1951:
2564:
1544:
3400:
2199:
1946:{\displaystyle {\begin{aligned}\cos \varphi &={\frac {(\mathbf {u} _{1}\times \mathbf {u} _{2})\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})}{|\mathbf {u} _{1}\times \mathbf {u} _{2}|\,|\mathbf {u} _{2}\times \mathbf {u} _{3}|}}\\\sin \varphi &={\frac {\mathbf {u} _{2}\cdot ((\mathbf {u} _{1}\times \mathbf {u} _{2})\times (\mathbf {u} _{2}\times \mathbf {u} _{3}))}{|\mathbf {u} _{2}|\,|\mathbf {u} _{1}\times \mathbf {u} _{2}|\,|\mathbf {u} _{2}\times \mathbf {u} _{3}|}},\end{aligned}}}
3236:
3229:
3286:
2559:{\displaystyle {\begin{aligned}\cos \varphi &={\frac {(\mathbf {u} _{1}\times \mathbf {u} _{2})\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})}{|\mathbf {u} _{1}\times \mathbf {u} _{2}|\,|\mathbf {u} _{2}\times \mathbf {u} _{3}|}}\\\sin \varphi &={\frac {|\mathbf {u} _{2}|\,\mathbf {u} _{1}\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})}{|\mathbf {u} _{1}\times \mathbf {u} _{2}|\,|\mathbf {u} _{2}\times \mathbf {u} _{3}|}},\end{aligned}}}
2184:
3222:
27:
2751:
1188:
1966:
872:
3785:
623:
3058:
2189:
This dihedral angle does not depend on the orientation of the chain (order in which the point are considered) — reversing this ordering consists of replacing each vector by its opposite vector, and exchanging the indices 1 and 3. Both operations do not change the cosine, but change the sign of the
2575:
3344:
The two types of terms can be combined so as to define four ranges of angle; 0° to ±30° synperiplanar (sp); 30° to 90° and −30° to −90° synclinal (sc); 90° to 150° and −90° to −150° anticlinal (ac); ±150° to 180° antiperiplanar (ap). The synperiplanar conformation is also known as the
1013:
2179:{\displaystyle \varphi =\operatorname {atan2} (\mathbf {u} _{2}\cdot ((\mathbf {u} _{1}\times \mathbf {u} _{2})\times (\mathbf {u} _{2}\times \mathbf {u} _{3})),|\mathbf {u} _{2}|\,(\mathbf {u} _{1}\times \mathbf {u} _{2})\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})).}
753:
3620:
407:
2770:
2746:{\displaystyle \varphi =\operatorname {atan2} (|\mathbf {u} _{2}|\,\mathbf {u} _{1}\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3}),(\mathbf {u} _{1}\times \mathbf {u} _{2})\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})).}
3316:
defines a half-plane. As explained above, when two such half-planes intersect (i.e., a set of four consecutively-bonded atoms), the angle between them is a dihedral angle. Dihedral angles are used to specify the
1183:{\displaystyle \cos \varphi ={\frac {(\mathbf {b} _{0}\times \mathbf {b} _{1})\cdot (\mathbf {b} _{0}\times \mathbf {b} _{2})}{|\mathbf {b} _{0}\times \mathbf {b} _{1}||\mathbf {b} _{0}\times \mathbf {b} _{2}|}}}
3087:
is the angle in the clockwise direction of the fourth atom compared to the first atom, while looking down the axis from the second atom to the third. Special cases (one may say the usual cases) are
4063:
Singh J, Hanson J, Heffernan R, Paliwal K, Yang Y, Zhou Y (August 2018). "Detecting
Proline and Non-Proline Cis Isomers in Protein Structures from Sequences Using Deep Residual Ensemble Learning".
2204:
1549:
376:
294:
666:
867:{\displaystyle \cos \varphi ={\frac {\left\vert \mathbf {n} _{\mathrm {A} }\cdot \mathbf {n} _{\mathrm {B} }\right\vert }{|\mathbf {n} _{\mathrm {A} }||\mathbf {n} _{\mathrm {B} }|}}}
1227:
3877:
Blondel, Arnaud; Karplus, Martin (7 Dec 1998). "New formulation for derivatives of torsion angles and improper torsion angles in molecular mechanics: Elimination of singularities".
922:
The absolute value is required in above formulas, as the planes are not changed when changing all coefficient signs in one equation, or replacing one normal vector by its opposite.
3780:{\displaystyle \cos \varphi ={\frac {\cos(\angle \mathrm {APB} )-\cos(\angle \mathrm {APC} )\cos(\angle \mathrm {BPC} )}{\sin(\angle \mathrm {APC} )\sin(\angle \mathrm {BPC} )}}}
1291:
1352:
3185:
3148:
1320:
1256:
618:{\displaystyle \cos \varphi ={\frac {\left\vert a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}\right\vert }{{\sqrt {a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}{\sqrt {a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}}}}
3111:
3614:
Given 3 faces of a polyhedron which meet at a common vertex P and have edges AP, BP and CP, the cosine of the dihedral angle between the faces containing APC and BPC is:
3932:
3903:
3085:
399:
1499:. In these cases, one is often interested in the half-planes defined by three consecutive points, and the dihedral angle between two consecutive such half-planes. If
720:
693:
3567:
Every polyhedron has a dihedral angle at every edge describing the relationship of the two faces that share that edge. This dihedral angle, also called the
3053:{\displaystyle (\mathbf {u} _{1}\times \mathbf {u} _{2})\times (\mathbf {u} _{2}\times \mathbf {u} _{3})=\mathbf {u} _{2}-\mathbf {u} _{1}=\mathbf {u} _{2}}
200:
main planes (commonly called "wings") are upwardly inclined to the lateral axis; when downwardly inclined they are said to be at a negative dihedral angle.
4198:
1526:
are three consecutive bond vectors, the intersection of the half-planes is oriented, which allows defining a dihedral angle that belongs to the interval
3823:
1379:, one may consider a chain of points and links between consecutive points. If the points are sequentially numbered and located at positions
4231:
3890:
3562:
4155:
Dunbrack, RL Jr; Karplus, M (May 1994). "Conformational analysis of the backbone-dependent rotamer preferences of protein sidechains".
