3282:
1813:
3101:
1271:
3320:
3336:
2724:
1808:{\displaystyle {\begin{aligned}\mathbf {r} &=a\cos t\mathbf {i} +a\sin t\mathbf {j} +bt\mathbf {k} \\\mathbf {v} &=-a\sin t\mathbf {i} +a\cos t\mathbf {j} +b\mathbf {k} \\\mathbf {a} &=-a\cos t\mathbf {i} -a\sin t\mathbf {j} +0\mathbf {k} \\|\mathbf {v} |&={\sqrt {(-a\sin t)^{2}+(a\cos t)^{2}+b^{2}}}={\sqrt {a^{2}+b^{2}}}\\|\mathbf {a} |&={\sqrt {(-a\sin t)^{2}+(a\cos t)^{2}}}=a\\s(t)&=\int _{0}^{t}{\sqrt {a^{2}+b^{2}}}d\tau ={\sqrt {a^{2}+b^{2}}}t\end{aligned}}}
3304:
3721:
3096:{\displaystyle {\begin{aligned}\mathbf {B} =\mathbf {T} \times \mathbf {N} &={\frac {1}{\sqrt {a^{2}+b^{2}}}}\left(b\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} -b\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +a\mathbf {k} \right)\\{\frac {d\mathbf {B} }{ds}}&={\frac {1}{a^{2}+b^{2}}}\left(b\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} +b\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +0\mathbf {k} \right)\end{aligned}}}
319:
2247:
343:
2478:
54:
46:
2004:
62:
1995:
2256:
2715:
310:
Helices can be either right-handed or left-handed. With the line of sight along the helix's axis, if a clockwise screwing motion moves the helix away from the observer, then it is called a right-handed helix; if towards the observer, then it is a left-handed helix. Handedness (or
2242:{\displaystyle {\frac {d\mathbf {r} }{ds}}=\mathbf {T} ={\frac {-a}{\sqrt {a^{2}+b^{2}}}}\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} +{\frac {a}{\sqrt {a^{2}+b^{2}}}}\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +{\frac {b}{\sqrt {a^{2}+b^{2}}}}\mathbf {k} }
1826:
873:
511:
3270:
is the distance an element of an airplane propeller would advance in one revolution if it were moving along a helix having an angle equal to that between the chord of the element and a plane perpendicular to the propeller axis; see also:
693:
2582:
3194:
2592:
2473:{\displaystyle {\frac {d\mathbf {T} }{ds}}=\kappa \mathbf {N} ={\frac {-a}{a^{2}+b^{2}}}\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} +{\frac {-a}{a^{2}+b^{2}}}\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +0\mathbf {k} }
1276:
907:, and changes of scale, all right-handed helices are equivalent to the helix defined above. The equivalent left-handed helix can be constructed in a number of ways, the simplest being to negate any one of the
2729:
3281:
1053:
742:
595:
389:
1122:
1196:
1258:
3319:
329:
of helices. One is left-handed and the other is right-handed. Each row compares the two helices from a different perspective. The chirality is a property of the object, not of the
1990:{\displaystyle \mathbf {r} (s)=a\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} +a\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +{\frac {bs}{\sqrt {a^{2}+b^{2}}}}\mathbf {k} }
2487:
3108:
315:) is a property of the helix, not of the perspective: a right-handed helix cannot be turned to look like a left-handed one unless it is viewed in a mirror, and vice versa.
3615:
Schmitt, J.-L.; Stadler, A.-M.; Kyritsakas, N.; Lehn, J.-M. (2003). "Helicity-Encoded
Molecular Strands: Efficient Access by the Hydrazone Route and Structural Features".
737:
384:
291:
if its principal normal makes a constant angle with a fixed line in space. It can be constructed by applying a transformation to the moving frame of a general helix.
3447:
590:
3549:
3303:
3335:
2710:{\displaystyle \mathbf {N} =-\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} -\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +0\mathbf {k} }
3659:
3440:
363:
3504:
3496:
265:
on a conic surface, with the distance to the apex an exponential function of the angle indicating direction from the axis.
969:
276:
if its tangent makes a constant angle with a fixed line in space. A curve is a general helix if and only if the ratio of
3221:
Some curves found in nature consist of multiple helices of different handedness joined together by transitions known as
375:
3745:
3366:
1068:
3958:
3794:
3386:
3204:
1137:
3546:
1211:
152:
3573:
3917:
3211:
566:
326:
312:
200:, "twisted, curved". A "filled-in" helix – for example, a "spiral" (helical) ramp – is a surface called a
3272:
904:
39:
3912:
3710:
3652:
3371:
230:
3861:
868:{\displaystyle {\begin{aligned}x(t)&=a\cos(t),\\y(t)&=a\sin(t),\\z(t)&=bt.\end{aligned}}}
3922:
961:
371:
3255:
is often modeled with helices or double helices, most often extending out of a circle such as the
506:{\displaystyle {\begin{aligned}x(t)&=\cos(t),\\y(t)&=\sin(t),\\z(t)&=t.\end{aligned}}}
3893:
3841:
3260:
3222:
1202:
3878:
889:
3856:
3846:
3804:
3700:
3591:
3469:
281:
245:
896:
and the real and imaginary parts of the function value give this plot three real dimensions.
3953:
3755:
3645:
3624:
3406:
3381:
3256:
688:{\displaystyle {\begin{aligned}r(t)&=1,\\\theta (t)&=t,\\h(t)&=t.\end{aligned}}}
299:
106:
2577:{\displaystyle \kappa =\left|{\frac {d\mathbf {T} }{ds}}\right|={\frac {|a|}{a^{2}+b^{2}}}}
3963:
3888:
3740:
3690:
3577:
3553:
3500:
3451:
3292:
878:
Another way of mathematically constructing a helix is to plot the complex-valued function
3189:{\displaystyle \tau =\left|{\frac {d\mathbf {B} }{ds}}\right|={\frac {b}{a^{2}+b^{2}}}.}
3819:
3705:
3376:
3326:
191:
3947:
3898:
330:
148:
3873:
3868:
3814:
3760:
3421:
3396:
3229:
295:
219:
175:
159:
35:
318:
3493:
3932:
3784:
3735:
3391:
3356:
3342:
3252:
3215:
359:
253:
183:
155:
144:
3906:
3809:
3789:
3637:
3472:
3416:
3361:
1059:
342:
3927:
3851:
3776:
3750:
3695:
3599:
3477:
3401:
1128:
517:
367:
277:
241:
3628:
3594:
53:
3883:
3799:
3411:
3288:
900:
233:) helices with the same axis, differing by a translation along the axis.
