3046:
177:
20:
2910:
and focus at the origin. Thus a limaçon can be defined as the inverse of a conic where the center of inversion is one of the foci. If the conic is a parabola then the inverse will be a cardioid, if the conic is a hyperbola then the corresponding limaçon will have an inner loop, and if the conic is an
1502:
2313:
1021:
we would, by changing the location of the origin, convert to the usual form of the equation of a centered trochoid. Note the change of independent variable at this point to make it clear that we are no longer using the default polar coordinate parameterization
2452:
132:
around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller circle is inside the larger circle. Thus, they belong to the family of curves called
2170:
616:
716:
498:
1252:
908:
1303:
814:
1168:
2541:
1981:
2181:
2873:
1015:
2324:
1553:
2046:
281:
2817:
2767:
2054:
1919:
345:
2908:
380:
1052:
944:
1673:
1850:
509:
1638:
2007:
1741:
1602:
2726:
1706:
1295:
1814:
1086:
2694:
2650:
2630:
2610:
2590:
2570:
1870:
1788:
1768:
622:
388:
3025:
1604:, the limaçon is a simple closed curve. However, the origin satisfies the Cartesian equation given above, so the graph of this equation has an
3064:
1179:
1497:{\displaystyle z=b\left(e^{it}+e^{2it}\right)=be^{3it \over 2}\left(e^{it \over 2}+e^{-it \over 2}\right)=2be^{3it \over 2}\cos {t \over 2},}
829:
727:
1094:
154:
Depending on the position of the point generating the curve, it may have inner and outer loops (giving the family its name), it may be
2467:
2988:
2657:
2308:{\displaystyle \left(b^{2}+{{a^{2}} \over 2}\right)\left(\pi -\arccos {b \over a}\right)+{3 \over 2}b{\sqrt {a^{2}-b^{2}}},}
1924:
3110:
147:
is the special case in which the point generating the roulette lies on the rolling circle; the resulting curve has a
3119:
3139:
2949:
2828:
2447:{\displaystyle \left(b^{2}+{{a^{2}} \over 2}\right)\left(\pi -2\arccos {b \over a}\right)+3b{\sqrt {a^{2}-b^{2}}}.}
952:
1872:
approaches 0, the loop fills up the outer curve and, in the limit, the limaçon becomes a circle traversed twice.
1513:
3030:
162:
2012:
3134:
2009:
this counts the area enclosed by the inner loop twice. In this case the curve crosses the origin at angles
2917:
2165:{\displaystyle \left(b^{2}+{{a^{2}} \over 2}\right)\arccos {b \over a}-{3 \over 2}b{\sqrt {a^{2}-b^{2}}},}
170:
242:
2781:
2731:
1883:
1567:
297:
2884:
350:
287:
3114:
1025:
3016:
611:{\displaystyle x=(b+a\cos \theta )\cos \theta ={a \over 2}+b\cos \theta +{a \over 2}\cos 2\theta ,}
1643:
916:
2775:
1823:
1817:
1559:
221:
148:
3020:
2984:
2980:
2973:
2944:
2458:
1258:
233:
213:
198:
134:
28:
1614:
2939:
1986:
1711:
1676:
1581:
205:. However, some insightful investigations regarding them had been undertaken earlier by the
121:
117:
76:
3101:
2699:
294:(thus introducing a point at the origin which in some cases is spurious), and substituting
1682:
1271:
166:
1793:
1065:
3083:
2926:
2679:
2635:
2615:
2595:
2575:
2555:
1855:
1773:
1753:
711:{\displaystyle y=(b+a\cos \theta )\sin \theta =b\sin \theta +{a \over 2}\sin 2\theta ;}
155:
2676:
is a limaçon. In fact, the pedal with respect to the origin of the circle with radius
3128:
820:
202:
1852:, the cusp expands to an inner loop, and the curve crosses itself at the origin. As
220:
contains specific geometric methods for producing limaçons. The curve was named by
3092:
2669:
503:
Applying the parametric form of the polar to
Cartesian conversion, we also have
209:
138:
1563:
493:{\displaystyle \left(x^{2}+y^{2}-ax\right)^{2}=b^{2}\left(x^{2}+y^{2}\right).}
185:
3003:
1744:
19:
3087:
1262:
181:
143:
63:
2656:
1247:{\displaystyle r^{1 \over 2}=(2b)^{1 \over 2}\cos {\frac {\theta }{2}},}
206:
129:
3004:
Weisstein, Eric W. "Limaçon." From MathWorld--A Wolfram Web
Resource.