4120:
Dunbrack, RL Jr.; Karplus, M (20 March 1993). "Backbone-dependent rotamer library for proteins. Application to side-chain prediction".
3971:
3550:
4262:
4039:
3308:
is defined as a particular example of a dihedral angle, describing the geometric relation of two parts of a molecule joined by a
135:
3457:
The figure at right illustrates the location of each of these angles (but it does not show correctly the way they are defined).
3861:
3584:
30:
Angle between two half-planes (α, β, pale blue) in a third plane (red) which cuts the line of intersection at right angles
3603:
3419:, C. Ramakrishnan, and V. Sasisekharan, is a way to visualize energetically allowed regions for backbone dihedral angles
157:
58:
54:
3987:
Ramachandran, G. N.; Ramakrishnan, C.; Sasisekharan, V. (1963). "Stereochemistry of polypeptide chain configurations".
3391:
For macromolecular usage the symbols T, C, G, G, A and A are recommended (ap, sp, +sc, −sc, +ac and −ac respectively).
4272:
4257:
300:
218:
20:
3380:
two planes can be specified in terms of the two central carbon atoms and either of the methyl carbon atoms. The
631:
4252:
3791:
3580:
3333:(a). Similarly, arrangements corresponding to angles between 30° and 150° or between −30° and −150° are called
3318:
3212:
2757:
4206:
1197:
3491:
is approximately 3.8 and 2.9 Å, respectively. The vast majority of the peptide bonds in proteins are
3399:
3208:
93:
2761:
1261:
209:
1325:
3153:
3116:
1296:
1232:
3831:
3576:
128:
88:
84:
3488:
3476:
3470:
3090:
1003:
belong respectively to the intersection line, the first half plane, and the second half plane. The
4106:
3235:
4180:
4088:
3416:
197:
177:
3228:
4172:
4137:
4080:
4045:
4035:
4004:
3967:
3857:
3432:
3412:
3264:
3070:
1496:
933:
whose boundaries are the same line. In this case, the half planes can be described by a point
384:
3529:
conformations. The stability of certain sidechain dihedral angles is affected by the values
4164:
4129:
4072:
4027:
3996:
3946:
3917:
3886:
189:
3285:
698:
671:
3591:
3301:
2193:
A simpler formula for the same dihedral angle is the following (the proof is given below)
1488:
1376:
185:
181:
121:
80:
76:
63:
1229:
In this case, switching the two half-planes gives the same result, and so does replacing
4267:
3595:
3572:
3322:
926:
173:
4031:
4000:
4246:
3599:
3309:
3064:
744:
196:. The planes of a flying machine are said to be at positive dihedral angle when both
4092:
3942:
3913:
4184:
3461:
3415:(also known as a Ramachandran diagram or a plot), originally developed in 1963 by
3575:
with respect to the polyhedron. An angle of 0° means the face normal vectors are
3937:
3908:
3803:
3384:-conformation shown above, with a dihedral angle of 60° is less stable than the
3221:
894:
111:
50:
4236:
3428:
1492:
930:
193:
161:
4076:
3941:, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "
3912:, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "
3950:
3921:
165:
26:
4133:
4084:
4022:
Richardson, J. S. (1981). "The
Anatomy and Taxonomy of Protein Structure".
4008:
3891:
10.1002/(SICI)1096-987X(19960715)17:9<1132::AID-JCC5>3.0.CO;2-T
4176:
4141:
4049:
168:, it is the clockwise angle between half-planes through two sets of three
3587:. An angle greater than 180° exists on concave portions of a polyhedron.
3313:
1293:
In chemistry (see below), we define a dihedral angle such that replacing
4168:
929:
can be and should be avoided when considering the dihedral angle of two
3517:). They tend to cluster near 180°, 60°, and −60°, which are called the
3496:
3436:
3404:
3579:
and the faces overlap each other, which implies that it is part of a
3377:
3290:
3275:
3254:
3325:
arrangements corresponding to angles between 0° and ±90° are called
3583:
polyhedron. An angle of 180° means the faces are parallel, as in a
3537:. For instance, there are direct steric interactions between the C
1957:
169:
153:
37:
25:
4026:. Advances in Protein Chemistry. Vol. 34. pp. 167–339.
3606:, the two quasiregular solids, and two quasiregular dual solids.
3549:
is near -60°. This is evident from statistical distributions in
3337:(c) and those between 0° and ±30° or ±150° and 180° are called
19:
This article is about the geometry term. For other uses, see
3545:
rotamer and the backbone nitrogen of the next residue when
668:
It can easily be observed that the angle is independent of
208:
When the two intersecting planes are described in terms of
3449:ψ (psi) is the angle in the chain N − C − C' − N (called
3329:(s), those corresponding to angles between ±90° and 180°
2756:
This can be deduced from previous formulas by using the
4239:
gives a step-by-step derivation of these exact values.
3623:
3156:
3119:
3093:
3073:
2773:
2578:
2202:
1969:
1547:
1328:
1299:
1264:
1235:
1200:
1016:
756:
701:
674:
634:
410:
387:
303:
221:
2190:
sine. Thus, together, they do not change the angle.
1354:
changes the sign of the angle, which can be between
192:, a dihedral angle represents the angle between two
184:
and two half-planes that have this line as a common
3594:polyhedron has the same value. This includes the 5
3506:The side chain dihedral angles are designated with
3443:ω (omega) is the angle in the chain C − C' − N − C,
3779:
3179:
3142:
3105:
3079:
3052:
2745:
2558:
2178:
1945:
1346:
1314:
1285:
1250:
1221:
1182:
866:
714:
687:
660:
617:
393:
370:
288:
3446:φ (phi) is the angle in the chain C' − N − C − C'
3480:case). The distance between the C atoms in the
2764:is zero if it contains twice the same vector:
3495:, though the peptide bond to the nitrogen of
3388:-conformation with a dihedral angle of 180°.
