
1159:
1150:
1147:
1141:
1133:
1106:
1100:
1092:
1083:
1077:
1069:
1056:
1055:
1054:
1037:
1034:
1031:
1025:
1022:
1019:
1013:
1010:
1007:
1001:
998:
992:
989:
986:
980:
972:
964:
962:
960:
941:
938:
930:
926:
917:
912:
901:
893:
889:
880:
875:
860:
859:
858:
841:
829:
823:
811:
789:
785:
781:
778:
755:
752:
744:
740:
731:
726:
715:
712:
707:
702:
695:
683:
679:
670:
669:
668:
651:
648:
640:
636:
627:
622:
611:
607:
600:
592:
583:
579:
572:
560:
553:
552:
551:
534:
531:
523:
519:
510:
505:
494:
486:
482:
473:
468:
453:
452:
451:
449:
437:
435:
415:
412:
409:
406:
402:
398:
395:
392:
389:
386:
379:
378:
377:
375:
372: +
371:
367:
363:
337:
334:
331:
328:
323:
319:
315:
312:
305:
302:
299:
296:
291:
287:
283:
280:
274:
265:
264:
263:
239:
233:
229:
226:
220:
217:
214:
211:
204:
200:
193:
189:
185:
180:
176:
170:
166:
163:
160:
157:
151:
142:
141:
140:
138:
122:
96:
92:
88:
85:
78:
77:
76:
74:
70:
66:
59:
57:
55:
51:
47:
43:
39:
35:
31:
27:
23:
19:
2937:
2932:
2924:
2919:
2870:
2863:
2747:
2692:
2682:
2678:
2674:
2651:
2637:
2625:
2571:
2418:
2251:
2131:
1897:
1726:in terms of
1661:
1565:
1400:
1185:
968:
956:
770:
666:
549:
446:
432:
373:
369:
365:
361:
359:
261:
114:
72:
64:
63:
25:
21:
15:
18:mathematics
2963:Categories
2911:References
2905:Retinotopy
2829:θ
2825:∂
2811:∂
2794:γ
2790:Λ
2756:γ
2727:γ
2723:Λ
2702:γ
2661:π
2548:ρ
2533:ν
2529:−
2523:ρ
2463:ν
2459:−
2395:ρ
2377:ρ
2357:−
2342:ρ
2306:λ
2140:ν
2086:ν
2082:−
2016:θ
2010:Θ
2001:ν
1991:θ
1981:Θ
1946:θ
1940:Θ
1919:θ
1867:θ
1858:ρ
1836:θ
1833:∂
1829:∂
1815:ρ
1812:∂
1808:∂
1775:θ
1772:∂
1768:∂
1743:ρ
1740:∂
1736:∂
1708:∂
1704:∂
1676:∂
1672:∂
1609:∂
1605:∂
1588:∂
1584:∂
1548:ρ
1545:∂
1540:Φ
1537:∂
1531:−
1522:θ
1519:∂
1511:∂
1481:θ
1478:∂
1473:Φ
1470:∂
1458:ρ
1455:∂
1447:∂
1418:
1377:∂
1372:Φ
1369:∂
1360:−
1351:θ
1348:∂
1340:
1334:∂
1304:θ
1301:∂
1296:Φ
1293:∂
1278:∂
1270:
1264:∂
1233:Φ
1211:θ
1165:∂
1157:∂
1151:−
1139:∂
1131:∂
1098:∂
1090:∂
1075:∂
1067:∂
927:θ
923:∂
909:∂
890:ρ
886:∂
872:∂
842:ρ
839:∂
835:∂
821:∂
817:∂
790:ρ
741:θ
737:∂
723:∂
693:∂
689:∂
637:θ
633:∂
619:∂
598:∂
590:∂
570:∂
566:∂
516:∂
502:∂
479:∂
465:∂
416:θ
407:ρ
335:θ
332:
324:ρ
303:θ
300:
292:ρ
221:
212:θ
167:
158:ρ
97:ρ
34:logarithm
2875:See also
2677:, where
2516:″
2370:′
2335:″
2203:′
2177:″
2069:′
2046:″
1984:″
50:harmonic
364:,
28:) is a
2820:
2252:where
2132:where
1505:
1502:
1499:
1496:
1493:
1490:
1328:
1325:
1322:
1319:
1316:
1313:
1125:
1122:
1119:
1116:
1113:
1110:
115:where
69:origin
218:atan2
38:angle
2592:and
2445:and
1758:and
1694:and
52:and
24:(or
2903:in
1433:):
1415:log
1337:log
1267:log
329:sin
297:cos
16:In
2965::
374:iy
164:ln
20:,
2844:0
2841:=
2833:2
2815:2
2804:+
2799:2
2762:1
2759:=
2683:n
2679:n
2675:n
2673:/
2638:n
2626:n
2600:y
2580:x
2557:0
2554:=
2551:)
2545:(
2542:P
2537:2
2526:)
2520:(
2513:P
2489:r
2467:2
2456:=
2453:d
2433:1
2430:=
2427:c
2404:0
2401:=
2398:)
2392:(
2389:P
2386:d
2383:+
2380:)
2374:(
2367:P
2363:)
2360:1
2354:c
2351:(
2348:+
2345:)
2339:(
2332:P
2302:r
2298:=
2295:)
2292:r
2289:(
2286:R
2266:d
2263:,
2260:c
2237:0
2234:=
2231:)
2228:r
2225:(
2222:R
2219:d
2216:+
2213:)
2210:r
2207:(
2200:R
2196:r
2193:c
2190:+
2187:)
2184:r
2181:(
2174:R
2168:2
2164:r
2110:0
2107:=
2104:)
2101:r
2098:(
2095:R
2090:2
2079:)
2076:r
2073:(
2066:R
2062:r
2059:+
2056:)
2053:r
2050:(
2043:R
2037:2
2033:r
2025:0
2022:=
2019:)
2013:(
2005:2
1997:+
1994:)
1988:(
1974:{
1949:)
1943:(
1937:)
1934:r
1931:(
1928:R
1925:=
1922:)
1916:,
1913:r
1910:(
1907:u
1878:0
1875:=
1872:)
1864:i
1861:+
1854:e
1850:(
1847:f
1843:)
1824:i
1821:+
1802:(
1711:y
1679:x
1647:0
1644:=
1641:)
1638:y
1635:i
1632:+
1629:x
1626:(
1623:f
1619:)
1612:y
1600:i
1597:+
1591:x
1578:(
1528:=
1514:P
1487:,
1464:=
1450:P
1421:R
1412:=
1409:P
1386:,
1380:r
1363:r
1357:=
1343:R
1310:,
1287:=
1281:r
1273:R
1258:r
1230:i
1226:e
1222:R
1219:=
1216:)
1208:i
1204:e
1200:r
1197:(
1194:f
1168:x
1160:v
1148:=
1142:y
1134:u
1107:,
1101:y
1093:v
1084:=
1078:x
1070:u
1041:)
1038:y
1035:,
1032:x
1029:(
1026:v
1023:i
1020:+
1017:)
1014:y
1011:,
1008:x
1005:(
1002:u
999:=
996:)
993:y
990:,
987:x
984:(
981:f
942:0
939:=
931:2
918:u
913:2
902:+
894:2
881:u
876:2
830:=
824:r
812:r
786:e
782:=
779:r
756:0
753:=
745:2
732:u
727:2
716:+
713:u
708:2
703:)
696:r
684:r
680:(
652:0
649:=
641:2
628:u
623:2
612:+
608:)
601:r
593:u
584:r
580:(
573:r
561:r
535:0
532:=
524:2
520:y
511:u
506:2
495:+
487:2
483:x
474:u
469:2
413:i
410:+
403:e
399:=
396:y
393:i
390:+
387:x
370:x
366:y
362:x
338:.
320:e
316:=
313:y
306:,
288:e
284:=
281:x
275:{
240:.
237:)
234:x
230:,
227:y
224:(
215:=
205:,
201:)
194:2
190:y
186:+
181:2
177:x
171:(
161:=
152:{
123:r
111:.
93:e
89:=
86:r
73:x
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