2673:
2661:
1605:
903:{\displaystyle z={a \over 2}+be^{i\theta }+{a \over 2}e^{2i\theta }.}
125:
809:{\displaystyle z=x+iy=(b+a\cos \theta )(\cos \theta +i\sin \theta )}
197:
The earliest formal research on limaçons is generally attributed to
176:
1163:{\displaystyle r=b(1+\cos \theta )=2b\cos ^{2}{\frac {\theta }{2}}}
2655:
175:
18:
2536:{\displaystyle 4(a+b)E\left({{2{\sqrt {ab}}} \over a+b}\right).}
2920:
of a circle with respect to a point on the circle is a limaçon.
1816:, the curve becomes a cardioid, and the indentation becomes a
232:
The equation (up to translation and rotation) of a limaçon in
3096:
2881:
which is the equation of a conic section with eccentricity
91:
2612:. Then the envelope of those circles whose center lies on
97:
3105:
2911:
ellipse then the corresponding limaçon will have no loop.
224:
when he used it as an example for finding tangent lines.
85:
2889:
2015:
1927:
919:
3060:, 2nd edition, page 708, John Wiley & Sons, 1984.
2887:
2831:
2784:
2734:
2702:
2682:
2638:
2618:
2598:
2578:
2558:
2470:
2327:
2184:
2057:
1989:
1976:{\textstyle \left(b^{2}+{{a^{2}} \over 2}\right)\pi }
1886:
1858:
1826:
1796:
1776:
1756:
1714:
1685:
1646:
1617:
1584:
1516:
1306:
1297:, the centered trochoid form of the equation becomes
1274:
1182:
1097:
1068:
1028:
955:
832:
730:
625:
512:
391:
353:
300:
245:
218:
Underweysung der
Messung (Instruction in Measurement)
100:
94:
3073:, Volume 2 (pages 51,56,273), Allyn and Bacon, 1965.
1790:, the indentation becomes more pronounced until, at
1640:, the area bounded by the curve is convex, and when
82:
88:
79:
3106:ENCYCLOPÉDIE DES FORMES MATHÉMATIQUES REMARQUABLES
2972:
2902:
2867:
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1975:
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1808:
1782:
1762:
1735:
1700:
1667:
1632:
1596:
1547:
1496:
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1162:
1080:
1046:
1009:
938:
902:
808:
710:
610:
492:
374:
339:
275:
2457:The circumference of the limaçon is given by a
819:yields this parameterization as a curve in the
1675:, the curve has an indentation bounded by two
2868:{\displaystyle r={1 \over {b+a\cos \theta }}}
2459:complete elliptic integral of the second kind
8:
1010:{\displaystyle z=be^{it}+{a \over 2}e^{2it}}
3120:"Limacon of Pascal" on PlanetPTC (Mathcad)
2966:
2964:
1548:{\displaystyle r=2b\cos {\theta \over 3}}
3115:Visual Dictionary of Special Plane Curves
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2048:, the area enclosed by the inner loop is
2028:
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1953:
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1181:
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969:
954:
923:
918:
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729:
683:
624:
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476:
463:
448:
435:
415:
402:
390:
352:
331:
318:
305:
299:
244:
3051:The Two-Year College Mathematics Journal
2175:the area enclosed by the outer loop is
2041:{\textstyle \pi \pm \arccos {b \over a}}
3026:MacTutor History of Mathematics Archive
2960:
180:Three limaçons: dimpled, with cusp (a
7:
3045:Jane Grossman and Michael Grossman.
1261:family of curves. This curve is the
913:If we were to shift horizontally by
3088:The MacTutor History of Mathematics
2778:with respect to the unit circle of
2318:and the area between the loops is
1562:family of curves. This curve is a
276:{\displaystyle r=b+a\cos \theta .}
14:
2975:A catalog of special plane curves
2812:{\displaystyle r=b+a\cos \theta }
2762:{\displaystyle r=b+a\cos \theta }
1914:{\displaystyle r=b+a\cos \theta }
1880:The area enclosed by the limaçon
340:{\displaystyle r^{2}=x^{2}+y^{2}}
2592:be a circle whose center is not
75:
2979:. Dover Publications. pp.
2925:A particular special case of a
2903:{\displaystyle {\tfrac {a}{b}}}
375:{\displaystyle r\cos \theta =x}
2715:
2703:
2486:
2474:
1730:
1715:
1566:, and is sometimes called the
1211:
1201:
1125:
1107:
1047:{\displaystyle \theta =\arg z}
803:
776:
773:
752:
653:
632:
540:
519:
184:), and looped. Not shown: the
137:; more specifically, they are
1:
939:{\textstyle -{\frac {1}{2}}a}
3053:, January 1982, pages 52–55.