3312:. Every set of three non-colinear atoms of a
129:
8:
4232:The Dihedral Angle in Woodworking at Tips.FM
4065:Journal of Chemical Information and Modeling
3854:Models for polymeric and anisotropic liquids
371:{\displaystyle a_{2}x+b_{2}y+c_{2}z+d_{2}=0}
289:{\displaystyle a_{1}x+b_{1}y+c_{1}z+d_{1}=0}
4024:Anatomy and Taxonomy of Protein Structures
661:{\displaystyle 0\leq \varphi \leq \pi /2.}
136:
122:
33:
3760:
3734:
3706:
3680:
3651:
3636:
3622:
3439:chain three dihedral angles are defined:
3169:
3155:
3132:
3118:
3092:
3072:
3044:
3039:
3029:
3024:
3011:
3006:
2996:
2991:
2975:
2970:
2960:
2955:
2942:
2937:
2927:
2922:
2906:
2901:
2891:
2886:
2873:
2868:
2858:
2853:
2834:
2829:
2819:
2814:
2798:
2793:
2783:
2778:
2772:
2728:
2723:
2713:
2708:
2692:
2687:
2677:
2672:
2656:
2651:
2641:
2636:
2623:
2618:
2616:
2611:
2605:
2600:
2594:
2577:
2541:
2535:
2530:
2520:
2515:
2509:
2508:
2503:
2497:
2492:
2482:
2477:
2471:
2460:
2455:
2445:
2440:
2427:
2422:
2420:
2415:
2409:
2404:
2398:
2395:
2367:
2361:
2356:
2346:
2341:
2335:
2334:
2329:
2323:
2318:
2308:
2303:
2297:
2286:
2281:
2271:
2266:
2250:
2245:
2235:
2230:
2223:
2203:
2201:
2161:
2156:
2146:
2141:
2125:
2120:
2110:
2105:
2100:
2095:
2089:
2084:
2078:
2063:
2058:
2048:
2043:
2027:
2022:
2012:
2007:
1991:
1986:
1968:
1928:
1922:
1917:
1907:
1902:
1896:
1895:
1890:
1884:
1879:
1869:
1864:
1858:
1857:
1852:
1846:
1841:
1835:
1821:
1816:
1806:
1801:
1785:
1780:
1770:
1765:
1749:
1744:
1740:
1712:
1706:
1701:
1691:
1686:
1680:
1679:
1674:
1668:
1663:
1653:
1648:
1642:
1631:
1626:
1616:
1611:
1595:
1590:
1580:
1575:
1568:
1548:
1546:
1338:
1333:
1327:
1306:
1301:
1298:
1274:
1269:
1263:
1242:
1237:
1234:
1199:
1172:
1166:
1161:
1151:
1146:
1140:
1135:
1129:
1124:
1114:
1109:
1103:
1092:
1087:
1077:
1072:
1056:
1051:
1041:
1036:
1029:
1015:
937:of their intersection, and three vectors
856:
849:
848:
843:
837:
832:
825:
824:
819:
813:
800:
799:
794:
783:
782:
777:
769:
755:
706:
700:
679:
673:
650:
633:
604:
599:
586:
581:
568:
563:
557:
549:
544:
531:
526:
513:
508:
502:
490:
480:
467:
457:
444:
434:
423:
409:
386:
356:
340:
324:
308:
302:
274:
258:
242:
226:
220:
3398:
3284:
1406:, etc. then bond vectors are defined by
3962:Anslyn, Eric; Dennis Dougherty (2006).
3815:
3407:, showing where ω, φ, & ψ refer to.
1487:, more generally. This is the case for
1222:{\displaystyle 0\leq \varphi <\pi .}
1005:dihedral angle of these two half planes
99:
69:
43:
36:
4199:"dihedral angle calculator polyhedron"
7:
3551:backbone-dependent rotamer libraries
3503:compared to other amino-acid pairs.
1538:. This dihedral angle is defined by
4237:Analysis of the 5 Regular Polyhedra
3563:Table of polyhedron dihedral angles
3966:. University Science. p. 95.
3938:Compendium of Chemical Terminology
3909:Compendium of Chemical Terminology
3879:Journal of Computational Chemistry
3767:
3764:
3761:
3757:
3741:
3738:
3735:
3731:
3713:
3710:
3707:
3703:
3687:
3684:
3681:
3677:
3658:
3655:
3652:
3648:
1286:{\displaystyle -\mathbf {b} _{0}.}
850:
826:
801:
784:
14:
3964:Modern Physical Organic Chemistry
3610:Law of cosines for dihedral angle
3353:-conformation; antiperiplanar as
1375:In some scientific areas such as
1347:{\displaystyle -\mathbf {b} _{0}}
919:is the product of their lengths.
172:, having two atoms in common. In
16:Angle between two planes in space
3296:as a function of dihedral angle.
3234:
3227:
3220:
3180:{\displaystyle \varphi =-\pi /3}
3143:{\displaystyle \varphi =+\pi /3}
3040:
3025:
3007:
2992:
2971:
2956:
2938:
2923:
2902:
2887:
2869:
2854:
2830:
2815:
2794:
2779:
2724:
2709:
2688:
2673:
2652:
2637:
2619:
2601:
2531:
2516:
2493:
2478:
2456:
2441:
2423:
2405:
2357:
2342:
2319:
2304:
2282:
2267:
2246:
2231:
2157:
2142:
2121:
2106:
2085:
2059:
2044:
2023:
2008:
1987:
1918:
1903:
1880:
1865:
1842:
1817:
1802:
1781:
1766:
1745:
1702:
1687:
1664:
1649:
1627:
1612:
1591:
1576:
1334:
1315:{\displaystyle \mathbf {b} _{0}}
1302:
1270:
1251:{\displaystyle \mathbf {b} _{0}}
1238:
1162:
1147:
1125:
1110:
1088:
1073:
1052:
1037:
844:
820:
795:
778:
3499:has an increased prevalence of
3771:
3754:
3745:
3728:
3717:
3700:
3691:
3674:
3662:
3645:
3035:
3017:
2987:
2984:
2966:
2948:
2918:
2915:
2897:
2879:
2849:
2846:
2840:
2810:
2804:
2774:
2737:
2734:
2704:
2698:
2668:
2662:
2632:
2612:
2595:
2591:
2542:
2510:
2504:
2472:
2466:
2436:
2416:
2399:
2368:
2336:
2330:
2298:
2292:
2262:
2256:
2226:
2170:
2167:
2137:
2131:
2101:
2096:
2079:
2072:
2069:
2039:
2033:
2003:
2000:
1982:
1929:
1897:
1891:
1859:
1853:
1836:
1830:
1827:
1797:
1791:
1761:
1758:
1713:
1681:
1675:
1643:
1637:
1607:
1601:
1571:
1173:
1141:
1136:
1104:
1098:
1068:
1062:
1032:
857:
838:
833:
814:
1:
4032:10.1016/S0065-3233(08)60520-3
4001:10.1016/S0022-2836(63)80023-6
3790:This can be deduced from the
3106:{\displaystyle \varphi =\pi }
2760:formula, and the fact that a
4122:Journal of Molecular Biology
3989:Journal of Molecular Biology
3245:according to dihedral angle
3063:Given the definition of the
3590:Every dihedral angle in an
4289:
3824:"Angle Between Two Planes"
3560:
3206:
401:between them is given by:
18:
4157:Nature Structural Biology
4107:"Side Chain Conformation"
3541:of the side chain in the
4263:Euclidean solid geometry
4077:10.1021/acs.jcim.8b00442
3792:spherical law of cosines
3604:Kepler–Poinsot polyhedra
3468:to be 180° (the typical
3213:Conformational isomerism
3080:{\displaystyle \varphi }
2758:vector quadruple product
394:{\displaystyle \varphi }
3951:10.1351/goldbook.D01730
3922:10.1351/goldbook.T06406
3852:Kröger, Martin (2005).
