2301:
1613:
1512:
1563:
1546:
1529:
1491:
1470:
1447:
1424:
337:
541:
2290:
2133:
141:
348:
2139:
1982:
651:
332:{\displaystyle {\begin{aligned}&x(\theta )=(R+r)\cos \theta \ -r\cos \left({\frac {R+r}{r}}\theta \right)\\&y(\theta )=(R+r)\sin \theta \ -r\sin \left({\frac {R+r}{r}}\theta \right)\end{aligned}}}
353:
146:
1360:
1950:
1005:
891:
536:{\displaystyle {\begin{aligned}&x(\theta )=r(k+1)\cos \theta -r\cos \left((k+1)\theta \right)\\&y(\theta )=r(k+1)\sin \theta -r\sin \left((k+1)\theta \right).\end{aligned}}}
1744:
1828:
1786:
1867:
1284:
702:
939:
1911:
1891:
1698:
1658:
1096:
1249:
129:
1411:
750:
1974:
1678:
1638:
1386:
1304:
1212:
1120:
1058:
1028:
819:
799:
779:
724:
100:
2285:{\displaystyle y=\left(R+r\right)\sin \theta -r\sin \left(\theta +\alpha \right)=\left(R+r\right)\sin \theta -r\sin \left({\frac {R+r}{r}}\theta \right)}
2128:{\displaystyle x=\left(R+r\right)\cos \theta -r\cos \left(\theta +\alpha \right)=\left(R+r\right)\cos \theta -r\cos \left({\frac {R+r}{r}}\theta \right)}
2509:
552:
20:
2397:
2492:
2414:
1312:
2550:
2317:
1919:
2545:
947:
2514:
827:
1709:
1490:
2332:
1797:
1755:
1601:
63:
1840:
1469:
1257:
1099:
132:
1511:
1446:
1423:
2500:
1562:
1545:
1528:
663:
2530:
1034:
897:
2504:
2470:
2393:
2389:
2382:
2337:
1215:
1896:
1876:
1683:
1643:
2362:
68:
2300:
1067:
19:
1225:
1061:
105:
1393:
732:
1959:
1663:
1623:
1612:
1371:
1289:
1197:
1105:
1043:
1013:
804:
784:
764:
709:
85:
2435:
2539:
2357:
74:
An epicycloid with a minor radius (R2) of 0 is a circle. This is a degenerate form.
2352:
2520:
1749:
By the definition of angle (which is the rate arc over radius), then we have that
2488:
2473:
2347:
2342:
1579:
67:—which rolls without slipping around a fixed circle. It is a particular kind of
54:
2531:
Historical note on the application of the epicycloid to the form of Gear Teeth
2367:
2425:
Epicycloids and
Blaschke products by Chunlei Cao, Alastair Fletcher, Zhuan Ye
2478:
1219:
646:{\displaystyle z(\theta )=r\left((k+1)e^{i\theta }-e^{i(k+1)\theta }\right)}
2327:
2309:
1590:
1586:
1460:
1437:
57:
produced by tracing the path of a chosen point on the circumference of a
42:
2525:
2448:
2322:
2305:
1597:
16:
Plane curve traced by a point on a circle rolled around another circle
58:
1703:
Since there is no sliding between the two cycles, then we have that
546:
This can be written in a more concise form using complex numbers as
2424:
2299:
1611:
23:
The red curve is an epicycloid traced as the small circle (radius
1700:
is the angle from the starting point to the tangential point.
1222:
of the space between the larger circle and a circle of radius
761:(Assuming the initial point lies on the larger circle.) When
1660:
is the angle from the tangential point to the moving point
1030:
is a positive integer, then the curve is closed, and has
2521:
Animation of
Epicycloids, Pericycloids and Hypocycloids
2142:
1985:
1962:
1922:
1899:
1879:
1843:
1800:
1758:
1712:
1686:
1666:
1646:
1626:
1396:
1374:
1315:
1292:
1260:
1228:
1200:
1108:
1070:
1046:
1016:
950:
900:
830:
807:
787:
767:
735:
712:
666:
555:
351:
144:
108:
88:
30:
rolls around the outside of the large circle (radius
1007:
larger in area than the original stationary circle.
