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909:
891:
447:
22:
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433:
Because of this equal-length property, the starting marks of 200m and 400m footraces are placed in staggered positions along a hyperbolic spiral. This ensures that the runners, restricted to their concentric lanes, all have equal-length paths to the finish line. For longer races where runners move to
3156:
2775:
1732:
315:. Newton asserted that the reverse was true: that conic sections were the only trajectories possible under an inverse-square law. Bernoulli criticized this step, observing that in the case of an inverse-cube law, multiple trajectories were possible, including both a
2619:
824:
2295:
1151:, but this is different in some important respects from the usual form of the hyperbolic spiral in the Euclidean plane. In particular, the corresponding curve in the hyperbolic plane does not have an asymptotic line.
323:, including the logarithmic and hyperbolic spirals, that all required an inverse-cube law. By 1720, Newton had resolved the controversy by proving that inverse-square laws always produce conic-section trajectories.
2106:
2472:
1143:
741:
577:
3324:{\displaystyle {\begin{aligned}A&={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}r(\varphi )^{2}\,d\varphi \\&={\frac {a}{2}}{\bigl (}r(\varphi _{1})-r(\varphi _{2}){\bigr )}.\end{aligned}}}
340:
3161:
2780:
1592:
1983:
1011:
1662:
2996:{\displaystyle {\begin{aligned}L&=a\int _{\varphi _{1}}^{\varphi _{2}}{\frac {\sqrt {1+\varphi ^{2}}}{\varphi ^{2}}}\,d\varphi \\&=a\left_{\varphi _{1}}^{\varphi _{2}}.\end{aligned}}}
1876:{\displaystyle \lim _{\varphi \to 0}x=a\lim _{\varphi \to 0}{\frac {\cos \varphi }{\varphi }}=\infty ,\qquad \lim _{\varphi \to 0}y=a\lim _{\varphi \to 0}{\frac {\sin \varphi }{\varphi }}=a,}
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2427:
1245:
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383:
149:
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113:
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obtained by tracing a point on this curve as it rolls along another curve; for instance, when a hyperbolic spiral rolls along a straight line, its center traces out a
221:
2006:
1459:
1391:
3407:
3358:
1913:
1419:
175:
2447:
636:
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The increasing pitch angle of the hyperbolic spiral, as a function of distance from its center, has led to the use of these spirals to model the shapes of some
429:
407:
201:
235:, and the hyperbolic spiral was first discovered by applying the equation of a hyperbola to polar coordinates. Hyperbolic spirals can also be generated as the
2046:
1355:, looking up or down from a viewpoint on the axis of the staircase. To model this projection mathematically, consider the central projection from point
489:, which in some cases have a similarly increasing pitch angle. However, this model does not provide a good fit to the shapes of all spiral galaxies. In
1069:
639:
312:
536:
277:) by reinterpreting the Cartesian coordinates of points on the given curve as polar coordinates of points on the polar curve. Varignon and later
1522:
319:(whose connection to the inverse-cube law was already observed by Newton) and a hyperbolic spiral. Cotes found a family of spirals, the
1597:
476:
4243:
4179:
4146:
4094:
4066:
4029:
3970:
3548:
3485:
26:
331:
258:. In cases where the name of these spirals might be ambiguous, their alternative name, reciprocal spirals, can be used instead.
1667:
118:
3754:
Ringermacher, Harry I.; Mead, Lawrence R. (July 2009), "A new formula describing the scaffold structure of spiral galaxies",
1351:
of a helix onto a plane perpendicular to the axis of the helix describes the view that one would see of the guardrail of a
3452:
An
Elementary Treatise on Analytic Geometry: Embracing Plane Geometry and an Introduction to Geometry of Three Dimensions
2614:{\displaystyle \kappa ={\frac {\varphi ^{4}}{a\left(\varphi ^{2}+1\right)^{3/2}}}={\frac {a^{3}}{r(a^{2}+r^{2})^{3/2}}}.}
965:
4329:
819:{\displaystyle x=a{\frac {\cos \varphi }{\varphi }},\qquad y=a{\frac {\sin \varphi }{\varphi }},\quad \varphi >0.}
2720:
2665:
1942:
462:
4378:
878:
and using the same equations produces a reflected copy of the spiral, and some sources treat these two copies as
1209:
3953:
Polezhaev, Andrey (2019), "Spirals, their types and peculiarities", in Tsuji, Kinko; Müller, Stefan C. (eds.),
3804:
3712:
4056:
2011:
2008:
between the tangent of any curve and the tangent of its corresponding polar circle. For the hyperbolic spiral
1298:
1257:
353:
4501:
1986:
1936:
1466:
502:
46:
3535:, in Barrallo, Javier; Friedman, Nathaniel; Maldonado, Juan Antonio; Martínez-Aroza, José; Sarhangi, Reza;
930:
180:
2383:
1017:
4496:
4294:
4236:
3925:
3635:
Guicciardini, Niccolò (1995), "Johann
Bernoulli, John Keill and the inverse problem of central forces",
2334:
228:
1163:
Hyperbolic spiral (blue) as image of an
Archimedean spiral (green) by inversion through a circle (red)
680:
644:
4445:
3900:
3813:
3775:
922:
3838:
Savchenko, S. S.; Reshetnikov, V. P. (September 2013), "Pitch angle variations in spiral galaxies",
855:
4506:
4153:
3398:
2122:
1248:
926:
829:
715:
278:
4195:
3447:
3379:
3122:
3087:
3060:
3033:
3006:
2631:
2300:
1177:
585:
86:
4477:
4425:
4111:
3976:
3849:
3765:
3737:
3729:
3610:
3567:
1348:
1252:
936:
615:
316:
308:
240:
54:
50:
4462:
4092:
Loria, Gino; Roever, W. H. (February 1919), "On certain constructions of descriptive geometry",
1159:
4133:, Mathematics and Its Applications, vol. 280–281, Springer Netherlands, pp. 260–335,
3995:
3664:(1710), "Extrait de la Réponse de M. Bernoulli à M. Herman, datée de Basle le 7 Octobre 1710",
3532:
3472:, Mathematics and Its Applications, vol. 280–281, Springer Netherlands, pp. 112–166,
3402:
3331:
That is, the area is proportional to the difference in radii, with constant of proportionality
4430:
4388:
4284:
4216:
4175:
4167:
4142:
4062:
4025:
4017:
3966:
3917:
3879:
3544:
3481:
2452:
721:
580:
320:
206:
81:
4052:
3802:
Kennicutt, R. C. Jr. (December 1981), "The shapes of spiral arms along the Hubble sequence",
3697:
1991:
1426:
1358:
4537:
4339:
4229:
4134:
4103:
3958:
3909:
3859:
3821:
3783:
3721:
3661:
3644:
3511:
3473:
3416:
2290:{\displaystyle \kappa ={\frac {r^{2}+2(r')^{2}-r\,r''}{\left(r^{2}+(r')^{2}\right)^{3/2}}}.}
1352:
1167:
1148:
506:
498:
472:
292:
282:
255:
62:
4076:
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4324:
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1884:
1339:
516:
Along with the
Archimedean and logarithmic spiral, the hyperbolic spiral has been used in
266:
251:
58:
3700:. Hammer dates this material to 1714, but it was not published until after Cotes's death.