1668:{\displaystyle a<b<2a}
158:-shaped, or it may be oval.
25:r = 2 + cos(π – θ)
23:Construction of the limaçon
2971:J. Dennis Lawrence (1972).
2660:Limaçon — pedal curve of a
1845:{\displaystyle 0<b<a}
3156:
2950:List of periodic functions
1558:making it a member of the
1507:or, in polar coordinates,
1257:making it a member of the
1770:is decreased relative to
286:This can be converted to
3031:University of St Andrews
2547:Relation to other curves
1088:, the polar equation is
120:formed by the path of a
1633:{\displaystyle b>2a}
2904:
2869:
2813:
2763:
2722:
2690:
2664:
2646:
2632:and that pass through
2626:
2606:
2586:
2566:
2537:
2448:
2309:
2166:
2042:
2003:
2002:{\displaystyle b<a}
1977:
1915:
1866:
1846:
1810:
1784:
1764:
1737:
1736:{\displaystyle (-a,0)}
1702:
1669:
1634:
1598:
1597:{\displaystyle b>a}
1549:
1498:
1291:
1248:
1164:
1082:
1048:
1011:
940:
904:
810:
712:
612:
494:
376:
341:
277:
189:
59:
16:Type of roulette curve
3047:"Dimple or no dimple"
2905:
2870:
2814:
2764:
2723:
2721:{\displaystyle (a,0)}
2691:
2659:
2647:
2627:
2607:
2587:
2567:
2538:
2449:
2310:
2167:
2043:
2004:
1978:
1916:
1867:
1847:
1811:
1785:
1765:
1738:
1703:
1670:
1635:
1599:
1550:
1499:
1292:
1249:
1165:
1083:
1049:
1012:
941:
905:
811:
713:
613:
495:
377:
342:
288:Cartesian coordinates
278:
179:
22:
3071:A Survey of Geometry
3017:Robertson, Edmund F.
2885:
2829:
2782:
2732:
2700:
2680:
2636:
2616:
2596:
2576:
2556:
2468:
2325:
2182:
2055:
2013:
1987:
1925:
1884:
1856:
1824:
1794:
1774:
1754:
1712:
1701:{\displaystyle b=2a}
1683:
1644:
1615:
1582:
1514:
1304:
1290:{\displaystyle a=2b}
1272:
1268:In the special case
1180:
1095:
1066:
1062:In the special case
1026:
953:
917:
830:
728:
623:
510:
389:
351:
298:
243:
3097:Mathematical curves
3066:pp. 725 – 726.
3015:O'Connor, John J.;
2728:has polar equation
1809:{\displaystyle b=a}
1608:or isolated point.
1081:{\displaystyle a=b}
2900:
2898:
2865:
2809:
2759:
2718:
2686:
2665:
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2622:
2602:
2582:
2562:
2533:
2444:
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1999:
1973:
1911:
1862:
1842:
1806:
1780:
1760:
1733:
1698:
1665:
1630:
1594:
1568:limaçon trisectrix
1545:
1494:
1287:
1244:
1160:
1078:
1044:
1007:
936:
900:
806:
708:
608:
490:
372:
337:
290:by multiplying by
273:
222:Gilles de Roberval
190:
135:centered trochoids
116:, is defined as a
108:, also known as a
60:
3140:Roulettes (curve)
3111:Limacon of Pascal
3102:Limaçon of Pascal
3084:Limacon of Pascal
2945:Centered trochoid
2897:
2863:
2689:{\displaystyle b}
2645:{\displaystyle P}
2625:{\displaystyle C}
2605:{\displaystyle P}
2585:{\displaystyle C}
2565:{\displaystyle P}
2524:
2510:
2439:
2398:
2363:
2300:
2270:
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2220:
2157:
2127:
2114:
2093:
2036:
1963:
1865:{\displaystyle b}
1783:{\displaystyle a}
1763:{\displaystyle b}
1677:inflection points
1543:
1489:
1472:
1435:
1409:
1384:
1259:sinusoidal spiral
1239:
1222:
1195:
1158:
989:
931:
879:
847:
691:
591:
563:
234:polar coordinates
128:when that circle
110:limaçon of Pascal
29:polar coordinates
3147:
3034:
3033:
3021:"Cartesian Oval"
3012:
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3001:
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1761:
1743:is a point of 0
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3124:
3080:
3063:Howard Anton.