202:
45:
179:
167:
17:
3567:
An
Explicit Parametrization of the Frenet Apparatus of the Slant Helix
378:
defines a particular helix; perhaps the simplest equations for one is
3668:
262:
3232:
are right-handed helices. The alpha helix in biology as well as the
3672:
3570:
3248:
3241:
3237:
3233:
317:
163:
60:
52:
44:
3310:
61:
3641:
171:
115:
31:
3441:
195:
294:
For more general helix-like space curves can be found, see
3203:
An example of a double helix in molecular biology is the
121:
3111:
2727:
2595:
2490:
2259:
2007:
1829:
1274:
1263:
A helix has constant non-zero curvature and torsion.
1214:
1140:
1071:
972:
740:
593:
387:
127:
124:
118:
3834:
3769:
3728:
3683:
1817:So a helix can be reparameterized as a function of
1048:{\displaystyle t\mapsto (a\cos t,a\sin t,bt),t\in }
112:
109:
49:(l-r) Tension, compression and torsion coil springs
3188:
3095:
2709:
2576:
2472:
2241:
1989:
1807:
1252:
1190:
1116:
1047:
867:
687:
505:
240:(i.e. one with constant radius) has constant band
30:This article is about the shape. For the shape of
731:) is described by the following parametrisation:
3240:forms of DNA are also right-handed helices. The
92:with arrowheads showing direction of increasing
1117:{\displaystyle A=T\cdot {\sqrt {a^{2}+b^{2}}},}
218:of a helix is the height of one complete helix
222:, measured parallel to the axis of the helix.
3653:
8:
3544:New special curves and developable surfaces.
960:) expressed in Cartesian coordinates as the
562:-axis, in a right-handed coordinate system.
1191:{\displaystyle {\frac {|a|}{a^{2}+b^{2}}},}
3766:
3660:
3646:
3638:
166:to a fixed axis. Helices are important in
3174:
3161:
3151:
3128:
3122:
3110:
3079:
3068:
3059:
3046:
3036:
3019:
3010:
2997:
2987:
2964:
2951:
2941:
2918:
2912:
2898:
2887:
2878:
2865:
2855:
2838:
2829:
2816:
2806:
2783:
2770:
2760:
2748:
2740:
2732:
2728:
2726:
2702:
2691:
2682:
2669:
2659:
2645:
2636:
2623:
2613:
2596:
2594:
2565:
2552:
2541:
2533:
2530:
2507:
2501:
2489:
2465:
2454:
2445:
2432:
2422:
2407:
2394:
2379:
2371:
2362:
2349:
2339:
2324:
2311:
2296:
2288:
2266:
2260:
2258:
2234:
2225:
2212:
2202:
2194:
2185:
2172:
2162:
2147:
2134:
2124:
2116:
2107:
2094:
2084:
2069:
2056:
2041:
2033:
2014:
2008:
2006:
1982:
1973:
1960:
1945:
1937:
1928:
1915:
1905:
1888:
1879:
1866:
1856:
1830:
1828:
1790:
1777:
1771:
1754:
1741:
1735:
1729:
1724:
1683:
1655:
1631:
1619:
1614:
1609:
1597:
1584:
1578:
1567:
1554:
1526:
1502:
1490:
1485:
1480:
1471:
1460:
1440:
1413:
1404:
1393:
1373:
1346:
1337:
1323:
1303:
1279:
1275:
1273:
1253:{\displaystyle {\frac {b}{a^{2}+b^{2}}}.}
1238:
1225:
1215:
1213:
1176:
1163:
1152:
1144:
1141:
1139:
1103:
1090:
1084:
1070:
971:
741:
739:
594:
592:
388:
386:
341:
3433:
3309:A natural left-handed helix, made by a
3277:
3454:, Henry George Liddell, Robert Scott,
1266:A helix is the vector-valued function
323:Two types of helix shown in comparison
584:, the same helix is parametrised by:
551:traces a right-handed helix of pitch
182:have helical substructures, known as
7:
3542:Izumiya, S. and Takeuchi, N. (2004)
558:(or slope 1) and radius 1 about the
27:Space curve that winds around a line
3210:An example of a conic helix is the
136:
25:
3531:Elementary Differential Geometry,
3518:Elementary Differential Geometry,
923:Arc length, curvature and torsion
884:as a function of the real number
3719:
3334:
3325:A charged particle in a uniform
3318:
3302:
3280:
3129:
3080:
3069:
3020:
2919:
2899:
2888:
2839:
2749:
2741:
2733:
2703:
2692:
2646:
2597:
2508:
2466:
2455:
2372:
2289:
2267:
2235:
2195:
2117:
2034:
2015:
1983:
1938:
1889:
1831:
1615:
1486:
1472:
1461:
1441:
1414:
1405:
1394:
1374:
1347:
1338:
1324:
1304:
1280:
143:) is a shape like a cylindrical
105:
346:A helix composed of sinusoidal
3505:Wolfram Demonstrations Project
2542:
2534:
1841:
1835:
1710:
1704:
1680:
1664:
1652:
1633:
1620:
1610:
1551:
1535:
1523:
1504:
1491:
1481:
1153:
1145:
1042:
1030:
1018:
979:
976:
842:
836:
823:
817:
798:
792:
779:
773:
754:
748:
665:
659:
636:
630:
607:
601:
483:
477:
464:
458:
442:
436:
423:
417:
401:
395:
1:
1821:, which must be unit-speed:
1999:The unit tangent vector is
927:A circular helix of radius
698:A circular helix of radius
229:consists of two (typically
3980:
3442:
2587:The unit normal vector is
196:
29:
3717:
3205:nucleic acid double helix
3329:following a helical path
338:Mathematical description
3456:A Greek-English Lexicon
3287:Crystal structure of a
3244:of DNA is left-handed.