3536:
3335:
1892:
1398:
3898:
Scott, Thomas R.; Noland, J. H. (1965), "Some stimulus dimensions of rotating spirals",
3817:
3779:
3692:
Harmonia
Mensurarum, Sive Analysis & Synthesis per Rationum & Angulorum Mensuras
157:
4403:
4289:
3003:
Here, the bracket notation means to calculate the formula within the brackets for both
2432:
1924:
621:
414:
392:
186:
4531:
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4044:
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3788:
3741:
1290:
517:
486:
304:
236:
66:
4457:
4452:
4398:
4344:
4107:
493:, it has been suggested that hyperbolic spirals are a good match for the design of
490:
300:
152:
3881:
A Popular Course of Pure and Mixed
Mathematics for the Use of Schools and Students
3468:
Drábek, Karel (1994), "Plane curves and constructions", in
Rektorys, Karel (ed.),
3515:
4516:
4368:
4319:
4138:
3962:
3957:, The Frontiers Collection, Springer International Publishing, pp. 91–112,
3929:
3687:
3477:
1171:
908:
890:
296:
77:
21:
4490:
4393:
4373:
4221:
3648:
3420:
3119:
The area of a sector (see diagram above) of a hyperbolic spiral with equation
1147:
It is also possible to use the polar equation to define a spiral curve in the
933:
in these coordinates by starting with its polar defining equation in the form
446:
4511:
4435:
4360:
4334:
4279:
3864:
2116:
1729:
The hyperbolic spiral approaches the origin as an asymptotic point. Because
1519:
The image under this projection of the helix with parametric representation
1174:
is a transformation of the plane that, in polar coordinates, maps the point
962:
and replacing its variables according to the
Cartesian-to-polar conversions
274:
232:
49:
that increases with distance from its center, unlike the constant angles of
3921:
638:. It can be represented in Cartesian coordinates by applying the standard
4467:
4383:
4129:
Kepr, Bořivoj (1994), "Differential geometry", in
Rektorys, Karel (ed.),
3696:. For the Cotes spirals, see pp. 30–35; the hyperbolic spiral is case 4,
453:
435:
286:
65:
and the starting arrangement of certain footraces, and is used to model
4202:, vol. II (2nd ed.), University of Calcutta, pp. 364–365
4115:
3733:
2429:
one gets the curvature of a hyperbolic spiral, in terms of the radius
2101:{\displaystyle \alpha =\tan ^{-1}\left(-{\frac {1}{\varphi }}\right).}
254:, and should not be confused with other kinds of spirals drawn in the
4252:
3913:
3424:
494:
468:
70:
42:
3725:
3429:
3825:
4260:
4256:
3854:
3770:
1923:
1338:
1158:
510:
269:
first studied the hyperbolic spiral in 1704, as an example of the
244:
30:
20:
33:
in the staircase project to hyperbolic spirals in its photograph.
1138:{\displaystyle {\sqrt {x^{2}+y^{2}}}\tan ^{-1}{\frac {y}{x}}=a.}
387:
a circular arc centered at the origin, continuing clockwise for
4225:
3605:
Johann
Bernoulli should not be confused with his older brother
3710:
Haines, C. R. (December 1977), "Old curves in a new setting",
3572:
Mémoires de l'Académie des sciences de l'Institut de France
3543:, Granada, Spain: University of Granada, pp. 521–528,
572:{\displaystyle r={\frac {a}{\varphi }},\quad \varphi >0}
4152:. For an equivalent formula for the direction angle (the
434:
the inside lane after the start, a different spiral (the
3506:
Hammer, Øyvind (2016), "15: The case of the staircase",
929:
of its Cartesian coordinates. However, one can obtain a
718:
for the Cartesian coordinates of this curve that treats
4156:
to the pitch angle) see Section 9.9, Theorem 1, p. 300
3984:; see especially Section 2.2, Hyperbolic spiral, p. 96
3541:
Meeting Alhambra, ISAMA-BRIDGES Conference Proceedings
2022:
1959:
1471:
1309:
1268:
1217:
1020:
992:
364:
3955:
Spirals and Vortices: In Culture, Nature, and Science
3510:, Springer International Publishing, pp. 65–68,
3338:
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356:
209:
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121:
89:
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4353:
4312:
4267:
4000:
Academic Press Dictionary of Science and Technology
1587:{\displaystyle (r\cos t,r\sin t,ct),\quad c\neq 0,}
4119:; for the central projection of a helix, see p. 51
3352:
3323:
3145:
3103:
3076:
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2995:
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2709:
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2613:
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2441:
2421:
2372:
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2146:
2100:
2035:
2000:
1977:
1928:Definition of sector (light blue) and pitch angle
1907:
1875:
1708:
1656:
1586:
1507:
1453:
1413:
1385:
1343:Hyperbolic spiral as central projection of a helix
1322:
1281:
1239:
1198:
1137:
1056:
1006:{\displaystyle \varphi =\tan ^{-1}{\tfrac {y}{x}}}
1005:
954:
870:
844:
818:
730:
704:
668:
630:
606:
571:
423:
401:
377:
215:
195:
169:
143:
107:
3841:Monthly Notices of the Royal Astronomical Society
3757:Monthly Notices of the Royal Astronomical Society
1657:{\displaystyle {\frac {dr}{d-ct}}(\cos t,\sin t)}
1831:
1806:
1762:
1737:
299:also wrote about this curve, in connection with
4087:
4085:
3893:
3891:
4217:Online exploration using JSXGraph (JavaScript)
3408:Transactions of the Royal Society of Edinburgh
2765:{\displaystyle (r(\varphi _{2}),\varphi _{2})}
2710:{\displaystyle (r(\varphi _{1}),\varphi _{1})}
273:obtained from another curve (in this case the
4237:
3568:"Nouvelle formation de Spirales – exemple II"
3309:
3261:
2628:The length of the arc of a hyperbolic spiral
1978:{\displaystyle \tan \alpha ={\tfrac {r'}{r}}}
8:
3948:
3946:
3944:
3942:
3940:
3938:
3671:
3454:(4th ed.), D. Van Nostrand, p. 232
738:as a parameter rather than as a coordinate:
925:, meaning that it cannot be defined from a
4350:
4244:
4230:
4222:
4061:, Cambridge University Press, p. 54,
3583:
3581:
1240:{\displaystyle ({\tfrac {1}{r}},\varphi )}
80:, a hyperbolic spiral can be described in
4058:Indra's Pearls: The Vision of Felix Klein
4011:
4009:
3863:
3853:
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3769:
3428:
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2213:
2201:
2174:
2167:
2159:
2124:
2080:
2060:
2048:
2021:
2013:
1993:
1958:
1944:
1894:
1846:
1834:
1809:
1777:
1765:
1740:
1734:
1709:{\displaystyle \rho ={\frac {dr}{d-ct}},}
1677:
1669:
1601:
1599:
1524:
1470:
1468:
1428:
1400:
1360:
1308:
1300:
1267:
1259:
1216:
1211:
1179:
1116:
1104:
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1046:
1033:
1027:
1019:
991:
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967:
938:
857:
831:
785:
754:
743:
723:
682:
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623:
587:
546:
538:
416:
394:
363:
355:
208:
188:
159:
128:
120:
88:
57:. As this curve widens, it approaches an
3682:
3680:
3533:"Hyperbolic spirals and spiral patterns"
3526:
3524:
3501:
3499:
3497:
3463:
3461:
3441:
3439:
2036:{\displaystyle r={\tfrac {a}{\varphi }}}
1323:{\displaystyle r={\tfrac {a}{\varphi }}}
1282:{\displaystyle r={\tfrac {\varphi }{a}}}
410:from any of its points, will end on the
378:{\displaystyle r={\tfrac {a}{\varphi }}}
144:{\displaystyle r={\frac {a}{\varphi }},}
4018:"Hyperbolic spiral (reciprocal spiral)"
3812:, American Astronomical Society: 1847,
3593:MacTutor History of Mathematics Archive
3403:"XXXV.—On the theory of rolling curves"
3370:
533:The hyperbolic spiral has the equation
250:Hyperbolic spirals are patterns in the
3622:
1508:{\displaystyle {\tfrac {d}{d-z}}(x,y)}
3448:"The reciprocal or hyperbolic spiral"
1716:which describes a hyperbolic spiral.
313:Newton's law of universal gravitation
303:'s discovery that bodies that follow
7:
3994:Morris, Christopher G., ed. (1992),
3609:, who made extensive studies of the
1937:vector calculus in polar coordinates
1057:{\textstyle r={\sqrt {x^{2}+y^{2}}}}
4053:"Inversions and the Riemann sphere"
3666:Mémoires de l'Académie des Sciences
2772:can be calculated by the integral:
2422:{\displaystyle r''=2a/\varphi ^{3}}
343:The staggered start of a 200m race
307:trajectories must be subject to an
61:. It can be found in the view up a
3384:, Baldwin and Cradock, p. 194
3381:A Treatise on Algebraical Geometry
2373:{\displaystyle r'=-a/\varphi ^{2}}
1798:
477:Archaeological Museum of Epidaurus
239:of Archimedean spirals, or as the
14:
4095:The American Mathematical Monthly
3508:The Perfect Shape: Spiral Stories
3057:, and to subtract the result for
2119:of any curve with polar equation
4303:
4131:Survey of Applicable Mathematics
3789:10.1111/j.1365-2966.2009.14950.x
3470:Survey of Applicable Mathematics
1293:) is the hyperbolic spiral with
914:Hyperbolic spiral: both branches
907:
889:
705:{\displaystyle y=r\sin \varphi }
669:{\displaystyle x=r\cos \varphi }
461:
445:
330:
27:Cathedral of St. John the Divine
4194:Ganguli, Surendramohan (1926),
4024:, CRC Press, pp. 222–223,
3690:(1722), Smith, Robertus (ed.),
1804:
1571:
1289:under this transformation (its
806:
775:
559:
520:on the perception of rotation.
151:for an arbitrary choice of the
4108:10.1080/00029890.1919.11998485
4002:, Academic Press, p. 1068
3884:, G. B. Whittaker, p. 436
3446:Bowser, Edward Albert (1882),
3304:
3291:
3282:
3269:
3223:
3216:
2759:
2743:
2730:
2724:
2704:
2688:
2675:
2669:
2588:
2561:
2254:
2242:
2198:
2186:
2141:
2135:
1838:
1813:
1769:
1744:
1651:
1627:
1565:
1526:
1502:
1490:
1448:
1430:
1380:
1362:
1234:
1213:
1193:
1181:
896:Hyperbolic spiral: branch for
871:{\displaystyle \varphi \neq 0}
640:polar-to-Cartesian conversions
601:
589:
438:of a circle) is used instead.
102:
90:
1:
2147:{\displaystyle r=r(\varphi )}
1335:Central projection of a helix
845:{\displaystyle \varphi >0}
826:Relaxing the constraint that
348:For a hyperbolic spiral with
4196:"289: The hyperbolic spiral"
4022:Handbook and Atlas of Curves
3516:10.1007/978-3-319-47373-4_15
3378:Waud, Samuel Wilkes (1835),
3146:{\displaystyle r=a/\varphi }
3104:{\displaystyle \varphi _{2}}
3077:{\displaystyle \varphi _{1}}
3050:{\displaystyle \varphi _{2}}
3023:{\displaystyle \varphi _{1}}
2655:{\displaystyle r=a/\varphi }
2324:{\displaystyle r=a/\varphi }
1199:{\displaystyle (r,\varphi )}
607:{\displaystyle (r,\varphi )}
227:. The same relation between
108:{\displaystyle (r,\varphi )}
4139:10.1007/978-94-015-8308-4_9
3963:10.1007/978-3-030-05798-5_4
3589:"Curves: Hyperbolic Spiral"
3478:10.1007/978-94-015-8308-4_4
955:{\displaystyle r\varphi =a}
921:The hyperbolic spiral is a
16:Spiral asymptotic to a line
4554:
4200:The Theory of Plane Curves
4174:, CRC Press, p. 143,
4016:Shikin, Eugene V. (2014),
3595:, University of St Andrews
1206:(excluding the origin) to
25:A spiral staircase in the
4301:
3878:Nicholson, Peter (1825),
3649:10.1080/00033799500200401
3421:10.1017/s008045680002247x
518:psychological experiments
4051:; Wright, David (2002),
3805:The Astronomical Journal
3713:The Mathematical Gazette
3531:Dunham, Douglas (2003),
2462:{\displaystyle \varphi }
1664:with the polar equation
731:{\displaystyle \varphi }
501:. It also describes the
262:History and applications
216:{\displaystyle \varphi }
53:or decreasing angles of
2001:{\displaystyle \alpha }
1454:{\displaystyle (x,y,z)}
1386:{\displaystyle (0,0,d)}
456:increases with distance
281:were interested in the
3674:, footnote 47, p. 554.