3289:Free energy diagram of
3187:, which are called the
1956:or, using the function
747:to the planes, one has
204:Mathematical background
176:, it is defined as the
4134:10.1006/jmbi.1993.1170
3781:
3474:case) or 0° (the rare
3408:
3319:molecular conformation
3297:
3209:Alkane stereochemistry
3181:
3144:
3107:
3081:
3054:
2747:
2560:
2180:
1947:
1348:
1316:
1287:
1252:
1223:
1184:
868:
716:
689:
662:
619:
395:
372:
290:
212:by the two equations
31:
3782:
3571:, is measured as the
3460:The planarity of the
3402:
3288:
3182:
3145:
3108:
3082:
3055:
2762:scalar triple product
2748:
2561:
2181:
1948:
1349:
1317:
1288:
1253:
1224:
1185:
869:
717:
715:{\displaystyle d_{2}}
690:
688:{\displaystyle d_{1}}
663:
620:
396:
373:
291:
210:Cartesian coordinates
29:
3621:
3279:sawhorse projection
3154:
3117:
3091:
3071:
2771:
2576:
2200:
1967:
1545:
1326:
1297:
1262:
1233:
1198:
1014:
754:
699:
672:
632:
408:
385:
381:the dihedral angle,
301:
219:
4209:on 25 November 2015
4169:10.1038/nsb0594-334
3361:; and synclinal as
3262:conformation (−60°)
3243:Configuration names
897:of the vectors and
609:
591:
573:
554:
536:
518:
158:intersecting planes
3777:
3464:usually restricts
3417:G. N. Ramachandran
3409:
3372:For example, with
3298:
3203:In stereochemistry
3177:
3140:
3103:
3077:
3067:, this means that
3050:
2743:
2556:
2554:
2176:
1943:
1941:
1371:In polymer physics
1344:
1312:
1283:
1248:
1219:
1180:
864:
725:Alternatively, if
712:
685:
658:
615:
595:
577:
559:
540:
522:
504:
391:
368:
286:
198:starboard and port
32:
4273:Planes (geometry)
4258:Protein structure
3775:
3433:protein structure
3413:Ramachandran plot
3283:
3282:
3265:Newman projection
2569:or equivalently,
2547:
2373:
1934:
1718:
1497:protein structure
1178:
862:
613:
610:
555:
190:higher dimensions
146:
145:
4280:
4219:
4218:
4216:
4214:
4205:. Archived from
4203:www.had2know.com
4195:
4189:
4188:
4152:
4146:
4145:
4117:
4111:
4110:
4103:
4097:
4096:
4071:(9): 2033–2042.
4060:
4054:
4053:
4019:
4013:
4012:
3984:
3978:
3977:
3959:
3953:
3930:
3924:
3901:
3895:
3894:
3885:(9): 1132–1141.
3874:
3868:
3867:
3849:
3843:
3842:
3840:
3839:
3830:. Archived from
3820:
3786:
3784:
3783:
3778:
3776:
3774:
3770:
3744:
3720:
3716:
3690:
3661:
3637:
3453:by Ramachandran)
3238:
3231:
3224:
3217:
3216:
3186:
3184:
3183:
3178:
3173:
3149:
3147:
3146:
3141:
3136:
3112:
3110:
3109:
3104:
3086:
3084:
3083:
3078:
3059:
3057:
3056:
3051:
3049:
3048:
3043:
3034:
3033:
3028:
3016:
3015:
3010:
3001:
3000:
2995:
2980:
2979:
2974:
2965:
2964:
2959:
2947:
2946:
2941:
2932:
2931:
2926:
2911:
2910:
2905:
2896:
2895:
2890:
2878:
2877:
2872:
2863:
2862:
2857:
2839:
2838:
2833:
2824:
2823:
2818:
2803:
2802:
2797:
2788:
2787:
2782:
2752:
2750:
2749:
2744:
2733:
2732:
2727:
2718:
2717:
2712:
2697:
2696:
2691:
2682:
2681:
2676:
2661:
2660:
2655:
2646:
2645:
2640:
2628:
2627:
2622:
2615:
2610:
2609:
2604:
2598:
2565:
2563:
2562:
2557:
2555:
2548:
2546:
2545:
2540:
2539:
2534:
2525:
2524:
2519:
2513:
2507:
2502:
2501:
2496:
2487:
2486:
2481:
2475:
2469:
2465:
2464:
2459:
2450:
2449:
2444:
2432:
2431:
2426:
2419:
2414:
2413:
2408:
2402:
2396:
2374:
2372:
2371:
2366:
2365:
2360:
2351:
2350:
2345:
2339:
2333:
2328:
2327:
2322:
2313:
2312:
2307:
2301:
2295:
2291:
2290:
2285:
2276:
2275:
2270:
2255:
2254:
2249:
2240:
2239:
2234:
2224:
2185:
2183:
2182:
2177:
2166:
2165:
2160:
2151:
2150:
2145:
2130:
2129:
2124:
2115:
2114:
2109:
2099:
2094:
2093:
2088:
2082:
2068:
2067:
2062:
2053:
2052:
2047:
2032:
2031:
2026:
2017:
2016:
2011:
1996:
1995:
1990:
1952:
1950:
1949:
1944:
1942:
1935:
1933:
1932:
1927:
1926:
1921:
1912:
1911:
1906:
1900:
1894:
1889:
1888:
1883:
1874:
1873:
1868:
1862:
1856:
1851:
1850:
1845:
1839:
1833:
1826:
1825:
1820:
1811:
1810:
1805:
1790:
1789:
1784:
1775:
1774:
1769:
1754:
1753:
1748:
1741:
1719:
1717:
1716:
1711:
1710:
1705:
1696:
1695:
1690:
1684:
1678:
1673:
1672:
1667:
1658:
1657:
1652:
1646:
1640:
1636:
1635:
1630:
1621:
1620:
1615:
1600:
1599:
1594:
1585:
1584:
1579:
1569:
1537:
1535:
1531:
1525:
1516:
1507:
1489:kinematic chains
1486:
1477:
1468:
1459:
1450:
1441:
1432:
1423:
1414:
1405:
1396:
1387:
1366:
1365:
1360:
1359:
1353:
1351:
1350:
1345:
1343:
1342:
1337:
1321:
1319:
1318:
1313:
1311:
1310:
1305:
1292:
1290:
1289:
1284:
1279:
1278:
1273:
1257:
1255:
1254:
1249:
1247:
1246:
1241:
1228:
1226:
1225:
1220:
1189:
1187:
1186:
1181:
1179:
1177:
1176:
1171:
1170:
1165:
1156:
1155:
1150:
1144:
1139:
1134:
1133:
1128:
1119:
1118:
1113:
1107:
1101:
1097:
1096:
1091:
1082:
1081:
1076:
1061:
1060:
1055:
1046:
1045:
1040:
1030:
1002:
989:
976:
963:
954:
945:
936:
918:
916:
907:
892:
873:
871:
870:
865:
863:
861:
860:
855:
854:
853:
847:
841:
836:
831:
830:
829:
823:
817:
811:
807:
806:
805:
804:
798:
789:
788:
787:
781:
770:
742:
733:
721:
719:
718:
713:
711:
710:
694:
692:
691:
686:
684:
683:
667:
665:
664:
659:
654:
624:
622:
621:
616:
614:
612:
611:
608:
603:
590:
585:
572:
567:
558:
556:
553:
548:
535:
530:
517:
512:
503:
500:
496:
495:
494:
485:
484:
472:
471:
462:
461:
449:
448:
439:
438:
424:
400:
398:
397:
392:
377:
375:
374:
369:
361:
360:
345:
344:
329:
328:
313:
312:
295:
293:
292:
287:
279:
278:
263:
262:
247:
246:
231:
230:
138:
131:
124:
34:
4288:
4287:
4283:
4282:
4281:
4279:
4278:
4277:
4253:Stereochemistry
4243:
4242:
4228:
4223:
4222:
4212:
4210:
4197:
4196:
4192:
4154:
4153:
4149:
4119:
4118:
4114:
4105:
4104:
4100:
4062:
4061:
4057:
4042:
4021:
4020:
4016:
3986:
3985:
3981:
3974:
3961:
3960:
3956:
3931:
3927:
3902:
3898:
3876:
3875:
3871:
3864:
3851:
3850:
3846:
3837:
3835:
3822:
3821:
3817:
3812:
3800:
3721:
3638:
3619:
3618:
3612:
3596:Platonic solids
3592:edge-transitive
3565:
3559:
3511:
3403:Depiction of a
3397:
3302:stereochemistry
3278:
3263:
3258:
3244:
3215:
3205:
3199:conformations.
3152:
3151:
3115:
3114:
3089:
3088:
3069:
3068:
3038:
3023:
3005:
2990:
2969:
2954:
2936:
2921:
2900:
2885:
2867:
2852:
2828:
2813:
2792:
2777:
2769:
2768:
2722:
2707:
2686:
2671:
2650:
2635:
2617:
2599:
2574:
2573:
2553:
2552:
2529:
2514:
2491:
2476:
2470:
2454:
2439:
2421:
2403:
2397:
2388:
2376:
2375:
2355:
2340:
2317:
2302:
2296:
2280:
2265:
2244:
2229:
2225:
2216:
2198:
2197:
2155:
2140:
2119:
2104:
2083:
2057:
2042:
2021:
2006:
1985:
1965:
1964:
1940:
1939:
1916:
1901:
1878:
1863:
1840:
1834:
1815:
1800:
1779:
1764:
1743:
1742:
1733:
1721:
1720:
1700:
1685:
1662:
1647:
1641:
1625:
1610:
1589:
1574:
1570:
1561:
1543:
1542:
1533:
1529:
1527:
1524:
1518:
1515:
1509:
1506:
1500:
1485:
1479:
1476:
1470:
1467:
1461:
1458:
1452:
1449:
1443:
1440:
1434:
1431:
1425:
1422:
1416:
1413:
1407:
1404:
1398:
1395:
1389:
1386:
1380:
1377:polymer physics
1373:
1363:
1362:
1357:
1355:
1332:
1324:
1323:
1300:
1295:
1294:
1268:
1260:
1259:
1236:
1231:
1230:
1196:
1195:
1160:
1145:
1123:
1108:
1102:
1086:
1071:
1050:
1035:
1031:
1012:
1011:
1007:is defined by
1001:
991:
988:
978:
975:
965:
962:
956:
953:
947:
944:
938:
934:
927:absolute values
915:
909:
906:
900:
898:
891:
884:
878:
842:
818:
812:
793:
776:
775:
771:
752:
751:
741:
735:
732:
726:
702:
697:
696:
675:
670:
669:
630:
629:
501:
486:
476:
463:
453:
440:
430:
429:
425:
406:
405:
383:
382:
352:
336:
320:
304:
299:
298:
270:
254:
238:
222:
217:
216:
206:
142:
109:
91:
87:
83:
79:
61:
57:
53:
38:Types of angles
24:
17:
12:
11:
5:
4286:
4284:
4276:
4275:
4270:
4265:
4260:
4255:
4245:
4244:
4241:
4240:
4234:
4227:
4226:External links
4224:
4221:
4220:
4190:
4147:
4112:
4098:
4055:
4040:
4014:
3979:
3973:978-1891389313
3972:
3954:
3943:Dihedral angle
3925:
3896:
3869:
3862:
3844:
3828:TutorVista.com
3814:
3813:
3811:
3808:
3807:
3806:
3799:
3796:
3788:
3787:
3773:
3769:
3766:
3763:
3759:
3756:
3753:
3750:
3747:
3743:
3740:
3737:
3733:
3730:
3727:
3724:
3719:
3715:
3712:
3709:
3705:
3702:
3699:
3696:
3693:
3689:
3686:
3683:
3679:
3676:
3673:
3670:
3667:
3664:
3660:
3657:
3654:
3650:
3647:
3644:
3641:
3635:
3632:
3629:
3626:
3611:
3608:
3600:Catalan solids
3573:internal angle
3558:
3555:
3509:
3455:
3454:
3447:
3444:
3396:
3393:
3323:Stereochemical
3281:
3280:
3267:
3246:
3240:
3239:
3232:
3225:
3204:
3201:
3176:
3172:
3168:
3165:
3162:
3159:
3139:
3135:
3131:
3128:
3125:
3122:
3102:
3099:
3096:
3076:
3061:
3060:
3047:
3042:
3037:
3032:
3027:
3022:
3019:
3014:
3009:
3004:
2999:
2994:
2989:
2986:
2983:
2978:
2973:
2968:
2963:
2958:
2953:
2950:
2945:
2940:
2935:
2930:
2925:
2920:
2917:
2914:
2909:
2904:
2899:
2894:
2889:
2884:
2881:
2876:
2871:
2866:
2861:
2856:
2851:
2848:
2845:
2842:
2837:
2832:
2827:
2822:
2817:
2812:
2809:
2806:
2801:
2796:
2791:
2786:
2781:
2776:
2754:
2753:
2742:
2739:
2736:
2731:
2726:
2721:
2716:
2711:
2706:
2703:
2700:
2695:
2690:
2685:
2680:
2675:
2670:
2667:
2664:
2659:
2654:
2649:
2644:
2639:
2634:
2631:
2626:
2621:
2614:
2608:
2603:
2597:
2593:
2590:
2587:
2584:
2581:
2567:
2566:
2551:
2544:
2538:
2533:
2528:
2523:
2518:
2512:
2506:
2500:
2495:
2490:
2485:
2480:
2474:
2468:
2463:
2458:
2453:
2448:
2443:
2438:
2435:
2430:
2425:
2418:
2412:
2407:
2401:
2394:
2391:
2389:
2387:
2384:
2381:
2378:
2377:
2370:
2364:
2359:
2354:
2349:
2344:
2338:
2332:
2326:
2321:
2316:
2311:
2306:
2300:
2294:
2289:
2284:
2279:
2274:
2269:
2264:
2261:
2258:
2253:
2248:
2243:
2238:
2233:
2228:
2222:
2219:
2217:
2215:
2212:
2209:
2206:
2205:
2187:
2186:
2175:
2172:
2169:
2164:
2159:
2154:
2149:
2144:
2139:
2136:
2133:
2128:
2123:
2118:
2113:
2108:
2103:
2098:
2092:
2087:
2081:
2077:
2074:
2071:
2066:
2061:
2056:
2051:
2046:
2041:
2038:
2035:
2030:
2025:
2020:
2015:
2010:
2005:
2002:
1999:
1994:
1989:
1984:
1981:
1978:
1975:
1972:
1954:
1953:
1938:
1931:
1925:
1920:
1915:
1910:
1905:
1899:
1893:
1887:
1882:
1877:
1872:
1867:
1861:
1855:
1849:
1844:
1838:
1832:
1829:
1824:
1819:
1814:
1809:
1804:
1799:
1796:
1793:
1788:
1783:
1778:
1773:
1768:
1763:
1760:
1757:
1752:
1747:
1739:
1736:
1734:
1732:
1729:
1726:
1723:
1722:
1715:
1709:
1704:
1699:
1694:
1689:
1683:
1677:
1671:
1666:
1661:
1656:
1651:
1645:
1639:
1634:
1629:
1624:
1619:
1614:
1609:
1606:
1603:
1598:
1593:
1588:
1583:
1578:
1573:
1567:
1564:
1562:
1560:
1557:
1554:
1551:
1550:
1522:
1513:
1504:
1483:
1474:
1465:
1456:
1447:
1438:
1429:
1420:
1411:
1402:
1393:
1384:
1372:
1369:
1341:
1336:
1331:
1309:
1304:
1282:
1277:
1272:
1267:
1245:
1240:
1218:
1215:
1212:
1209:
1206:
1203:
1194:and satisfies
1192:
1191:
1175:
1169:
1164:
1159:
1154:
1149:
1143:
1138:
1132:
1127:
1122:
1117:
1112:
1106:
1100:
1095:
1090:
1085:
1080:
1075:
1070:
1067:
1064:
1059:
1054:
1049:
1044:
1039:
1034:
1028:
1025:
1022:
1019:
999:
986:
973:
960:
951:
942:
913:
904:
889:
882:
875:
874:
859:
852:
846:
840:
835:
828:
822:
816:
810:
803:
797:
792:
786:
780:
774:
768:
765:
762:
759:
739:
730:
709:
705:
682:
678:
657:
653:
649:
646:
643:
640:
637:
628:and satisfies
626:
625:
607:
602:
598:
594:
589:
584:
580:
576:
571:
566:
562:
552:
547:
543:
539:
534:
529:
525:
521:
516:
511:
507:
499:
493:
489:
483:
479:
475:
470:
466:
460:
456:
452:
447:
443:
437:
433:
428:
422:
419:
416:
413:
390:
379:
378:
367:
364:
359:
355:
351:
348:
343:
339:
335:
332:
327:
323:
319:
316:
311:
307:
296:
285:
282:
277:
273:
269:
266:
261:
257:
253:
250:
245:
241:
237:
234:
229:
225:
205:
202:
174:solid geometry
150:dihedral angle
144:
143:
141:
140:
133:
126:
118:
115:
114:
102:
101:
97:
96:
72:
71:
70:2D angle pairs
67:
66:
46:
45:
41:
40:
15:
13:
10:
9:
6:
4:
3:
2:
4285:
4274:
4271:
4269:
4266:
4264:
4261:
4259:
4256:
4254:
4251:
4250:
4248:
4238:
4235:
4233:
4230:
4229:
4225:
4208:
4204:
4200:
4194:
4191:
4186:
4182:
4178:
4174:
4170:
4166:
4163:(5): 334–40.