3042:
3040:Further reading
3037:
3014:
3013:
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2962:
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2698:
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2678:
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2574:
2573:
2572:be a point and
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2370:
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2011:
2010:
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855:
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195:
167:algebraic curve
165:rational plane
161:A limaçon is a
78:
74:
51:
48:
45:
44:
42:
32:
24:
17:
12:
11:
5:
3153:
3151:
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3135:Quartic curves
3127:
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3099:
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3078:External links
3076:
3075:
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3061:
3056:Howard Anton.
3054:
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2927:Cartesian oval
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214:Albrecht Dürer
199:Étienne Pascal
194:
191:
118:roulette curve
114:Pascal's Snail
15:
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9:
6:
4:
3:
2:
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3069:Howard Eves.
3068:
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2933:
2929:is a limaçon.
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2675:
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2652:is a limaçon.
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2019:
2016:
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1990:
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1934:
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1116:
1113:
1110:
1104:
1101:
1098:
1091:
1090:
1089:
1075:
1072:
1069:
1058:Special cases
1057:
1055:
1041:
1038:
1035:
1032:
1029:
1002:
999:
996:
992:
986:
983:
978:
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863:
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856:
852:
849:
844:
841:
836:
833:
826:
825:
824:
822:
821:complex plane
800:
797:
794:
791:
788:
785:
782:
779:
770:
767:
764:
761:
758:
755:
749:
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743:
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737:
734:
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723:
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705:
702:
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531:
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427:
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394:
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384:
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369:
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332:
328:
324:
319:
315:
311:
306:
302:
293:
289:
270:
267:
264:
261:
258:
255:
252:
249:
246:
239:
238:
237:
236:has the form
235:
227:
225:
223:
219:
215:
211:
208:
204:
203:Blaise Pascal
200:
192:
187:
183:
178:
174:
172:
168:
164:
159:
157:
152:
150:
146:
145:
140:
136:
131:
127:
123:
119:
115:
111:
105:
73:
69:
65:
40:
36:
30:
21:
3070:
3057:
3050:
3024:
3010:
2999:
2974:
2456:
2317:
2174:
1879:
1749:
1708:, the point
1610:
1577:
1557:
1506:
1267:
1256:
1172:
1061:
1020:
912:
818:
720:
502:
291:
285:
231:
217:
201:, father of
196:
160:
153:
142:
139:epitrochoids
113:
109:
71:
67:
61:
38:
34:
31:' origin at
2696:and center
1876:Measurement
210:Renaissance
124:fixed to a
3129:Categories
2956:References
1564:trisectrix
382:to obtain
216:. Dürer's
163:bicircular
2860:θ
2857:
2807:θ
2804:
2757:θ
2754:
2427:−
2388:
2379:−
2376:π
2288:−
2242:
2236:−
2233:π
2145:−
2117:−
2104:
2026:
2020:±
2017:π
1971:π
1909:θ
1906:
1745:curvature
1719:−
1538:θ
1533:
1479:
1423:−
1234:θ
1229:
1153:θ
1148:
1123:θ
1120:
1039:
1030:θ
921:−
893:θ
864:θ
801:θ
798:
786:θ
783:
771:θ
768:
703:θ
697:
678:θ
675:
663:θ
660:
651:θ
648:
603:θ
597:
578:θ
575:
550:θ
547:
538:θ
535:
422:−
364:θ
361:
268:θ
265:
228:Equations
3058:Calculus
2940:Roulette
2934:See also
2918:conchoid
1263:cardioid
946:, i.e.,
182:cardioid
144:cardioid
64:geometry
3093:Limaçon
2981:113–118
2776:inverse
1983:. When
212:artist
193:History
188:limaçon
72:limacon
68:limaçon
55:
43:
2987:
2674:circle
2662:circle
2385:arccos
2239:arccos
2101:arccos
2023:arccos
1820:. For
1606:acnode
207:German
186:convex
171:degree
141:. The
126:circle
2672:of a
2670:pedal
1679:. At
1611:When
1578:When
156:heart
130:rolls
122:point
41:) = (
27:with
2985:ISBN
2916:The
2774:The
2552:Let
1994:<
1837:<
1831:<
1818:cusp
1657:<
1651:<
1622:>
1589:>
1574:Form
1560:rose
1173:or
347:and
149:cusp
66:, a
57:, 0)
3113:at
3104:at
3095:at
3086:at
2854:cos
2801:cos
2751:cos
1921:is
1903:cos
1750:As
1530:cos
1476:cos
1226:cos
1139:cos
1117:cos
1036:arg
795:sin
780:cos
765:cos
694:sin
672:sin
657:sin
645:cos
594:cos
572:cos
544:cos
532:cos
358:cos
262:cos
173:4.