3354:
3325:
3147:
3105:
3078:
3051:
3024:
2997:
2766:
2711:
2656:
2615:
2469:of any of its points:
2463:
2443:
2423:
2374:
2325:
2291:
2148:
2102:
2037:
2002:
1979:
1932:
1909:
1877:
1710:
1658:
1588:
1509:
1455:
1423:This will map a point
1415:
1387:
1344:
1324:
1283:
1241:
1200:
1164:
1139:
1058:
1007:
956:
931:trigonometric equation
872:
846:
820:
732:
706:
670:
632:
608:
573:
425:
403:
379:
217:
197:
171:
145:
109:
34:
4166:Rutter, J.W. (2018),
3865:10.1093/mnras/stt1627
3694:(in Latin), Cambridge
3355:
3326:
3148:
3106:
3079:
3052:
3025:
2998:
2767:
2712:
2657:
2616:
2464:
2444:
2424:
2375:
2326:
2292:
2149:
2103:
2038:
2003:
1980:
1939:one gets the formula
1927:
1910:
1878:
1711:
1659:
1589:
1510:
1456:
1416:
1388:
1342:
1325:
1284:
1242:
1201:
1162:
1140:
1059:
1008:
957:
873:
847:
821:
733:
707:
671:
633:
609:
574:
426:
404:
380:
311:, such as the one in
229:Cartesian coordinates
218:
198:
172:
146:
110:
71:architectural volutes
24:
3901:Psychological Review
3399:Maxwell, James Clerk
3336:
3157:
3123:
3088:
3084:from the result for
3061:
3034:
3007:
2776:
2721:
2666:
2632:
2473:
2453:
2433:
2384:
2335:
2331:and its derivatives
2301:
2158:
2123:
2047:
2012:
1992:
1943:
1893:
1733:
1668:
1598:
1523:
1467:
1427:
1399:
1359:
1299:
1258:
1247:and vice versa. The
1210:
1178:
1070:
1018:
966:
937:
923:transcendental curve
856:
830:
742:
722:
681:
645:
622:
586:
537:
529:Coordinate equations
497:from columns of the
415:
393:
354:
223:it is also called a
207:
187:
158:
119:
87:
4154:complementary angle
3996:"Hyperbolic spiral"
3818:1981AJ.....86.1847K
3780:2009MNRAS.397..164R
3672:Guicciardini (1995)
3625:, pp. 119–120.
3353:{\displaystyle a/2}
3212:
2985:
2824:
2662:between the points
2043:the pitch angle is
1908:{\displaystyle y=a}
1414:{\displaystyle z=0}
927:polynomial equation
882:of a single curve.
716:parametric equation
452:The pitch angle of
279:James Clerk Maxwell
241:central projections
55:Archimedean spirals
51:logarithmic spirals
4172:Geometry of Curves
3611:logarithmic spiral
3350:
3321:
3319:
3184:
3143:
3101:
3074:
3047:
3020:
2993:
2991:
2877:
2796:
2762:
2707:
2652:
2611:
2459:
2439:
2419:
2370:
2321:
2297:From the equation
2287:
2144:
2098:
2033:
2031:
1998:
1975:
1973:
1933:
1905:
1873:
1845:
1820:
1776:
1751:
1706:
1654:
1584:
1505:
1488:
1451:
1411:
1383:
1349:central projection
1345:
1320:
1318:
1279:
1277:
1253:Archimedean spiral
1237:
1226:
1196:
1165:
1135:
1054:
1003:
1001:
952:
868:
842:
816:
728:
702:
666:
628:
604:
569:
421:
399:
375:
373:
317:logarithmic spiral
309:inverse-square law
213:
193:
170:{\displaystyle a.}
167:
141:
105:
35:
4525:
4524:
4414:
4413:
3688:Cotesium, Rogerum
3662:Bernoulli, Johann
3637:Annals of Science
3257:
3182:
2949:
2909:
2905:
2855:
2844:
2606:
2540:
2442:{\displaystyle r}
2282:
2088:
2030:
1972:
1883:the curve has an
1862:
1830:
1805:
1793:
1761:
1736:
1701:
1625:
1487:
1317:
1276:
1225:
1124:
1098:
1052:
1000:
801:
770:
631:{\displaystyle a}
581:polar coordinates
554:
505:up the axis of a
424:{\displaystyle x}
402:{\displaystyle a}
372:
231:would describe a
225:reciprocal spiral
196:{\displaystyle r}
183:relation between
136:
82:polar coordinates
39:hyperbolic spiral
4545:
4351:
4330:Boerdijk–Coxeter
4307:
4306:
4246:
4239:
4232:
4223:
4204:
4203:
4191:
4185:
4184:
4163:
4157:
4151:
4126:
4120:
4118:
4089:
4080:
4079:
4049:Series, Caroline
4041:
4035:
4034:
4013:
4004:
4003:
3991:
3985:
3983:
3950:
3933:
3932:
3914:10.1037/h0022204
3895:
3886:
3885:
3875:
3869:
3868:
3867:
3857:
3848:(2): 1074–1083,
3835:
3829:
3828:
3799:
3793:
3792:
3791:
3773:
3751:
3745:
3744:
3720:(418): 262–266,
3707:
3701:
3695:
3684:
3675:
3669:
3658:
3652:
3651:
3632:
3626:
3620:
3614:
3603:
3597:
3596:
3585:
3576:
3575:
3564:Varignon, Pierre
3560:
3554:
3553:
3528:
3519:
3518:
3503:
3492:
3490:
3465:
3456:
3455:
3443:
3434:
3433:
3432:
3395:
3386:
3385:
3375:
3361:
3359:
3357:
3356:
3351:
3346:
3330:
3328:
3327:
3322:
3320:
3313:
3312:
3303:
3302:
3281:
3280:
3265:
3264:
3258:
3250:
3242:
3231:
3230:
3211:
3210:
3209:
3199:
3198:
3197:
3183:
3175:
3152:
3150:
3149:
3144:
3139:
3110:
3108:
3107:
3102:
3100:
3099:
3083:
3081:
3080:
3075:
3073:
3072:
3056:
3054:
3053:
3048:
3046:
3045:
3029:
3027:
3026:
3021:
3019:
3018:
3002:
3000:
2999:
2994:
2992:
2984:
2983:
2982:
2972:
2971:
2970:
2960:
2956:
2955:
2951:
2950:
2948:
2947:
2932:
2910:
2904:
2903:
2888:
2887:
2867:
2856:
2854:
2853:
2843:
2842:
2827:
2826:
2823:
2822:
2821:
2811:
2810:
2809:
2771:
2769:
2768:
2763:
2758:
2757:
2742:
2741:
2716:
2714:
2713:
2708:
2703:
2702:
2687:
2686:
2661:
2659:
2658:
2653:
2648:
2620:
2618:
2617:
2612:
2607:
2605:
2604:
2603:
2599:
2586:
2585:
2573:
2572:
2556:
2555:
2546:
2541:
2539:
2538:
2537:
2533:
2524:
2520:
2513:
2512:
2493:
2492:
2483:
2468:
2466:
2465:
2460:
2449:or of the angle
2448:
2446:
2445:
2440:
2428:
2426:
2425:
2420:
2418:
2417:
2408:
2394:
2379:
2377:
2376:
2371:
2369:
2368:
2359:
2345:
2330:
2328:
2327:
2322:
2317:
2296:
2294:
2293:
2288:
2283:
2281:
2280:
2276:
2267:
2263:
2262:
2261:
2252:
2238:
2237:
2222:
2221:
2206:
2205:
2196:
2179:
2178:
2168:
2153:
2151:
2150:
2145:
2107:
2105:
2104:
2099:
2094:
2090:
2089:
2081:
2068:
2067:
2042:
2040:
2039:
2034:
2032:
2023:
2007:
2005:
2004:
1999:
1984:
1982:
1981:
1976:
1974:
1968:
1960:
1931:
1916:
1914:
1912:
1911:
1906:
1882:
1880:
1879:
1874:
1863:
1858:
1847:
1844:
1819:
1794:
1789:
1778:
1775:
1750:
1715:
1713:
1712:
1707:
1702:
1700:
1686:
1678:
1663:
1661:
1660:
1655:
1626:
1624:
1610:
1602:
1593:
1591:
1590:
1585:
1516:
1514:
1512:
1511:
1506:
1489:
1486:
1472:
1460:
1458:
1457:
1452:
1422:
1420:
1418:
1417:
1412:
1392:
1390:
1389:
1384:
1353:spiral staircase
1331:
1329:
1327:
1326:
1321:
1319:
1310:
1288:
1286:
1285:
1280:
1278:
1269:
1246:
1244:
1243:
1238:
1227:
1218:
1205:
1203:
1202:
1197:
1168:Circle inversion
1149:hyperbolic plane
1144:
1142:
1141:
1136:
1125:
1117:
1112:
1111:
1099:
1097:
1096:
1084:
1083:
1074:
1065:
1063:
1061:
1060:
1055:
1053:
1051:
1050:
1038:
1037:
1028:
1012:
1010:
1009:
1004:
1002:
993:
987:
986:
961:
959:
958:
953:
911:
902:
893:
877:
875:
874:
869:
851:
849:
848:
843:
825:
823:
822:
817:
802:
797:
786:
771:
766:
755:
737:
735:
734:
729:
713:
711:
709:
708:
703:
675:
673:
672:
667:
637:
635:
634:
629:
613:
611:
610:
605:
578:
576:
575:
570:
555:
547:
507:spiral staircase
503:perspective view
499:Corinthian order
473:Corinthian order
465:
449:
432:
430:
428:
427:
422:
409:
408:
406:
405:
400:
386:
384:
382:
381:
376:
374:
365:
334:
293:Johann Bernoulli
256:hyperbolic plane
222:
220:
219:
214:
202:
200:
199:
194:
176:
174:
173:
168:
150:
148:
147:
142:
137:
129:
115:by the equation
114:
112:
111:
106:
63:spiral staircase
4553:
4552:
4548:
4547:
4546:
4544:
4543:
4542:
4528:
4527:
4526:
4521:
4410:
4364:
4349:
4308:
4304:
4299:
4263:
4250:
4213:
4208:
4207:
4193:
4192:
4188:
4182:
4165:
4164:
4160:
4149:
4128:
4127:
4123:
4091:
4090:
4083:
4069:
4043:
4042:
4038:
4032:
4015:
4014:
4007:
3993:
3992:
3988:
3973:
3952:
3951:
3936:
3897:
3896:
3889:
3877:
3876:
3872:
3837:
3836:
3832:
3801:
3800:
3796:
3753:
3752:
3748:
3726:10.2307/3617399
3709:
3708:
3704:
3686:
3685:
3678:
3660:
3659:
3655:
3634:
3633:
3629:
3621:
3617:
3607:Jacob Bernoulli
3604:
3600:
3587:
3586:
3579:
3562:
3561:
3557:
3551:
3530:
3529:
3522:
3505:
3504:
3495:
3488:
3467:
3466:
3459:
3445:
3444:
3437:
3397:
3396:
3389:
3377:
3376:
3372:
3367:
3334:
3333:
3332:
3318:
3317:
3294:
3272:
3240:
3239:
3222:
3201:
3189:
3167:
3155:
3154:
3121:
3120:
3117:
3091:
3086:
3085:
3064:
3059:
3058:
3037:
3032:
3031:
3010:
3005:
3004:
2990:
2989:
2974:
2962:
2939:
2924:
2920:
2895:
2882:
2878:
2865:
2864:
2845:
2834:
2813:
2801:
2786:
2774:
2773:
2749:
2733:
2719:
2718:
2694:
2678:
2664:
2663:
2630:
2629:
2626:
2587:
2577:
2564:
2557:
2547:
2504:
2503:
2499:
2498:
2494:
2484:
2471:
2470:
2451:
2450:
2431:
2430:
2409:
2387:
2382:
2381:
2360:
2338:
2333:
2332:
2299:
2298:
2253:
2245:
2229:
2228:
2224:
2223:
2214:
2197:
2189:
2170:
2169:
2156:
2155:
2121:
2120:
2113:
2076:
2072:
2056:
2045:
2044:
2010:
2009:
1990:
1989:
1961:
1941:
1940:
1929:
1922:
1891:
1890:
1888:
1885:asymptotic line
1848:
1779:
1731:
1730:
1727:
1722:
1687:
1679:
1666:
1665:
1611:
1603:
1596:
1595:
1521:
1520:
1476:
1465:
1464:
1462:
1425:
1424:
1397:
1396:
1394:
1393:onto the image
1357:
1356:
1337:
1297:
1296:
1294:
1256:
1255:
1208:
1207:
1176:
1175:
1157:
1100:
1088:
1075:
1068:
1067:
1042:
1029:
1016:
1015:
1014:
975:
964:
963:
935:
934:
919:
918:
917:
916:
915:
912:
904:
903:
897:
894:
854:
853:
828:
827:
787:
756:
740:
739:
720:
719:
679:
678:
676:
643:
642:
620:
619:
584:
583:
535:
534:
531:
526:
487:spiral galaxies
483:
482:
481:
480:
479:
475:capital in the
466:
458:
457:
450:
413:
412:
411:
391:
390:
388:
352:
351:
349:
346:
345:
344:
342:
337:
336:
335:
321:Cotes's spirals
267:Pierre Varignon
264:
252:Euclidean plane
205:
204:
185:
184:
179:Because of the
156:
155:
117:
116:
85:
84:
67:spiral galaxies
59:asymptotic line
17:
12:
11:
5:
4551:
4549:
4541:
4540:
4530:
4529:
4523:
4522:
4520:
4519:
4514:
4509:
4504:
4499:
4494:
4487:
4486:
4485:
4475:
4470:
4465:
4460:
4455:
4450:
4449:
4448:
4443:
4438:
4428:
4422:
4420:
4416:
4415:
4412:
4411:
4409:
4408:
4407:
4406:
4396:
4391:
4386:
4381:
4376:
4371:
4366:
4362:
4357:
4355:
4348:
4347:
4342:
4337:
4332:
4327:
4322:
4316:
4314:
4310:
4309:
4302:
4300:
4298:
4297:
4292:
4287:
4282:
4277:
4271:
4269:
4265:
4264:
4251:
4249:
4248:
4241:
4234:
4226:
4220:
4219:
4212:
4211:External links
4209:
4206:
4205:
4186:
4180:
4168:"Theorem 7.11"
4158:
4147:
4121:
4081:
4067:
4045:Mumford, David
4036:
4030:
4005:
3986:
3971:
3934:
3908:(5): 344–357,
3887:
3870:
3830:
3826:10.1086/113064
3794:
3764:(1): 164–171,
3746:
3702:
3676:
3670:. As cited by
3653:
3643:(6): 537–575,
3627:
3615:
3598:
3577:
3555:
3549:
3520:
3493:
3486:
3457:
3435:
3415:(5): 519–540,
3387:
3369:
3368:
3366:
3363:
3349:
3345:
3341:
3316:
3311:
3306:
3301:
3297:
3293:
3290:
3287:
3284:
3279:
3275:
3271:
3268:
3263:
3256:
3253:
3248:
3245:
3243:
3241:
3238:
3235:
3229:
3225:
3221:
3218:
3215:
3208:
3204:
3196:
3192:
3187:
3181:
3178:
3173:
3170:
3168:
3166:
3163:
3162:
3142:
3138:
3134:
3131:
3128:
3116:
3113:
3098:
3094:
3071:
3067:
3044:
3040:
3017:
3013:
2988:
2981:
2977:
2969:
2965:
2959:
2954:
2946:
2942:
2938:
2935:
2930:
2927:
2923:
2919:
2916:
2913:
2908:
2902:
2898:
2894:
2891:
2885:
2881:
2876:
2873:
2870:
2868:
2866:
2863:
2860:
2852:
2848:
2841:
2837:
2833:
2830:
2820:
2816:
2808:
2804:
2799:
2795:
2792:
2789:
2787:
2785:
2782:
2781:
2761:
2756:
2752:
2748:
2745:
2740:
2736:
2732:
2729:
2726:
2706:
2701:
2697:
2693:
2690:
2685:
2681:
2677:
2674:
2671:
2651:
2647:
2643:
2640:
2637:
2625:
2622:
2610:
2602:
2598:
2594:
2590:
2584:
2580:
2576:
2571:
2567:
2563:
2560:
2554:
2550:
2544:
2536:
2532:
2528:
2523:
2519:
2516:
2511:
2507:
2502:
2497:
2491:
2487:
2481:
2478:
2458:
2438:
2416:
2412:
2407:
2403:
2400:
2397:
2393:
2390:
2367:
2363:
2358:
2354:
2351:
2348:
2344:
2341:
2320:
2316:
2312:
2309:
2306:
2286:
2279:
2275:
2271:
2266:
2260:
2256:
2251:
2248:
2244:
2241:
2236:
2232:
2227:
2220:
2217:
2212:
2209:
2204:
2200:
2195:
2192:
2188:
2185:
2182:
2177:
2173:
2166:
2163:
2143:
2140:
2137:
2134:
2131:
2128:
2112:
2109:
2097:
2093:
2087:
2084:
2079:
2075:
2071:
2066:
2063:
2059:
2055:
2052:
2029:
2026:
2020:
2017:
1997:
1971:
1967:
1964:
1957:
1954:
1951:
1948:
1921:
1918:
1904:
1901:
1898:
1872:
1869:
1866:
1861:
1857:
1854:
1851:
1843:
1840:
1837:
1833:
1829:
1826:
1823:
1818:
1815:
1812:
1808:
1803:
1800:
1797:
1792:
1788:
1785:
1782:
1774:
1771:
1768:
1764:
1760:
1757:
1754:
1749:
1746:
1743:
1739:
1726:
1723:
1721:
1718:
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467:
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420:
398:
371:
368:
362:
359:
339:
338:
329:
328:
327:
326:
325:
263:
260:
237:inverse curves
212:
192:
166:
163:
140:
135:
132:
127:
124:
104:
101:
98:
95:
92:
31:helical curves
15:
13:
10:
9:
6:
4:
3:
2:
4550:
4539:
4536:
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4518:
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4187:
4183:
4181:9781482285673
4177:
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4148:9789401583084
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4068:9781107717190
4064:
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4031:9781498710671
4027:
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4019:
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3997:
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3972:9783030057985
3968:
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3646:
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3638:
3631:
3628:
3624:
3623:Hammer (2016)
3619:
3616:
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3608:
3602:
3599:
3594:
3590:
3584:
3582:
3578:
3573:
3569:
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3556:
3552:
3550:84-930669-1-5
3546:
3542:
3538:
3537:Séquin, Carlo
3534:
3527:
3525:
3521:
3517:
3513:
3509:
3502:
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3494:
3489:
3487:9789401583084
3483:
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3277:
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3254:
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3213:
3206:
3202:
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3112:
3096:
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3015:
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2806:
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2734:
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2649:
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2641:
2638:
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2623:
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2608:
2600:
2596:
2592:
2582:
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2574:
2569:
2565:
2558:
2552:
2548:
2542:
2534:
2530:
2526:
2521:
2517:
2514:
2509:
2505:
2500:
2495:
2489:
2485:
2479:
2476:
2456:
2436:
2414:
2410:
2405:
2401:
2398:
2395:
2391:
2388:
2365:
2361:
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2318:
2314:
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2307:
2304:
2284:
2277:
2273:
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2258:
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2246:
2239:
2234:
2230:
2225:
2218:
2215:
2210:
2207:
2202:
2193:
2190:
2183:
2180:
2175:
2171:
2164:
2161:
2138:
2132:
2129:
2126:
2118:
2110:
2108:
2095:
2091:
2085:
2082:
2077:
2073:
2069:
2064:
2061:
2057:
2053:
2050:
2027:
2024:
2018:
2015:
1995:
1988:
1969:
1965:
1962:
1955:
1952:
1949:
1946:
1938:
1926:
1919:
1917:
1902:
1899:
1896:
1886:
1870:
1867:
1864:
1859:
1855:
1852:
1849:
1841:
1835:
1827:
1824:
1821:
1816:
1810:
1801:
1795:
1790:
1786:
1783:
1780:
1772:
1766:
1758:
1755:
1752:
1747:
1741:
1724:
1719:
1717:
1703:
1697:
1694:
1691:
1688:
1683:
1680:
1674:
1671:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1621:
1618:
1615:
1612:
1607:
1604:
1594:is the curve
1581:
1578:
1575:
1572:
1568:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1517:
1499:
1496:
1493:
1483:
1480:
1477:
1473:
1445:
1442:
1439:
1436:
1433:
1408:
1405:
1402:
1377:
1374:
1371:
1368:
1365:
1354:
1350:
1341:
1334:
1332:
1314:
1311:
1305:
1302:
1292:
1291:inverse curve
1273:
1270:
1264:
1261:
1254:
1250:
1231:
1228:
1222:
1219:
1190:
1187:
1184:
1173:
1169:
1161:
1154:
1152:
1150:
1145:
1132:
1129:
1126:
1121:
1118:
1113:
1108:
1105:
1101:
1093:
1089:
1085:
1080:
1076:
1047:
1043:
1039:
1034:
1030:
1024:
1021:
997:
994:
988:
983:
980:
976:
972:
969:
949:
946:
943:
940:
932:
928:
924:
910:
900:
892:
883:
881:
865:
862:
859:
839:
836:
833:
813:
810:
807:
803:
798:
794:
791:
788:
782:
779:
776:
772:
767:
763:
760:
757:
751:
748:
745:
725:
717:
699:
696:
693:
690:
687:
684:
663:
660:
657:
654:
651:
648:
641:
625:
617:
598:
595:
592:
582:
566:
563:
560:
556:
551:
548:
543:
540:
528:
524:Constructions
523:
521:
519:
514:
512:
508:
504:
500:
496:
492:
488:
478:
474:
470:
464:
455:
448:
439:
437:
418:
396:
369:
366:
360:
357:
341:
333:
324:
322:
318:
314:
310:
306:
305:conic section
302:
298:
294:
290:
288:
284:
280:
276:
272:
268:
261:
259:
257:
253:
248:
246:
242:
238:
234:
230:
226:
210:
190:
182:
177:
164:
161:
154:
138:
133:
130:
125:
122:
99:
96:
93:
83:
79:
74:
72:
68:
64:
60:
56:
52:
48:
44:
41:is a type of
40:
32:
28:
23:
19:
4489:
4440:
4354:Biochemistry
4199:
4189:
4171:
4161:
4130:
4124:
4102:(2): 45–53,
4099:
4093:
4057:
4039:
4021:
3999:
3989:
3954:
3905:
3899:
3880:
3873:
3845:
3839:
3833:
3809:
3803:
3797:
3761:
3755:
3749:
3717:
3711:
3705:
3691:
3665:
3656:
3640:
3636:
3630:
3618:
3601:
3592:
3571:
3558:
3540:
3507:
3491:; see p. 138
3469:
3451:
3412:
3406:
3380:
3373:
3118:
2627:
2114:
1934:
1728:
1518:
1346:
1170:through the
1166:
1146:
920:
898:
879:
714:obtaining a
618:coefficient
532:
515:
491:architecture
484:
347:
301:Isaac Newton
291:
270:
265:
249:
224:
178:
153:scale factor
75:
38:
36:
18:
4502:Pitch angle
4478:Logarithmic
4426:Archimedean
4389:Polyproline
3115:Sector area
1987:pitch angle
1920:Pitch angle
1172:unit circle
513:structure.
297:Roger Cotes
271:polar curve
78:plane curve
47:pitch angle
4491:On Spirals
4441:Hyperbolic
3365:References
2624:Arc length
1725:Asymptotes
1720:Properties
181:reciprocal
29:. Several
4512:Spirangle
4507:Theodorus
4446:Poinsot's
4436:Epispiral
4280:Curvature
4275:Algebraic
3981:150149152
3930:614277135
3855:1309.4308
3771:0908.0892
3742:189050097
3296:φ
3286:−
3274:φ
3237:φ
3220:φ
3203:φ
3191:φ
3186:∫
3141:φ
3093:φ
3066:φ
3039:φ
3012:φ
2976:φ
2964:φ
2941:φ
2926:φ
2918:
2907:φ
2897:φ
2884:−
2862:φ
2847:φ
2836:φ
2815:φ
2803:φ
2798:∫
2751:φ
2735:φ
2696:φ
2680:φ
2650:φ
2506:φ
2486:φ
2477:κ
2457:φ
2411:φ
2362:φ
2350:−
2319:φ
2208:−
2162:κ
2139:φ
2117:curvature
2111:Curvature
2086:φ
2078:−
2070:
2062:−
2051:α
2028:φ
1996:α
1953:α
1950:
1889:equation
1860:φ
1856:φ
1853:
1839:→
1836:φ
1814:→
1811:φ
1799:∞
1791:φ
1787:φ
1784:
1770:→
1767:φ
1745:→
1742:φ
1692:−
1672:ρ
1646:
1634:
1616:−
1576:≠
1551:
1536:
1481:−
1315:φ
1295:equation
1271:φ
1232:φ
1191:φ
1155:Inversion
1114:
1106:−
989:
981:−
970:φ
944:φ
863:≠
860:φ
834:φ
808:φ
799:φ
795:φ
792:
768:φ
764:φ
761:
726:φ
700:φ
697:
664:φ
661:
599:φ
561:φ
552:φ
509:or other
370:φ
350:equation
283:roulettes
275:hyperbola
233:hyperbola
211:φ
134:φ
100:φ
4532:Category
4468:Involute
4463:Fermat's
4404:Collagen
4340:Symmetry
3926:ProQuest
3668:: 521–33
3574:: 94–103
3566:(1704),
3539:(eds.),
3401:(1849),
2392:″
2343:′
2250:′
2219:″
2194:′
1985:for the
1966:′
1461:to the
1066:giving:
880:branches
454:NGC 4622
436:involute
287:tractrix
4538:Spirals
4497:Padovan
4431:Cotes's
4419:Spirals
4325:Antenna
4313:Helices
4285:Gallery
4261:helices
4253:Spirals
4116:2973138
4077:3558870
3922:5318086
3814:Bibcode
3776:Bibcode
3734:3617399
3430:2250749
3427::
511:helical
495:volutes
469:Volutes
389:length
245:helixes
45:with a
4483:Golden
4399:Triple
4379:Double
4345:Triple
4295:Topics
4268:Curves
4257:curves
4178:
4145:
4114:
4075:
4065:
4028:
3979:
3969:
3928:
3920:
3740:
3732:
3547:
3484:
3425:Zenodo
1463:point
1395:plane
1251:of an
901:> 0
431:-axis.