2719:The binormal vector is
567:cylindrical coordinates
176:two intertwined helices
65:The right-handed helix
3629:10.1002/hlca.200390137
3617:Helvetica Chimica Acta
3565:Menninger, T. (2013),
3367:Boerdijk–Coxeter helix
3289:folded molecular helix
3273:pitch angle (aviation)
3190:
3097:
2711:
2578:
2474:
2243:
1991:
1809:
1254:
1192:
1118:
1049:
869:
689:
507:
355:
334:
261:, may be defined as a
174:molecule is formed as
96:
58:
50:
40:Helix (disambiguation)
38:. For other uses, see
3372:Circular polarization
3259:, so as to represent
3191:
3098:
2712:
2579:
2475:
2251:The normal vector is
2244:
1992:
1810:
1255:
1193:
1119:
1050:
870:
690:
523:increases, the point
508:
376:Cartesian coordinates
370:space. The following
345:
325:. This shows the two
321:
64:
56:
48:
32:deoxyribonucleic acid
3109:
2725:
2593:
2488:
2257:
2005:
1827:
1272:
1212:
1138:
1069:
970:
738:
591:
385:
287:A curve is called a
268:A curve is called a
210:Properties and types
3503:" by Sándor Kabai,
3223:tendril perversions
1734:
962:parametric equation
147:or the thread of a
3592:Weisstein, Eric W.
3576:2018-02-05 at the
3552:2016-03-04 at the
3499:2008-04-30 at the
3470:Weisstein, Eric W.
3450:2012-10-16 at the
3261:octave equivalency
3214:roller coaster at
3186:
3093:
3091:
2707:
2574:
2470:
2239:
1987:
1805:
1803:
1720:
1250:
1188:
1114:
1045:
865:
863:
685:
683:
503:
501:
356:
335:
257:, also known as a
151:. It is a type of
97:
59:
51:
3941:
3940:
3830:
3829:
3181:
3142:
3066:
3065:
3017:
3016:
2971:
2932:
2885:
2884:
2836:
2835:
2790:
2789:
2689:
2688:
2643:
2642:
2572:
2521:
2482:Its curvature is
2452:
2451:
2414:
2369:
2368:
2331:
2280:
2232:
2231:
2192:
2191:
2154:
2153:
2114:
2113:
2076:
2075:
2028:
1980:
1979:
1935:
1934:
1886:
1885:
1796:
1760:
1689:
1603:
1573:
1245:
1183:
1109:
274:cylindrical helix
16:(Redirected from
3971:
3959:Geometric shapes
3767:
3746:Boerdijk–Coxeter
3723:
3722:
3662:
3655:
3648:
3639:
3633:
3632:
3623:(5): 1598–1624.
3612:
3606:
3605:
3604:
3587:
3581:
3563:
3557:
3540:
3534:
3527:
3521:
3514:
3508:
3490:
3484:
3483:
3482:
3465:
3459:
3445:
3444:
3438:
3407:Seashell surface
3382:Helical symmetry
3338:
3322:
3306:
3284:
3257:circle of fifths
3218:amusement park.
3195:
3193:
3192:
3187:
3182:
3180:
3179:
3178:
3166:
3165:
3152:
3147:
3143:
3141:
3133:
3132:
3123:
3102:
3100:
3099:
3094:
3092:
3088:
3084:
3083:
3072:
3067:
3064:
3063:
3051:
3050:
3041:
3037:
3023:
3018:
3015:
3014:
3002:
3001:
2992:
2988:
2972:
2970:
2969:
2968:
2956:
2955:
2942:
2933:
2931:
2923:
2922:
2913:
2907:
2903:
2902:
2891:
2886:
2883:
2882:
2870:
2869:
2860:
2856:
2842:
2837:
2834:
2833:
2821:
2820:
2811:
2807:
2791:
2788:
2787:
2775:
2774:
2765:
2761:
2752:
2744:
2736:
2716:
2714:
2713:
2708:
2706:
2695:
2690:
2687:
2686:
2674:
2673:
2664:
2660:
2649:
2644:
2641:
2640:
2628:
2627:
2618:
2614:
2600:
2583:
2581:
2580:
2575:
2573:
2571:
2570:
2569:
2557:
2556:
2546:
2545:
2537:
2531:
2526:
2522:
2520:
2512:
2511:
2502:
2479:
2477:
2476:
2471:
2469:
2458:
2453:
2450:
2449:
2437:
2436:
2427:
2423:
2415:
2413:
2412:
2411:
2399:
2398:
2388:
2380:
2375:
2370:
2367:
2366:
2354:
2353:
2344:
2340:
2332:
2330:
2329:
2328:
2316:
2315:
2305:
2297:
2292:
2281:
2279:
2271:
2270:
2261:
2248:
2246:
2245:
2240:
2238:
2233:
2230:
2229:
2217:
2216:
2207:
2203:
2198:
2193:
2190:
2189:
2177:
2176:
2167:
2163:
2155:
2152:
2151:
2139:
2138:
2129:
2125:
2120:
2115:
2112:
2111:
2099:
2098:
2089:
2085:
2077:
2074:
2073:
2061:
2060:
2051:
2050:
2042:
2037:
2029:
2027:
2019:
2018:
2009:
1996:
1994:
1993:
1988:
1986:
1981:
1978:
1977:
1965:
1964:
1955:
1954:
1946:
1941:
1936:
1933:
1932:
1920:
1919:
1910:
1906:
1892:
1887:
1884:
1883:
1871:
1870:
1861:
1857:
1834:
1820:
1814:
1812:
1811:
1806:
1804:
1797:
1795:
1794:
1782:
1781:
1772:
1761:
1759:
1758:
1746:
1745:
1736:
1733:
1728:
1690:
1688:
1687:
1660:
1659:
1632:
1623:
1618:
1613:
1604:
1602:
1601:
1589:
1588:
1579:
1574:
1572:
1571:
1559:
1558:
1531:
1530:
1503:
1494:
1489:
1484:
1475:
1464:
1444:
1417:
1408:
1397:
1377:
1350:
1341:
1327:
1307:
1283:
1259:
1257:
1256:
1251:
1246:
1244:
1243:
1242:
1230:
1229:
1216:
1197:
1195:
1194:
1189:
1184:
1182:
1181:
1180:
1168:
1167:
1157:
1156:
1148:
1142:
1123:
1121:
1120:
1115:
1110:
1108:
1107:
1095:
1094:
1085:
1054:
1052:
1051:
1046:
959:
952:
951:
949:
948:
943:
940:
930:
918:
914:
910:
895:
892:). The value of
887:
883:
874:
872:
871:
866:
864:
730:
723:
722:
720:
719:
714:
711:
701:
694:
692:
691:
686:
684:
583:
561:
557:
550:
522:
512:
510:
509:
504:
502:
353:
349:
300:spherical spiral
199:
198:
138:
134:
133:
130:
129:
126:
123:
120:
117:
114:
111:
95:
91:
80:
21:
3979:
3978:
3974:
3973:
3972:
3970:
3969:
3968:
3944:
3943:
3942:
3937:
3826:
3780:
3765:
3724:
3720:
3715:
3679:
3666:
3636:
3614:
3613:
3609:
3590:
3589:
3588:
3584:
3578:Wayback Machine
3571:arXiv:1302.3175
3564:
3560:
3554:Wayback Machine
3541:
3537:
3528:
3524:
3515:
3511:
3501:Wayback Machine
3491:
3487:
3468:
3467:
3466:
3462:
3452:Wayback Machine
3439:
3435:
3431:
3426:
3352:
3345:
3339:
3330:
3323:
3314:
3307:
3298:
3285:
3268:geometric pitch
3201:
3170:
3157:
3156:
3134:
3124:
3118:
3107:
3106:
3105:Its torsion is
3090:
3089:
3055:
3042:
3006:
2993:
2977:
2973:
2960:
2947:
2946:
2934:
2924:
2914:
2909:
2908:
2874:
2861:
2825:
2812:
2796:
2792:
2779:
2766:
2753:
2723:
2722:
2678:
2665:
2632:
2619:
2591:
2590:
2561:
2548:
2547:
2532:
2513:
2503:
2497:
2486:
2485:
2441:
2428:
2403:
2390:
2389:
2381:
2358:
2345:
2320:
2307:
2306:
2298:
2272:
2262:
2255:
2254:
2221:
2208:
2181:
2168:
2143:
2130:
2103:
2090:
2065:
2052:
2043:
2020:
2010:
2003:
2002:
1969:
1956:
1947:
1924:
1911:
1875:
1862:
1825:
1824:
1818:
1802:
1801:
1786:
1773:
1750:
1737:
1713:
1698:
1697:
1679:
1651:
1624:
1606:
1605:
1593:
1580:
1563:
1550:
1522:
1495:
1477:
1476:
1418:
1410:
1409:
1351:
1343:
1342:
1284:
1270:
1269:
1234:
1221:
1220:
1210:
1209:
1172:
1159:
1158:
1143:
1136:
1135:
1099:
1086:
1067:
1066:
968:
967:
954:
944:
941:
936:
935:
933:
932:
928:
925:
916:
912:
908:
893:
890:Euler's formula
885:
879:
862:
861:
845:
830:
829:
801:
786:
785:
757:
736:
735:
725:
715:
712:
707:
706:
704:
703:
699:
682:
681:
668:
653:
652:
639:
624:
623:
610:
589:
588:
569:
559:
552:
524:
520:
500:
499:
486:
471:
470:
445:
430:
429:
404:
383:
382:
372:parametrisation
362:, a helix is a
351:
347:
340:
308:
212:
190:comes from the
108:
104:
93:
82:
66:
57:A machine screw
43:
28:
23:
22:
15:
12:
11:
5:
3977:
3975:
3967:
3966:
3961:
3956:
3946:
3945:
3939:
3938:
3936:
3935:
3930:
3925:
3920:
3915:
3910:
3903:
3902:
3901:
3891:
3886:
3881:
3876:
3871:
3866:
3865:
3864:
3859:
3854:
3844:
3838:
3836:
3832:
3831:
3828:
3827:
3825:
3824:
3823:
3822:
3812:
3807:
3802:
3797:
3792:
3787:
3782:
3778:
3773:
3771:
3764:
3763:
3758:
3753:
3748:
3743:
3738:
3732:
3730:
3726:
3725:
3718:
3716:
3714:
3713:
3708:
3703:
3698:
3693:
3687:
3685:
3681:
3680:
3667:
3665:
3664:
3657:
3650:
3642:
3635:
3634:
3607:
3582:
3558:
3535:
3522:
3509:
3485:
3460:
3432:
3430:
3427:
3425:
3424:
3419:
3414:
3409:
3404:
3399:
3394:
3389:
3384:
3379:
3377:Collagen helix
3374:
3369:
3364:
3359:
3353:
3351:
3348:
3347:
3346:
3340:
3333:
3331:
3327:magnetic field
3324:
3317:
3315:
3308:
3301:
3299:
3286:
3279:
3228:Most hardware
3200:
3197:
3185:
3177:
3173:
3169:
3164:
3160:
3155:
3150:
3146:
3140:
3137:
3131:
3127:
3121:
3117:
3114:
3087:
3082:
3078:
3075:
3071:
3062:
3058:
3054:
3049:
3045:
3040:
3035:
3032:
3029:
3026:
3022:
3013:
3009:
3005:
3000:
2996:
2991:
2986:
2983:
2980:
2976:
2967:
2963:
2959:
2954:
2950:
2945:
2940:
2937:
2935:
2930:
2927:
2921:
2917:
2911:
2910:
2906:
2901:
2897:
2894:
2890:
2881:
2877:
2873:
2868:
2864:
2859:
2854:
2851:
2848:
2845:
2841:
2832:
2828:
2824:
2819:
2815:
2810:
2805:
2802:
2799:
2795:
2786:
2782:
2778:
2773:
2769:
2764:
2759:
2756:
2754:
2751:
2747:
2743:
2739:
2735:
2731:
2730:
2705:
2701:
2698:
2694:
2685:
2681:
2677:
2672:
2668:
2663:
2658:
2655:
2652:
2648:
2639:
2635:
2631:
2626:
2622:
2617:
2612:
2609:
2606:
2603:
2599:
2568:
2564:
2560:
2555:
2551:
2544:
2540:
2536:
2529:
2525:
2519:
2516:
2510:
2506:
2500:
2496:
2493:
2468:
2464:
2461:
2457:
2448:
2444:
2440:
2435:
2431:
2426:
2421:
2418:
2410:
2406:
2402:
2397:
2393:
2387:
2384:
2378:
2374:
2365:
2361:
2357:
2352:
2348:
2343:
2338:
2335:
2327:
2323:
2319:
2314:
2310:
2304:
2301:
2295:
2291:
2287:
2284:
2278:
2275:
2269:
2265:
2237:
2228:
2224:
2220:
2215:
2211:
2206:
2201:
2197:
2188:
2184:
2180:
2175:
2171:
2166:
2161:
2158:
2150:
2146:
2142:
2137:
2133:
2128:
2123:
2119:
2110:
2106:
2102:
2097:
2093:
2088:
2083:
2080:
2072:
2068:
2064:
2059:
2055:
2049:
2046:
2040:
2036:
2032:
2026:
2023:
2017:
2013:
1985:
1976:
1972:
1968:
1963:
1959:
1953:
1950:
1944:
1940:
1931:
1927:
1923:
1918:
1914:
1909:
1904:
1901:
1898:
1895:
1891:
1882:
1878:
1874:
1869:
1865:
1860:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1833:
1800:
1793:
1789:
1785:
1780:
1776:
1770:
1767:
1764:
1757:
1753:
1749:
1744:
1740:
1732:
1727:
1723:
1719:
1716:
1714:
1712:
1709:
1706:
1703:
1700:
1699:
1696:
1693:
1686:
1682:
1678:
1675:
1672:
1669:
1666:
1663:
1658:
1654:
1650:
1647:
1644:
1641:
1638:
1635:
1630:
1627:
1625:
1622:
1617:
1612:
1608:
1607:
1600:
1596:
1592:
1587:
1583:
1577:
1570:
1566:
1562:
1557:
1553:
1549:
1546:
1543:
1540:
1537:
1534:
1529:
1525:
1521:
1518:
1515:
1512:
1509:
1506:
1501:
1498:
1496:
1493:
1488:
1483:
1479:
1478:
1474:
1470:
1467:
1463:
1459:
1456:
1453:
1450:
1447:
1443:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1419:
1416:
1412:
1411:
1407:
1403:
1400:
1396:
1392:
1389:
1386:
1383:
1380:
1376:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1352:
1349:
1345:
1344:
1340:
1336:
1333:
1330:
1326:
1322:
1319:
1316:
1313:
1310:
1306:
1302:
1299:
1296:
1293:
1290:
1287:
1285:
1282:
1278:
1277:
1261:
1260:
1249:
1241:
1237:
1233:
1228:
1224:
1219:
1199:
1198:
1187:
1179:
1175:
1171:
1166:
1162:
1155:
1151:
1147:
1125:
1124:
1113:
1106:
1102:
1098:
1093:
1089:
1083:
1080:
1077:
1074:
1056:
1055:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
924:
921:
876:
875:
860:
857:
854:
851:
848:
846:
844:
841:
838:
835:
832:
831:
828:
825:
822:
819:
816:
813:
810:
807:
804:
802:
800:
797:
794:
791:
788:
787:
784:
781:
778:
775:
772:
769:
766:
763:
760:
758:
756:
753:
750:
747:
744:
743:
696:
695:
680:
677:
674:
671:
669:
667:
664:
661:
658:
655:
654:
651:
648:
645:
642:
640:
638:
635:
632:
629:
626:
625:
622:
619:
616:
613:
611:
609:
606:
603:
600:
597:
596:
514:
513:
498:
495:
492:
489:
487:
485:
482:
479:
476:
473:
472:
469:
466:
463:
460:
457:
454:
451:
448:
446:
444:
441:
438:
435:
432:
431:
428:
425:
422:
419:
416:
413:
410:
407:
405:
403:
400:
397:
394:
391:
390:
339:
336:
307:
304:
238:circular helix
211:
208:
162:at a constant
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3976:
3965:
3962:
3960:
3957:
3955:
3952:
3951:
3949:
3934:
3931:
3929:
3926:
3924:
3921:
3919:
3916:
3914:
3911:
3909:
3908:
3904:
3900:
3897:
3896:
3895:
3892:
3890:
3887:
3885:
3882:
3880:
3877:
3875:
3872:
3870:
3867:
3863:
3860:
3858:
3855:
3853:
3850:
3849:
3848:
3845:
3843:
3840:
3839:
3837:
3833:
3821:
3818:
3817:
3816:
3813:
3811:
3808:
3806:
3803:
3801:
3798:
3796:
3793:
3791:
3788:
3786:
3783:
3781:
3775:
3774:
3772:
3768:
3762:
3759:
3757:
3754:
3752:
3749:
3747:
3744:
3742:
3739:
3737:
3734:
3733:
3731:
3727:
3712:
3709:
3707:
3704:
3702:
3699:
3697:
3694:
3692:
3689:
3688:
3686:
3682:
3678:
3674:
3670:
3663:
3658:
3656:
3651:
3649:
3644:
3643:
3640:
3630:
3626:
3622:
3618:
3611:
3608:
3602:
3601:
3596:
3593:
3586:
3583:
3579:
3575:
3572:
3568:
3562:
3559:
3556:, 28:153–163.