4162:
4158:
4151:
4148:
4143:
4139:
4135:
4131:
4128:(2): 543–74.
4127:
4123:
4116:
4113:
4108:
4102:
4099:
4094:
4090:
4086:
4082:
4078:
4074:
4070:
4066:
4059:
4056:
4051:
4047:
4043:
4041:9780120342341
4037:
4033:
4029:
4025:
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4015:
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3994:
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3955:
3952:
3948:
3944:
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3934:
3929:
3926:
3923:
3919:
3915:
3914:Torsion angle
3911:
3910:
3905:
3900:
3897:
3892:
3888:
3884:
3880:
3873:
3870:
3865:
3859:
3855:
3848:
3845:
3834:on 2020-10-28
3833:
3829:
3825:
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3809:
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3342:
3340:
3336:
3332:
3328:
3324:
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3315:
3311:
3310:chemical bond
3307:
3306:torsion angle
3303:
3295:
3293:
3287:
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3274:
3271:
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3256:
3253:
3250:
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3137:
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3074:
3066:
3065:cross product
3045:
3030:
3020:
3012:
3002:
2997:
2981:
2976:
2961:
2951:
2943:
2933:
2928:
2912:
2907:
2892:
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2843:
2835:
2825:
2820:
2807:
2799:
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2767:
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2763:
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2683:
2678:
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2647:
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2218:
2213:
2210:
2207:
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2195:
2194:
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2173:
2162:
2152:
2147:
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2126:
2116:
2111:
2090:
2075:
2064:
2054:
2049:
2036:
2028:
2018:
2013:
1997:
1992:
1979:
1976:
1973:
1970:
1963:
1962:
1961:
1959:
1936:
1923:
1913:
1908:
1885:
1875:
1870:
1847:
1822:
1812:
1807:
1794:
1786:
1776:
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1659:
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1539:
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1512:
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1490:
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1473:
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1368:
1339:
1329:
1307:
1280:
1275:
1265:
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1213:
1210:
1207:
1204:
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1167:
1157:
1152:
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1120:
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1020:
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920:
912:
908:| |
903:
896:
888:
885: ·
881:
808:
790:
772:
766:
763:
760:
757:
750:
749:
748:
746:
745:normal vector
738:
729:
723:
707:
703:
680:
676:
655:
651:
647:
644:
641:
638:
635:
605:
600:
596:
592:
587:
582:
578:
574:
569:
564:
560:
550:
545:
541:
537:
532:
527:
523:
519:
514:
509:
505:
497:
491:
487:
481:
477:
473:
468:
464:
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454:
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445:
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435:
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426:
420:
417:
414:
411:
404:
403:
402:
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362:
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349:
346:
341:
337:
333:
330:
325:
321:
317:
314:
309:
305:
297:
283:
280:
275:
271:
267:
264:
259:
255:
251:
248:
243:
239:
235:
232:
227:
223:
215:
214:
213:
211:
203:
201:
199:
195:
191:
187:
183:
179:
175:
171:
167:
163:
159:
155:
151:
139:
134:
132:
127:
125:
120:
119:
117:
116:
113:
110:
108:
104:
103:
98:
95:
92:
90:
89:Supplementary
86:
85:Complementary
82:
78:
74:
73:
68:
65:
62:
60:
56:
52:
48:
47:
42:
39:
35:
28:
22:
4211:. Retrieved
4207:the original
4202:
4193:
4160:
4156:
4150:
4125:
4121:
4115:
4101:
4068:
4064:
4058:
4023:
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3992:
3988:
3982:
3963:
3957:
3936:
3928:
3907:
3899:
3882:
3878:
3872:
3856:. Springer.
3853:
3847:
3836:. Retrieved
3832:the original
3827:
3818:
3789:
3613:
3589:
3577:antiparallel
3568:
3566:
3546:
3542:
3538:
3534:
3530:
3526:
3522:
3518:
3514:
3507:
3505:
3500:
3492:
3485:
3481:
3475:
3469:
3465:
3462:peptide bond
3459:
3456:
3450:
3431:residues in
3424:
3420:
3410:
3390:
3385:
3381:
3373:
3371:
3366:
3362:
3358:
3354:
3350:
3346:
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3338:
3334:
3330:
3326:
3305:
3299:
3291:
3272:
3269:
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3192:
3188:
3062:
2755:
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2188:
1955:
1519:
1510:
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1480:
1471:
1462:
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1426:
1417:
1408:
1399:
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1381:
1374:
1193:
1004:
996:
992:
983:
979:
970:
966:
957:
948:
939:
925:However the
924:
921:
910:
901:
886:
879:
876:
736:
727:
724:
627:
380:
207:
156:between two
149:
147:
106:
105:
75:
49:
3804:Atropisomer
1493:amino acids
931:half planes
895:dot product
194:hyperplanes
162:half-planes
94:Transversal
4247:Categories
4213:25 October
3863:3540262105
3838:2018-07-06
3810:References
3581:degenerate
3569:face angle
3561:See also:
3429:amino acid
3339:periplanar
3207:See also:
964:such that
3758:∠
3752:
3732:∠
3726:
3704:∠
3698:
3678:∠
3672:
3666:−
3649:∠
3643:
3631:φ
3628:
3598:, the 13
3167:π
3164:−
3158:φ
3130:π
3121:φ
3101:π
3095:φ
3075:φ
3021:⋅
3003:×
2952:⋅
2934:×
2913:−
2883:⋅
2865:×
2826:×
2808:×
2790:×
2720:×
2702:⋅
2684:×
2648:×
2630:⋅
2589:
2580:φ
2527:×
2489:×
2452:×
2434:⋅
2386:φ
2383:
2353:×
2315:×
2278:×
2260:⋅
2242:×
2214:φ
2211:
2153:×
2135:⋅
2117:×
2055:×
2037:×
2019:×
1998:⋅
1980:
1971:φ
1914:×
1876:×
1813:×
1795:×
1777:×
1756:⋅
1731:φ
1728:
1698:×
1660:×
1623:×
1605:⋅
1587:×
1559:φ
1556:
1330:−
1266:−
1214:π
1208:φ
1205:≤
1158:×
1121:×
1084:×
1066:⋅
1048:×
1024:φ
1021:
791:⋅
764:φ
761:
648:π
645:≤
642:φ
639:≤
418:φ
415:
389:φ
166:chemistry
100:3D angles
64:Spherical
44:2D angles
4093:52031431
4085:30118602
4009:13990617
3995:: 95–9.