2381:
2284:
2127:
1968:
1956:From the figure, we see the position of the point
1944:
1905:
1885:
1861:
1822:
1780:
1738:
1692:
1672:
1652:
1632:
1405:
1380:
1354:
1298:
1278:
1243:
1206:
1114:
1090:
1052:
1022:
999:
933:
885:
813:
793:
773:
744:
718:
696:
645:
535:
331:
123:
94:
1834:From these two conditions, we get the identity
1355:{\displaystyle R\leq {\overline {OP}}\leq R+2r}
1945:{\displaystyle \alpha ={\frac {R}{r}}\theta }
8:
1873:By calculating, we get the relation between
2436:Epicycloid Evolute - from Wolfram MathWorld
1218:, then the curve never closes, and forms a
1168:total rotations of outer rolling circle =
1133:complete the 1st repeating pattern :
1000:{\displaystyle {\frac {(k+1)(k+2)}{k^{2}}}}
1306:on the small circle varies up and down as
2256:
2141:
2099:
1984:
1961:
1929:
1921:
1898:
1878:
1842:
1805:
1799:
1763:
1757:
1730:
1717:
1711:
1685:
1665:
1645:
1625:
1395:
1373:
1322:
1314:
1291:
1261:
1259:
1227:
1199:
1107:
1080:
1069:
1045:
1015:
989:
951:
949:
899:
874:
829:
806:
786:
766:
734:
711:
665:
614:
598:
554:
352:
350:
299:
209:
145:
143:
107:
87:
1124:
18:
2510:MacTutor History of Mathematics Archive
2407:
1416:
886:{\displaystyle A=(k+1)(k+2)\pi r^{2},}
135:for the curve can be given by either:
1184:Count the animation rotations to see
7:
1578:The epicycloid is a special kind of
1739:{\displaystyle \ell _{R}=\ell _{r}}
102:, and the larger circle has radius
2493:The Wolfram Demonstrations Project
1823:{\displaystyle \ell _{r}=\alpha r}
1781:{\displaystyle \ell _{R}=\theta R}
14:
2384:A catalog of special plane curves
1862:{\displaystyle \theta R=\alpha r}
82:If the smaller circle has radius
1561:
1544:
1527:
1510:
1489:
1468:
1445:
1422:
1279:{\displaystyle {\overline {OP}}}
944:It means that the epicycloid is
781:is a positive integer, the area
2388:. Dover Publications. pp.
1620:We assume that the position of
1585:An epicycle with one cusp is a
981:
969:
966:
954:
922:
910:
864:
852:
849:
837:
706:the smaller circle has radius
688:
673:
630:
618:
591:
579:
565:
559:
515:
503:
474:
462:
453:
447:
428:
416:
387:
375:
366:
360:
267:
255:
249:
243:
177:
165:
159:
153:
1:
1976:on the small circle clearly.
1286:from the origin to the point
729:the larger circle has radius
2447:Pietrocola, Giorgio (2005).
2304:Animated gif with turtle in
1413:= diameter of small circle .
1388:= radius of large circle and
1332:
1271:
697:{\displaystyle \theta \in ,}
2380:J. Dennis Lawrence (1972).
2567:
2318:List of periodic functions
1640:is what we want to solve,
934:{\displaystyle s=8(k+1)r.}
2515:University of St Andrews
1906:{\displaystyle \theta }
1886:{\displaystyle \alpha }
1693:{\displaystyle \theta }
1653:{\displaystyle \alpha }
1128:To close the curve and
1037:(i.e., sharp corners).
821:of this epicycloid are
2313:
2286:
2129:
1970:
1946:
1907:
1887:
1863:
1824:
1782:
1740:
1694:
1674:
1654:
1634:
1617:
1596:An epicycloid and its
1407:
1382:
1356:
1300:
1280:
1245:
1208:
1116:
1092:
1054:
1024:
1001:
935:
887:
815:
795:
775:
746:
720:
698:
647:
537:
333:
125:
96:
38:
2333:Deferent and epicycle
2303:
2287:
2130:
1971:
1947:
1908:
1888:
1864:
1825:
1783:
1741:
1695:
1675:
1655:
1635:
1615:
1408:
1383:
1357:
1301:
1281:
1246:
1209:
1117:
1102:, then the curve has
1093:
1091:{\displaystyle k=p/q}
1055:
1025:
1002:
936:
888:
816:
796:
776:
747:
721:
699:
648:
538:
334:
126:
97:
22:
2526:Spirograph -- GeoFun
2501:Robertson, Edmund F.