3555:
3551:
3548:
3545:
3539:
3536:
3532:
3526:
3523:
3519:
3513:
3510:
3506:
3502:
3498:
3495:
3489:
3486:
3480:
3479:
3474:
3471:
3464:
3461:
3457:
3453:
3449:
3446:
3437:
3434:
3428:
3423:
3420:
3418:
3415:
3413:
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3388:
3385:
3383:
3380:
3378:
3375:
3373:
3370:
3368:
3365:
3363:
3360:
3358:
3355:
3354:
3349:
3344:
3337:
3332:
3328:
3321:
3316:
3312:
3305:
3300:
3297:
3294:
3290:
3283:
3278:
3276:
3274:
3269:
3266:In aviation,
3264:
3262:
3258:
3254:
3250:
3245:
3243:
3239:
3235:
3231:
3230:screw threads
3226:
3224:
3219:
3217:
3213:
3208:
3206:
3198:
3196:
3183:
3175:
3171:
3167:
3162:
3158:
3153:
3148:
3144:
3138:
3135:
3125:
3119:
3115:
3112:
3103:
3085:
3076:
3073:
3060:
3056:
3052:
3047:
3043:
3038:
3033:
3030:
3027:
3024:
3011:
3007:
3003:
2998:
2994:
2989:
2984:
2981:
2978:
2974:
2965:
2961:
2957:
2952:
2948:
2943:
2938:
2936:
2928:
2925:
2915:
2904:
2895:
2892:
2879:
2875:
2871:
2866:
2862:
2857:
2852:
2849:
2846:
2843:
2830:
2826:
2822:
2817:
2813:
2808:
2803:
2800:
2797:
2793:
2784:
2780:
2776:
2771:
2767:
2762:
2757:
2755:
2745:
2737:
2720:
2717:
2699:
2696:
2683:
2679:
2675:
2670:
2666:
2661:
2656:
2653:
2650:
2637:
2633:
2629:
2624:
2620:
2615:
2610:
2607:
2604:
2601:
2588:
2585:
2566:
2562:
2558:
2553:
2549:
2538:
2527:
2523:
2517:
2514:
2504:
2498:
2494:
2491:
2483:
2480:
2462:
2459:
2446:
2442:
2438:
2433:
2429:
2424:
2419:
2416:
2408:
2404:
2400:
2395:
2391:
2385:
2382:
2376:
2363:
2359:
2355:
2350:
2346:
2341:
2336:
2333:
2325:
2321:
2317:
2312:
2308:
2302:
2299:
2293:
2285:
2282:
2276:
2273:
2263:
2252:
2249:
2226:
2222:
2218:
2213:
2209:
2204:
2199:
2186:
2182:
2178:
2173:
2169:
2164:
2159:
2156:
2148:
2144:
2140:
2135:
2131:
2126:
2121:
2108:
2104:
2100:
2095:
2091:
2086:
2081:
2078:
2070:
2066:
2062:
2057:
2053:
2047:
2044:
2038:
2030:
2024:
2021:
2011:
2000:
1997:
1974:
1970:
1966:
1961:
1957:
1951:
1948:
1942:
1929:
1925:
1921:
1916:
1912:
1907:
1902:
1899:
1896:
1893:
1880:
1876:
1872:
1867:
1863:
1858:
1853:
1850:
1847:
1844:
1838:
1822:
1815:
1798:
1791:
1787:
1783:
1778:
1774:
1768:
1765:
1762:
1755:
1751:
1747:
1742:
1738:
1730:
1725:
1721:
1717:
1715:
1707:
1701:
1694:
1691:
1684:
1676:
1673:
1670:
1667:
1661:
1656:
1648:
1645:
1642:
1639:
1636:
1628:
1626:
1598:
1594:
1590:
1585:
1581:
1575:
1568:
1564:
1560:
1555:
1547:
1544:
1541:
1538:
1532:
1527:
1519:
1516:
1513:
1510:
1507:
1499:
1497:
1468:
1465:
1457:
1454:
1451:
1448:
1445:
1437:
1434:
1431:
1428:
1425:
1422:
1420:
1401:
1398:
1390:
1387:
1384:
1381:
1378:
1370:
1367:
1364:
1361:
1358:
1355:
1353:
1334:
1331:
1328:
1320:
1317:
1314:
1311:
1308:
1300:
1297:
1294:
1291:
1288:
1286:
1267:
1264:
1247:
1239:
1235:
1231:
1226:
1222:
1217:
1208:
1207:
1206:
1204:
1185:
1177:
1173:
1169:
1164:
1160:
1149:
1134:
1133:
1132:
1130:
1111:
1104:
1100:
1096:
1091:
1087:
1081:
1078:
1075:
1072:
1065:
1064:
1063:
1061:
1039:
1036:
1033:
1027:
1024:
1021:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
973:
966:
965:
964:
963:
958:
947:
939:
922:
920:
906:
902:
897:
891:
882:
858:
855:
852:
849:
847:
839:
833:
826:
820:
814:
811:
808:
805:
803:
795:
789:
782:
776:
770:
767:
764:
761:
759:
751:
745:
734:
733:
732:
729:
718:
710:
678:
675:
672:
670:
662:
656:
649:
646:
643:
641:
633:
627:
620:
617:
614:
612:
604:
598:
587:
586:
585:
581:
577:
573:
568:
563:
556:
548:
544:
540:
536:
532:
528:
519:
496:
493:
490:
488:
480:
474:
467:
461:
455:
452:
449:
447:
439:
433:
426:
420:
414:
411:
408:
406:
398:
392:
381:
380:
379:
377:
373:
369:
365:
361:
344:
337:
332:
328:
324:
320:
316:
314:
305:
303:
301:
297:
292:
290:
285:
284:is constant.