169:of
112:or
70:or
62:In
3131::
3049:,
3029:,
3023:,
3019:,
2983:.
2963:^
2819:is
2668:A
2461::
1747:.
1570:.
1265:.
1054:.
823::
151:.
37:,
2993:.
2895:b
2892:a
2851:a
2848:+
2845:b
2841:1
2836:=
2833:r
2798:a
2795:+
2792:b
2789:=
2786:r
2769:.
2748:a
2745:+
2742:b
2739:=
2736:r
2716:)
2713:0
2710:,
2707:a
2704:(
2684:b
2640:P
2620:C
2600:P
2580:C
2560:P
2531:.
2527:)
2521:b
2518:+
2515:a
2508:b
2505:a
2500:2
2494:(
2490:E
2487:)
2484:b
2481:+
2478:a
2475:(
2472:4
2442:.
2435:2
2431:b
2422:2
2418:a
2412:b
2409:3
2406:+
2402:)
2396:a
2393:b
2382:2
2372:(
2367:)
2361:2
2355:2
2351:a
2344:+
2339:2
2335:b
2330:(
2303:,
2296:2
2292:b
2283:2
2279:a
2273:b
2268:2
2265:3
2260:+
2256:)
2250:a
2247:b
2229:(
2224:)
2218:2
2212:2
2208:a
2201:+
2196:2
2192:b
2187:(
2160:,
2153:2
2149:b
2140:2
2136:a
2130:b
2125:2
2122:3
2112:a
2109:b
2097:)
2091:2
2085:2
2081:a
2074:+
2069:2
2065:b
2060:(
2034:a
2031:b
1997:a
1991:b
1967:)
1961:2
1955:2
1951:a
1944:+
1939:2
1935:b
1930:(
1900:a
1897:+
1894:b
1891:=
1888:r
1860:b
1840:a
1834:b
1828:0
1804:a
1801:=
1798:b
1778:a
1758:b
1731:)
1728:0
1725:,
1722:a
1716:(
1696:a
1693:2
1690:=
1687:b
1663:a
1660:2
1654:b
1648:a
1628:a
1625:2
1619:b
1592:a
1586:b
1541:3
1527:b
1524:2
1521:=
1518:r
1492:,
1487:2
1484:t
1470:2
1466:t
1463:i
1460:3
1454:e
1450:b
1447:2
1444:=
1440:)
1433:2
1429:t
1426:i
1417:e
1413:+
1407:2
1403:t
1400:i
1394:e
1389:(
1382:2
1378:t
1375:i
1372:3
1366:e
1362:b
1359:=
1355:)
1349:t
1346:i
1343:2
1339:e
1335:+
1330:t
1327:i
1323:e
1318:(
1314:b
1311:=
1308:z
1285:b
1282:2
1279:=
1276:a
1242:,
1237:2
1220:2
1217:1
1212:)
1208:b
1205:2
1202:(
1199:=
1193:2
1190:1
1185:r
1156:2
1143:2
1135:b
1132:2
1129:=
1126:)
1114:+
1111:1
1108:(
1105:b
1102:=
1099:r
1076:b
1073:=
1070:a
1042:z
1033:=
1017:,
1003:t
1000:i
997:2
993:e
987:2
984:a
979:+
974:t
971:i
967:e
963:b
960:=
957:z
934:a
929:2
926:1
898:.
890:i
887:2
883:e
877:2
874:a
869:+
861:i
857:e
853:b
850:+
845:2
842:a
837:=
834:z
804:)
792:i
789:+
777:(
774:)
762:a
759:+
756:b
753:(
750:=
747:y
744:i
741:+
738:x
735:=
732:z
706:;
700:2
689:2
686:a
681:+
669:b
666:=
654:)
642:a
639:+
636:b
633:(
630:=
627:y
606:,
600:2
589:2
586:a
581:+
569:b
566:+
561:2
558:a
553:=
541:)
529:a
526:+
523:b
520:(
517:=
514:x
488:.
484:)
478:2
474:y
470:+
465:2
461:x
456:(
450:2
446:b
442:=
437:2
432:)
428:x
425:a
417:2
413:y
409:+
404:2
400:x
395:(
370:x
367:=
355:r
333:2
329:y
325:+
320:2
316:x
312:=
307:2
303:r
292:r
271:.
259:a
256:+
253:b
250:=
247:r
104:/
101:n
98:ɒ
95:s
92:ə
89:m
86:ɪ
83:l
80:ˈ
77:/
52:2
49:/
46:1
39:y
35:x
33:(
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