2140:
1983:
1960:
1920:
1897:
1877:
1841:
1798:
1756:
1710:
1684:
1664:
1644:
1624:
1394:
1372:
1313:
1290:
1258:
1244:{\displaystyle R+2r}
1226:
1198:
1106:
1100:irreducible fraction
1068:
1044:
1014:
948:
898:
828:
805:
785:
765:
733:
710:
664:
553:
349:
142:
133:parametric equations
124:{\displaystyle R=kr}
106:
86:
2499:O'Connor, John J.;
2491:" by Michael Ford,
1418:Epicycloid examples
757:Area and Arc Length
2471:Weisstein, Eric W.
2314:
2282:
2125:
1966:
1942:
1903:
1883:
1859:
1820:
1778:
1736:
1690:
1670:
1650:
1630:
1618:
1406:{\displaystyle 2r}
1403:
1378:
1352:
1296:
1276:
1241:
1204:
1112:
1088:
1050:
1020:
997:
931:
883:
811:
791:
771:
745:{\displaystyle kr}
742:
716:
694:
643:
533:
531:
329:
327:
121:
92:
39:
2551:Roulettes (curve)
2399:978-0-486-60288-2
2390:161, 168–170, 175
2338:Epicyclic gearing
2272:
2115:
1969:{\displaystyle p}
1937:
1673:{\displaystyle p}
1633:{\displaystyle p}
1589:, two cusps is a
1381:{\displaystyle R}
1335:
1299:{\displaystyle p}
1274:
1216:irrational number
1207:{\displaystyle k}
1182:
1181:
1115:{\displaystyle p}
1053:{\displaystyle k}
1023:{\displaystyle k}
995:
814:{\displaystyle s}
794:{\displaystyle A}
774:{\displaystyle k}
719:{\displaystyle r}
315:
281:
225:
191:
95:{\displaystyle r}
2558:
2546:Algebraic curves
2517:
2484:
2483:
2457:
2456:
2444:
2438:
2433:
2427:
2422:
2416:
2412:
2403:
2387:
2363:Roulette (curve)
2291:
2289:
2288:
2283:
2281:
2277:
2273:
2268:
2257:
2229:
2225:
2207:
2203:
2167:
2163:
2134:
2132:
2131:
2126:
2124:
2120:
2116:
2111:
2100:
2072:
2068:
2050:
2046:
2010:
2006:
1975:
1973:
1972:
1967:
1951:
1949:
1948:
1943:
1938:
1930:
1912:
1910:
1909:
1904:
1892:
1890:
1889:
1884:
1868:
1866:
1865:
1860:
1829:
1827:
1826:
1821:
1810:
1809:
1787:
1785:
1784:
1779:
1768:
1767:
1745:
1743:
1742:
1737:
1735:
1734:
1722:
1721:
1699:
1697:
1696:
1691:
1679:
1677:
1676:
1671:
1659:
1657:
1656:
1651:
1639:
1637:
1636:
1631:
1616:sketch for proof
1573:
1565:
1556:
1548:
1539:
1531:
1522:
1514:
1501:
1493:
1480:
1472:
1457:
1449:
1434:
1426:
1412:
1410:
1409:
1404:
1387:
1385:
1384:
1379:
1361:
1359:
1358:
1353:
1336:
1331:
1323:
1305:
1303:
1302:
1297:
1285:
1283:
1282:
1277:
1275:
1270:
1262:
1250:
1248:
1247:
1242:
1213:
1211:
1210:
1205:
1191:
1187:
1177:
1162:
1158:
1147:
1143:
1125:
1121:
1119:
1118:
1113:
1097:
1095:
1094:
1089:
1084:
1059:
1057:
1056:
1051:
1033:
1029:
1027:
1026:
1021:
1006:
1004:
1003:
998:
996:
994:
993:
984:
952:
940:
938:
937:
932:
892:
890:
889:
884:
879:
878:
820:
818:
817:
812:
800:
798:
797:
792:
780:
778:
777:
772:
751:
749:
748:
743:
725:
723:
722:
717:
703:
701:
700:
695:
652:
650:
649:
644:
642:
638:
637:
636:
606:
605:
542:
540:
539:
534:
532:
525:
521:
442:
438:
434:
355:
338:
336:
335:
330:
328:
324:
320:
316:
311:
300:
279:
238:
234:
230:
226:
221:
210:
189:
148:
130:
128:
127:
122:
101:
99:
98:
93:
36:
29:
2566:
2565:
2561:
2560:
2559:
2557:
2556:
2555:
2536:
2535:
2498:
2469:
2468:
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2441:
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2423:
2419:
2413:
2409:
2400:
2379:
2376:
2298:
2258:
2255:
2251:
2215:
2211:
2193:
2189:
2153:
2149:
2138:
2137:
2101:
2098:
2094:
2058:
2054:
2036:
2032:
1996:
1992:
1981:
1980:
1958:
1957:
1918:
1917:
1895:
1894:
1875:
1874:
1839:
1838:
1801:
1796:
1795:
1759:
1754:
1753:
1726:
1713:
1708:
1707:
1682:
1681:
1662:
1661:
1642:
1641:
1622:
1621:
1610:
1574:
1568:
1566:
1557:
1551:
1549:
1540:
1534:
1532:
1523:
1517:
1515:
1506:
1496:
1494:
1485:
1475:
1473:
1464:
1452:
1450:
1441:
1429:
1427:
1392:
1391:
1370:
1369:
1324:
1311:
1310:
1288:
1287:
1263:
1256:
1255:
1224:
1223:
1196:
1195:
1189:
1185:
1169:
1160:
1153:
1145:
1138:
1104:
1103:
1066:
1065:
1062:rational number
1042:
1041:
1031:
1012:
1011:
985:
953:
946:
945:
896:
895:
870:
826:
825:
803:
802:
801:and arc length
783:
782:
763:
762:
759:
731:
730:
708:
707:
662:
661:
610:
594:
578:
574:
551:
550:
530:
529:
502:
498:
440:
439:
415:
411:
347:
346:
326:
325:
301:
298:
294:
236:
235:
211:
208:
204:
140:
139:
104:
103:
84:
83:
80:
31:
24:
17:
12:
11:
5:
2564:
2562:
2554:
2553:
2548:
2538:
2537:
2534:
2533:
2528:
2523:
2518:
2496:
2485:
2464:
2463:External links
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497:
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479:
476:
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470:
467:
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458:
455:
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449:
446:
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441:
437:
433:
430:
427:
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421:
418:
414:
410:
407:
404:
401:
398:
395:
392:
389:
386:
383:
380:
377:
374:
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368:
365:
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359:
356:
354:
340:
339:
323:
319:
314:
310:
307:
304:
297:
293:
290:
287:
284:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
248:
245:
242:
239:
237:
233:
229:
224:
220:
217:
214:
207:
203:
200:
197:
194:
188:
185:
182:
179:
176:
173:
170:
167:
164:
161:
158:
155:
152:
149:
147:
120:
117:
114:
111:
91:
79:
76:
15:
13:
10:
9:
6:
4:
3:
2:
2563:
2552:
2549:
2547:
2544:
2543:
2541:
2532:
2529:
2527:
2524:
2522:
2519:
2516:
2512:
2511:
2506:
2502:
2497:
2494:
2490:
2486:
2481:
2480:
2475:
2472:
2467:
2466:
2462:
2454:
2450:
2449:"Tartapelago"
2443:
2440:
2437:
2432:
2429:
2426:
2421:
2418:
2415:
2411:
2408:
2401:
2395:
2391:
2386:
2385:
2378:
2377:
2373:
2369:
2366:
2364:
2361:
2359:
2358:Multibrot set
2356:
2354:
2351:
2349:
2346:
2344:
2341:
2339:
2336:
2334:
2331:
2329:
2326:
2324:
2321:
2319:
2316:
2315:
2311:
2307:
2302:
2295:
2278:
2274:
2269:
2265:
2262:
2259:
2252:
2248:
2245:
2242:
2239:
2236:
2233:
2230:
2226:
2222:
2219:
2216:
2212:
2208:
2204:
2200:
2197:
2194:
2190:
2186:
2183:
2180:
2177:
2174:
2171:
2168:
2164:
2160:
2157:
2154:
2150:
2146:
2143:
2136:
2121:
2117:
2112:
2108:
2105:
2102:
2095:
2091:
2088:
2085:
2082:
2079:
2076:
2073:
2069:
2065:
2062:
2059:
2055:
2051:
2047:
2043:
2040:
2037:
2033:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2007:
2003:
2000:
1997:
1993:
1989:
1986:
1979:
1978:
1977:
1963:
1939:
1934:
1931:
1926:
1923:
1916:
1915:
1914:
1900:
1880:
1856:
1853:
1850:
1847:
1844:
1837:
1836:
1835:
1817:
1814:
1811:
1806:
1802:
1794:
1793:
1792:
1775:
1772:
1769:
1764:
1760:
1752:
1751:
1750:
1731:
1727:
1723:
1718:
1714:
1706:
1705:
1704:
1701:
1687:
1667:
1647:
1627:
1614:
1607:
1605:
1603:
1599:
1594:
1592:
1588:
1583:
1581:
1571:
1564:
1559:
1554:
1547:
1542:
1537:
1530:
1525:
1521:= 2.1 = 21/10
1520:
1513:
1508:
1505:
1504:quatrefoiloid
1499:
1492:
1487:
1484:
1478:
1471:
1466:
1463:
1462:
1455:
1448:
1443:
1440:
1439:
1432:
1425:
1420:
1417:
1400:
1397:
1390:
1375:
1368:
1367:
1366:
1349:
1346:
1343:
1340:
1337:
1328:
1325:
1319:
1316:
1309:
1308:
1307:
1293:
1267:
1264:
1254:The distance
1252:
1238:
1235:
1232:
1229:
1221:
1217:
1201:
1192:
1176:
1172:
1167:
1166:
1156:
1152:
1151:
1141:
1137:
1136:
1132:
1131:
1127:
1126:
1123:
1109:
1101:
1098:expressed as
1085:
1081:
1077:
1074:
1071:
1063:
1047:
1038:
1036:
1017:
1008:
990:
986:
978:
975:
972:
963:
960:
957:
928:
925:
919:
916:
913:
907:
904:
901:
894:
880:
875:
871:
867:
861:
858:
855:
846:
843:
840:
834:
831:
824:
823:
822:
808:
788:
768:
756:
739:
736:
728:
713:
705:
691:
685:
682:
679:
676:
670:
667:
659:
658:
657:
639:
633:
627:
624:
621:
615:
611:
607:
602:
599:
595:
588:
585:
582:
575:
571:
568:
562:
556:
549:
548:
547:
526:
522:
518:
512:
509:
506:
499:
495:
492:
489:
486:
483:
480:
477:
471:
468:
465:
459:
456:
450:
444:
435:
431:
425:
422:
419:
412:
408:
405:
402:
399:
396:
393:
390:
384:
381:
378:
372:
369:
363:
357:
345:
344:
343:
321:
317:
312:
308:
305:
302:
295:
291:
288:
285:
282:
276:
273:
270:
264:
261:
258:
252:
246:
240:
231:
227:
222:
218:
215:
212:
205:
201:
198:
195:
192:
186:
183:
180:
174:
171:
168:
162:
156:
150:
138:
137:
136:
134:
118:
115:
112:
109:
89:
77:
75:
72:
70:
66:
65:
60:
56:
52:
49:(also called
48:
44:
34:
27:
21:
2508:
2505:"Epicycloid"
2477:
2474:"Epicycloid"
2452:
2442:
2431:
2420:
2410:
2383:
2353:Hypotrochoid
1955:
1872:
1833:
1790:
1748:
1702:
1619:
1595:
1584:
1577:
1572:= 7.2 = 36/5
1569:
1555:= 5.5 = 11/2
1552:
1538:= 3.8 = 19/5
1535:
1518:
1503:
1497:
1482:
1476:
1459:
1453:
1436:
1430:
1364:
1253:
1220:dense subset
1193:
1183:
1174:
1170:
1154:
1139:
1039:
1009:
943:
760:
655:
545:
341:
81:
73:
62:
51:hypercycloid
50:
46:
40:
32:
25:
2348:Hypocycloid
2343:Epitrochoid
1913:, which is
1580:epitrochoid
131:, then the
61:—called an
55:plane curve
2540:Categories
2489:Epicycloid
2374:References
2368:Spirograph
1483:trefoiloid
1178:rotations
1163:rotations
1148:rotations
660:the angle
47:epicycloid
2479:MathWorld
2275:θ
2249:
2240:−
2237:θ
2234:
2201:α
2195:θ
2187:
2178:−
2175:θ
2172:
2118:θ
2092:
2083:−
2080:θ
2077:
2044:α
2038:θ
2030:
2021:−
2018:θ
2015:
1940:θ
1924:α
1901:θ
1881:α
1854:α
1845:θ
1815:α
1803:ℓ
1773:θ
1761:ℓ
1728:ℓ
1715:ℓ
1688:θ
1648:α
1338:≤
1333:¯
1320:≤
1272:¯
868:π
686:π
671:∈
668:θ
634:θ
608:−
603:θ
563:θ
519:θ
496:
487:−
484:θ
481:
451:θ
432:θ
409:
400:−
397:θ
394:
364:θ
318:θ
292:
283:−
277:θ
274:
247:θ
228:θ
202:
193:−
187:θ
184:
157:θ
78:Equations
2328:Cyclogon
2310:Cardioid
2296:See also
1591:nephroid
1587:cardioid
1461:nephroid
1438:cardioid
1122:cusps.