3798:See also
3602:, the 4
3557:Geometry
3423:against
3395:Proteins
3314:molecule
107:Dihedral
81:Vertical
77:Adjacent
59:Exterior
55:Interior
21:Dihedral
4185:9157373
4177:7664040
4142:8464064
4050:7020376
3497:proline
3489:isomers
3437:protein
3435:. In a
3405:protein
3294:-butane
893:is the
152:is the
4183:
4175:
4140:
4091:
4083:
4048:
4038:
4007:
3970:
3860:
3585:tiling
3543:gauche
3527:gauche
3525:, and
3523:gauche
3378:butane
3363:gauche
3335:clinal
3276:Butane
3260:gauche
3257:in the
3255:Butane
3197:gauche
3195:, and
3193:gauche
1460:, and
917:|
899:|
877:where
188:. In
4268:Angle
4181:S2CID
4089:S2CID
3933:IUPAC
3904:IUPAC
3519:trans
3513:(chi-
3493:trans
3482:trans
3471:trans
3359:trans
3349:- or
3341:(p).
3189:trans
2586:atan2
1977:atan2
1958:atan2
1495:in a
1322:with
1258:with
180:of a
178:union
170:atoms
164:. In
154:angle
112:Solid
51:Right
4215:2015
4173:PMID
4138:PMID
4081:PMID
4046:PMID
4036:ISBN
4005:PMID
3968:ISBN
3858:ISBN
3533:and
3484:and
3386:anti
3367:skew
3355:anti
3331:anti
3304:, a
3211:and
3150:and
1517:and
1361:and
1211:<
990:and
955:and
743:are
734:and
695:and
186:edge
182:line
4165:doi
4130:doi
4126:230
4073:doi
4028:doi
3997:doi
3947:doi
3945:".
3918:doi
3916:".
3887:doi
3749:sin
3723:sin
3695:cos
3669:cos
3640:cos
3625:cos
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3486:cis
3477:cis
3427:of
3382:syn
3365:or
3357:or
3351:cis
3347:syn
3327:syn
3300:In
3270:syn
3249:syn
2380:sin
2208:cos
1725:sin
1553:cos
1491:or
1475:i+1
1018:cos
758:cos
412:cos
160:or
4249::
4201:.
4179:.
4171:.
4159:.
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4087:.
4079:.
4069:58
4067:.
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3881:.
3826:.
3794:.
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3451:φ′
3411:A
3369:.
3321:.
3273:n-
3252:n-
3191:,
3113:,
1960:,
1532:,
1528:(−
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995:+
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969:+
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722:.
656:2.
148:A
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4109:.
4095:.
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4011:.
3999::
3993:7
3976:.
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3893:.
3889::
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3768:C
3765:P
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3755:(
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3646:(
3634:=
3547:ψ
3539:γ
3535:ψ
3531:φ
3515:n
3510:n
3508:χ
3466:ω
3425:φ
3421:ψ
3376:-
3374:n
3292:n
3175:3
3171:/
3161:=
3138:3
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3098:=
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2393:=
2369:|
2363:3
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2331:|
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2148:2
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2112:1
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2065:3
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2050:2
2045:u
2040:(
2034:)
2029:2
2024:u
2014:1
2009:u
2004:(
2001:(
1993:2
1988:u
1983:(
1974:=
1937:,
1930:|
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1909:2
1904:u
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1572:(
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1534:π
1530:π
1523:3
1520:u
1514:2
1511:u
1505:1
1502:u
1484:i
1481:r
1478:−
1472:r
1469:=
1466:i
1463:u
1457:2
1454:r
1451:−
1448:3
1445:r
1442:=
1439:2
1436:u
1430:1
1427:r
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1421:2
1418:r
1415:=
1412:1
1409:u
1403:3
1400:r
1394:2
1391:r
1385:1
1382:r
1364:π
1358:π
1356:−
1340:0
1335:b
1308:0
1303:b
1281:.
1276:0
1271:b
1244:0
1239:b
1217:.
1202:0
1190:,
1174:|
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1126:b
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1099:)
1094:2
1089:b
1079:0
1074:b
1069:(
1063:)
1058:1
1053:b
1043:0
1038:b
1033:(
1027:=
1000:2
997:b
993:P
987:1
984:b
980:P
974:0
971:b
967:P
961:2
958:b
952:1
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943:0
940:b
935:P
914:B
911:n
905:A
902:n
890:B
887:n
883:A
880:n
858:|
851:B
845:n
839:|
834:|
827:A
821:n
815:|
809:|
802:B
796:n
785:A
779:n
773:|
767:=
740:B
737:n
731:A
728:n
708:2
704:d
681:1
677:d
652:/
636:0
606:2
601:2
597:c
593:+
588:2
583:2
579:b
575:+
570:2
565:2
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546:1
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538:+
533:2
528:1
524:b
520:+
515:2
510:1
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498:|
492:2
488:c
482:1
478:c
474:+
469:2
465:b
459:1
455:b
451:+
446:2
442:a
436:1
432:a
427:|
421:=
366:0
363:=
358:2
354:d
350:+
347:z
342:2
338:c
334:+
331:y
326:2
322:b
318:+
315:x
310:2
306:a
284:0
281:=
276:1
272:d
268:+
265:z
260:1
256:c
252:+
249:y
244:1
240:b
236:+
233:x
228:1
224:a
137:e
130:t
123:v
23:.
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