283:
279:
275:
271:
270:general helix
266:
264:
260:
256:
255:
249:
247:
244:and constant
243:
239:
234:
232:
228:
223:
221:
217:
209:
207:
205:
204:
193:
189:
185:
184:alpha helices
181:
177:
173:
169:
165:
161:
160:tangent lines
157:
154:
150:
149:machine screw
146:
142:
132:
102:
90:
86:
78:
74:
70:
63:
55:
47:
41:
37:
33:
19:
3905:
3770:Biochemistry
3676:
3620:
3616:
3610:
3598:
3585:
3566:
3561:
3543:
3538:
3530:
3529:O'Neill, B.
3525:
3517:
3516:O'Neill, B.
3512:
3494:Double Helix
3488:
3476:
3463:
3458:, on Perseus
3455:
3436:
3422:Triple helix
3397:Helical axis
3295:
3291:reported by
3267:
3265:
3246:
3227:
3220:
3209:
3202:
3104:
2721:
2718:
2589:
2586:
2484:
2481:
2253:
2250:
2001:
1998:
1823:
1816:
1268:
1265:
1262:
1200:
1126:
1057:
956:
945:
937:
926:
919:components.
905:translations
898:
880:
877:
727:
716:
708:
697:
579:
575:
571:
564:
554:
546:
542:
538:
534:
530:
526:
515:
357:
333:(view-angle)
322:
309:
296:space spiral
293:
288:
286:
273:
269:
267:
259:conic spiral
258:
252:
250:
237:
235:
227:double helix
226:
224:
215:
213:
201:
187:
140:
100:
98:
88:
84:
76:
72:
68:
36:Double helix
3918:Pitch angle
3894:Logarithmic
3842:Archimedean
3805:Polyproline
3547:Turk J Math
3392:Helix angle
3357:Alpha helix
3343:coil spring
3253:pitch space
3216:Cedar Point
899:Except for
368:dimensional
360:mathematics
331:perspective
327:chiralities
289:slant helix
254:conic helix
186:. The word
178:, and many
156:space curve
145:coil spring
3948:Categories
3907:On Spirals
3857:Hyperbolic
3533:1961 pg 74
3520:1961 pg 72
3473:"Helicoid"
3429:References
3417:Superhelix
3362:Arc spring
3341:A helical
1060:arc length
953:(or pitch
931:and slope
724:(or pitch
702:and slope
354:components
306:Handedness
3928:Spirangle
3923:Theodorus
3862:Poinsot's
3852:Epispiral
3696:Curvature
3691:Algebraic
3600:MathWorld
3478:MathWorld
3402:Hemihelix
3212:Corkscrew
3113:τ
3034:
2985:
2853:
2844:−
2804:
2746:×
2657:
2651:−
2611:
2605:−
2492:κ
2420:
2383:−
2337:
2300:−
2286:κ
2160:
2082:
2045:−
1903:
1854:
1766:τ
1722:∫
1674:
1646:
1637:−
1545:
1517:
1508:−
1455:
1446:−
1435:
1426:−
1388:
1368:
1359:−
1318:
1298:
1129:curvature
1082:⋅
1028:∈
1004:
989:
977:↦
901:rotations
815:
771:
628:θ
518:parameter
456:
415:
313:chirality
278:curvature
242:curvature
231:congruent
170:, as the
3884:Involute
3879:Fermat's
3820:Collagen
3756:Symmetry
3574:Archived
3550:Archived
3497:Archived
3448:Archived
3412:Solenoid
3387:Helicity
3350:See also
3199:Examples
298:; e.g.,
203:helicoid
180:proteins
3954:Helices
3913:Padovan
3847:Cotes's
3835:Spirals
3741:Antenna
3729:Helices
3701:Gallery
3677:helices
3669:Spirals
3595:"Helix"
3311:climber
1203:torsion
1058:has an
950:
934:
721:
705:
516:As the
282:torsion
246:torsion
168:biology
141:helices
18:Helices
3964:Curves
3899:Golden
3815:Triple
3795:Double
3761:Triple
3711:Topics
3684:Curves
3673:curves
3296:et al.
3242:Z form
1201:and a
263:spiral
153:smooth
139:
71:, sin
34:, see
3874:Euler
3869:Doyle
3810:Super
3785:Alpha
3736:Angle
3313:plant
3249:music
888:(see
366:in 3-
364:curve
216:pitch
194:word
192:Greek
188:helix
164:angle
158:with
101:helix
67:(cos
3933:Ulam
3889:List
3790:Beta
3751:Hemi
3706:List
3675:and
3443:ἕλιξ
3293:Lehn
3236:and
350:and
220:turn
214:The
197:ἕλιξ
83:0 ≤
81:for
3625:doi
3247:In
3031:sin
2982:cos
2850:cos
2801:sin
2654:sin
2608:cos
2417:sin
2334:cos
2157:cos
2079:sin
1900:sin
1851:cos
1671:cos
1643:sin
1542:cos
1514:sin
1452:sin
1432:cos
1385:cos
1365:sin
1315:sin
1295:cos
1205:of
1131:of
1062:of
1001:sin
986:cos
915:or
812:sin
768:cos
565:In
453:sin
412:cos
374:in
358:In
280:to
272:or
172:DNA
137:pl.
87:≤ 4
3950::
3800:Pi
3779:10
3671:,
3621:86
3619:.
3597:.
3569:.
3475:.
3275:.
3263:.
3251:,
3225:.
3207:.
2584:.
1127:a
957:πb
911:,
903:,
728:πb
578:,
574:,
549:))
541:),
533:),
302:.
251:A
248:.
236:A
225:A
206:.
135:;
116:iː
99:A
75:,
3777:3
3661:e
3654:t
3647:v
3631:.