43:spiral
4458:Euler
4453:Doyle
4394:Super
4369:Alpha
4320:Angle
4112:JSTOR
3977:S2CID
3850:arXiv
3766:arXiv
3738:S2CID
3730:JSTOR
3698:p. 34
1935:From
1887:with
1249:image
616:scale
471:on a
76:As a
4517:Ulam
4473:List
4374:Beta
4335:Hemi
4290:List
4259:and
4176:ISBN
4143:ISBN
4063:ISBN
4026:ISBN
3967:ISBN
3918:PMID
3545:ISBN
3482:ISBN
3153:is:
3030:and
2717:and
2380:and
2115:The
1347:The
1013:and
837:>
811:>
677:and
614:and
579:for
564:>
295:and
203:and
69:and
4135:doi
4104:doi
3959:doi
3910:doi
3860:doi
3846:436
3822:doi
3784:doi
3762:397
3722:doi
3645:doi
3512:doi
3474:doi
3417:doi
2154:is
2058:tan
1947:tan
1850:sin
1832:lim
1807:lim
1781:cos
1763:lim
1738:lim
1643:sin
1631:cos
1548:sin
1533:cos
1102:tan
977:tan
852:to
789:sin
758:cos
694:sin
658:cos
243:of
4534::
4384:Pi
4363:10
4255:,
4198:,
4170:,
4141:,
4110:,
4100:26
4098:,
4084:^
4073:MR
4071:,
4055:,
4047:;
4020:,
4008:^
3998:,
3975:,
3965:,
3937:^
3924:,
3916:,
3906:72
3904:,
3890:^
3858:,
3844:,
3820:,
3810:86
3808:,
3782:,
3774:,
3760:,
3736:,
3728:,
3718:61
3716:,
3679:^
3641:52
3639:,
3591:,
3580:^
3570:,
3523:^
3496:^
3480:,
3460:^
3450:,
3438:^
3423:,
3413:16
3411:,
3405:,
3390:^
3111:.
2915:ln
814:0.
289:.
247:.
73:.
37:A
4361:3
4245:e
4238:t
4231:v
4137::
4106::
3961::
3912::
3862::
3852::
3824::
3816::
3786::
3778::
3768::
3724::
3647::
3613:.
3514::
3476::
3419::
3360:.
3348:2
3344:/
3340:a
3315:.
3310:)
3305:)
3300:2
3292:(
3289:r
3283:)
3278:1
3270:(
3267:r
3262:(
3255:2
3252:a
3247:=
3234:d
3228:2
3224:)
3217:(
3214:r
3207:2
3195:1
3180:2
3177:1
3172:=
3165:A
3137:/
3133:a
3130:=
3127:r
3097:2
3070:1
3043:2
3016:1
2987:.
2980:2
2968:1
2958:]
2953:)
2945:2
2937:+
2934:1
2929:+
2922:(
2912:+
2901:2
2893:+
2890:1
2880:[
2875:a
2872:=
2859:d
2851:2
2840:2
2832:+
2829:1
2819:2
2807:1
2794:a
2791:=
2784:L
2760:)
2755:2
2747:,
2744:)
2739:2
2731:(
2728:r
2725:(
2705:)
2700:1
2692:,
2689:)
2684:1
2676:(
2673:r
2670:(
2646:/
2642:a
2639:=
2636:r
2609:.
2601:2
2597:/
2593:3
2589:)
2583:2
2579:r
2575:+
2570:2
2566:a
2562:(
2559:r
2553:3
2549:a
2543:=
2535:2
2531:/
2527:3
2522:)
2518:1
2515:+
2510:2
2501:(
2496:a
2490:4
2480:=
2437:r
2415:3
2406:/
2402:a
2399:2
2396:=
2389:r
2366:2
2357:/
2353:a
2347:=
2340:r
2315:/
2311:a
2308:=
2305:r
2285:.
2278:2
2274:/
2270:3
2265:)
2259:2
2255:)
2247:r
2243:(
2240:+
2235:2
2231:r
2226:(
2216:r
2211:r
2203:2
2199:)
2191:r
2187:(
2184:2
2181:+
2176:2
2172:r
2165:=
2142:)
2136:(
2133:r
2130:=
2127:r
2096:.
2092:)
2083:1
2074:(
2065:1
2054:=
2025:a
2019:=
2016:r
1970:r
1963:r
1956:=
1930:α
1915:.
1903:a
1900:=
1897:y
1871:,
1868:a
1865:=
1842:0
1828:a
1825:=
1822:y
1817:0
1802:,
1796:=
1773:0
1759:a
1756:=
1753:x
1748:0
1704:,
1698:t
1695:c
1689:d
1684:r
1681:d
1675:=
1652:)
1649:t
1640:,
1637:t
1628:(
1622:t
1619:c
1613:d
1608:r
1605:d
1582:,
1579:0
1573:c
1569:,
1566:)
1563:t
1560:c
1557:,
1554:t
1545:r
1542:,
1539:t
1530:r
1527:(
1515:.
1503:)
1500:y
1497:,
1494:x
1491:(
1484:z
1478:d
1474:d
1449:)
1446:z
1443:,
1440:y
1437:,
1434:x
1431:(
1421:.
1409:0
1406:=
1403:z
1381:)
1378:d
1375:,
1372:0
1369:,
1366:0
1363:(
1330:.
1312:a
1306:=
1303:r
1274:a
1265:=
1262:r
1235:)
1229:,
1223:r
1220:1
1214:(
1194:)
1188:,
1185:r
1182:(
1133:.
1130:a
1127:=
1122:x
1119:y
1109:1
1094:2
1090:y
1086:+
1081:2
1077:x
1064:,
1048:2
1044:y
1040:+
1035:2
1031:x
1025:=
1022:r
998:x
995:y
984:1
973:=
950:a
947:=
941:r
899:φ
866:0
840:0
804:,
783:a
780:=
777:y
773:,
752:a
749:=
746:x
712:,
691:r
688:=
685:y
655:r
652:=
649:x
626:a
602:)
596:,
593:r
590:(
567:0
557:,
549:a
544:=
541:r
419:x
397:a
385:,
367:a
361:=
358:r
191:r
165:.
162:a
139:,
131:a
126:=
123:r
103:)
97:,
94:r
91:(
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