69:roulette
64:epicycle
43:geometry
2323:Cycloid
2306:MSWLogo
1602:similar
1598:evolute
656:where
53:) is a
2495:, 2007
2453:Maecla
2396:
1680:, and
1365:where
1214:is an
1064:, say
280:
190:
59:circle
1791:and
1608:Proof
1060:is a
1035:cusps
726:, and
45:, an
2394:ISBN
1893:and
1600:are
1502:; a
1481:; a
1458:; a
1435:; a
1188:and
342:or:
35:= 3)
28:= 1)
2246:sin
2231:sin
2184:sin
2169:sin
2089:cos
2074:cos
2027:cos
2012:cos
1500:= 4
1479:= 3
1456:= 2
1433:= 1
1194:If
1159:to
1157:= 0
1144:to
1142:= 0
1040:If
1010:If
493:sin
478:sin
406:cos
391:cos
289:sin
271:sin
199:cos
181:cos
41:In
2542::
2513:,
2507:,
2503:,
2476:.
2451:.
2392:.
1604:.
1593:.
1582:.
1251:.
1173:+
71:.
2487:"
2482:.
2455:.
2402:.
2312:)
2308:(
2279:)
2270:r
2266:r
2263:+
2260:R
2253:(
2243:r
2227:)
2223:r
2220:+
2217:R
2213:(
2209:=
2205:)
2198:+
2191:(
2181:r
2165:)
2161:r
2158:+
2155:R
2151:(
2147:=
2144:y
2122:)
2113:r
2109:r
2106:+
2103:R
2096:(
2086:r
2070:)
2066:r
2063:+
2060:R
2056:(
2052:=
2048:)
2041:+
2034:(
2024:r
2008:)
2004:r
2001:+
1998:R
1994:(
1990:=
1987:x
1964:p
1952:.
1935:r
1932:R
1927:=
1869:.
1857:r
1851:=
1848:R
1830:.
1818:r
1812:=
1807:r
1776:R
1770:=
1765:R
1732:r
1724:=
1719:R
1668:p
1628:p
1570:k
1553:k
1536:k
1519:k
1498:k
1477:k
1454:k
1431:k
1401:r
1398:2
1376:R
1350:r
1347:2
1344:+
1341:R
1329:P
1326:O
1317:R
1294:p
1268:P
1265:O
1239:r
1236:2
1233:+
1230:R
1202:k
1190:q
1186:p
1175:q
1171:p
1161:p
1155:α
1146:q
1140:θ
1110:p
1086:q
1082:/
1078:p
1075:=
1072:k
1048:k
1032:k
1018:k
991:2
987:k
982:)
979:2
976:+
973:k
970:(
967:)
964:1
961:+
958:k
955:(
929:.
926:r
923:)
920:1
917:+
914:k
911:(
908:8
905:=
902:s
881:,
876:2
872:r
865:)
862:2
859:+
856:k
853:(
850:)
847:1
844:+
841:k
838:(
835:=
832:A
809:s
789:A
769:k
752:.
740:r
737:k
714:r
692:,
689:]
683:2
680:,
677:0
674:[
640:)
631:)
628:1
625:+
622:k
619:(
616:i
612:e
600:i
596:e
592:)
589:1
586:+
583:k
580:(
576:(
572:r
569:=
566:)
560:(
557:z
527:.
523:)
516:)
513:1
510:+
507:k
504:(
500:(
490:r
475:)
472:1
469:+
466:k
463:(
460:r
457:=
454:)
448:(
445:y
436:)
429:)
426:1
423:+
420:k
417:(
413:(
403:r
388:)
385:1
382:+
379:k
376:(
373:r
370:=
367:)
361:(
358:x
322:)
313:r
309:r
306:+
303:R
296:(
286:r
268:)
265:r
262:+
259:R
256:(
253:=
250:)
244:(
241:y
232:)
223:r
219:r
216:+
213:R
206:(
196:r
178:)
175:r
172:+
169:R
166:(
163:=
160:)
154:(
151:x
119:r
116:k
113:=
110:R
90:r
37:.
33:R
26:r
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