3627::
3603:.
3580:.
3507:.
3492:"
3481:.
3238:B
3234:A
3184:.
3176:2
3172:b
3168:+
3163:2
3159:a
3154:b
3149:=
3145:|
3139:s
3136:d
3130:B
3126:d
3120:|
3116:=
3086:)
3081:k
3077:0
3074:+
3070:j
3061:2
3057:b
3053:+
3048:2
3044:a
3039:s
3028:b
3025:+
3021:i
3012:2
3008:b
3004:+
2999:2
2995:a
2990:s
2979:b
2975:(
2966:2
2962:b
2958:+
2953:2
2949:a
2944:1
2939:=
2929:s
2926:d
2920:B
2916:d
2905:)
2900:k
2896:a
2893:+
2889:j
2880:2
2876:b
2872:+
2867:2
2863:a
2858:s
2847:b
2840:i
2831:2
2827:b
2823:+
2818:2
2814:a
2809:s
2798:b
2794:(
2785:2
2781:b
2777:+
2772:2
2768:a
2763:1
2758:=
2750:N
2742:T
2738:=
2734:B
2704:k
2700:0
2697:+
2693:j
2684:2
2680:b
2676:+
2671:2
2667:a
2662:s
2647:i
2638:2
2634:b
2630:+
2625:2
2621:a
2616:s
2602:=
2598:N
2567:2
2563:b
2559:+
2554:2
2550:a
2543:|
2539:a
2535:|
2528:=
2524:|
2518:s
2515:d
2509:T
2505:d
2499:|
2495:=
2467:k
2463:0
2460:+
2456:j
2447:2
2443:b
2439:+
2434:2
2430:a
2425:s
2409:2
2405:b
2401:+
2396:2
2392:a
2386:a
2377:+
2373:i
2364:2
2360:b
2356:+
2351:2
2347:a
2342:s
2326:2
2322:b
2318:+
2313:2
2309:a
2303:a
2294:=
2290:N
2283:=
2277:s
2274:d
2268:T
2264:d
2236:k
2227:2
2223:b
2219:+
2214:2
2210:a
2205:b
2200:+
2196:j
2187:2
2183:b
2179:+
2174:2
2170:a
2165:s
2149:2
2145:b
2141:+
2136:2
2132:a
2127:a
2122:+
2118:i
2109:2
2105:b
2101:+
2096:2
2092:a
2087:s
2071:2
2067:b
2063:+
2058:2
2054:a
2048:a
2039:=
2035:T
2031:=
2025:s
2022:d
2016:r
2012:d
1984:k
1975:2
1971:b
1967:+
1962:2
1958:a
1952:s
1949:b
1943:+
1939:j
1930:2
1926:b
1922:+
1917:2
1913:a
1908:s
1897:a
1894:+
1890:i
1881:2
1877:b
1873:+
1868:2
1864:a
1859:s
1848:a
1845:=
1842:)
1839:s
1836:(
1832:r
1819:s
1799:t
1792:2
1788:b
1784:+
1779:2
1775:a
1769:=
1763:d
1756:2
1752:b
1748:+
1743:2
1739:a
1731:t
1726:0
1718:=
1711:)
1708:t
1705:(
1702:s
1695:a
1692:=
1685:2
1681:)
1677:t
1668:a
1665:(
1662:+
1657:2
1653:)
1649:t
1640:a
1634:(
1629:=
1621:|
1616:a
1611:|
1599:2
1595:b
1591:+
1586:2
1582:a
1576:=
1569:2
1565:b
1561:+
1556:2
1552:)
1548:t
1539:a
1536:(
1533:+
1528:2
1524:)
1520:t
1511:a
1505:(
1500:=
1492:|
1487:v
1482:|
1473:k
1469:0
1466:+
1462:j
1458:t
1449:a
1442:i
1438:t
1429:a
1423:=
1415:a
1406:k
1402:b
1399:+
1395:j
1391:t
1382:a
1379:+
1375:i
1371:t
1362:a
1356:=
1348:v
1339:k
1335:t
1332:b
1329:+
1325:j
1321:t
1312:a
1309:+
1305:i
1301:t
1292:a
1289:=
1281:r
1248:.
1240:2
1236:b
1232:+
1227:2
1223:a
1218:b
1186:,
1178:2
1174:b
1170:+
1165:2
1161:a
1154:|
1150:a
1146:|
1112:,
1105:2
1101:b
1097:+
1092:2
1088:a
1079:T
1076:=
1073:A
1043:]
1040:T
1037:,
1034:0
1031:[
1025:t
1022:,
1019:)
1016:t
1013:b
1010:,
1007:t
998:a
995:,
992:t
983:a
980:(
974:t
955:2
946:b
942:/
938:a
929:a
917:z
913:y
909:x
894:x
886:x
881:e
859:.
856:t
853:b
850:=
843:)
840:t
837:(
834:z
827:,
824:)
821:t
818:(
809:a
806:=
799:)
796:t
793:(
790:y
783:,
780:)
777:t
774:(
765:a
762:=
755:)
752:t
749:(
746:x
726:2
717:b
713:/
709:a
700:a
679:.
676:t
673:=
666:)
663:t
660:(
657:h
650:,
647:t
644:=
637:)
634:t
631:(
621:,
618:1
615:=
608:)
605:t
602:(
599:r
582:)
580:h
576:θ
572:r
570:(
560:z
555:π
553:2
547:t
545:(
543:z
539:t
537:(
535:y
531:t
529:(
527:x
525:(
521:t
497:.
494:t
491:=
484:)
481:t
478:(
475:z
468:,
465:)
462:t
459:(
450:=
443:)
440:t
437:(
434:y
427:,
424:)
421:t
418:(
409:=
402:)
399:t
396:(
393:x
352:y
348:x
131:/
128:s
125:k
122:ɪ
119:l
113:h
110:ˈ
107:/
103:(
94:t
89:π
85:t
79:)
77:t
73:t
69:t
42:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
