27:
3397:
1751:
1363:
2572:
515:
901:
3254:
263:, in all directions, constitute a two-parameter uncountable infinity of rotation vectors encoding the same rotation as the zero vector. These facts must be taken into account when inverting the exponential map, that is, when finding a rotation vector that corresponds to a given rotation matrix. The exponential map is
1546:
1165:
3035:
2414:
2155:
320:
757:
3220:
2934:
3103:
3392:{\displaystyle {\begin{aligned}\theta &=2\arccos r\\{\boldsymbol {\omega }}&={\begin{cases}{\dfrac {\mathbf {v} }{\sin {\tfrac {\theta }{2}}}},&{\text{if }}\theta \neq 0\\0,&{\text{otherwise}}.\end{cases}}\end{aligned}}}
2779:
1746:{\displaystyle R=I+\left(\theta -{\frac {\theta ^{3}}{3!}}+{\frac {\theta ^{5}}{5!}}-\cdots \right)\mathbf {K} +\left({\frac {\theta ^{2}}{2!}}-{\frac {\theta ^{4}}{4!}}+{\frac {\theta ^{6}}{6!}}-\cdots \right)\mathbf {K} ^{2}\,,}
2015:
2010:
1840:
2919:
1104:
1358:{\displaystyle R=\exp(\theta \mathbf {K} )=\sum _{k=0}^{\infty }{\frac {(\theta \mathbf {K} )^{k}}{k!}}=I+\theta \mathbf {K} +{\frac {1}{2!}}(\theta \mathbf {K} )^{2}+{\frac {1}{3!}}(\theta \mathbf {K} )^{3}+\cdots }
1039:
601:
3466:
2676:
2567:{\displaystyle \log R={\begin{cases}0&{\text{if }}\theta =0\\{\dfrac {\theta }{2\sin \theta }}\left(R-R^{\mathsf {T}}\right)&{\text{if }}\theta \neq 0{\text{ and }}\theta \in (-\pi ,\pi )\end{cases}}}
214:
3146:
3259:
679:, is an efficient algorithm for rotating a Euclidean vector, given a rotation axis and an angle of rotation. In other words, Rodrigues' formula provides an algorithm to compute the exponential map from
929:
There are several ways to represent a rotation. It is useful to understand how different representations relate to one another, and how to convert between them. Here the unit vector is denoted
510:{\displaystyle (\mathrm {axis} ,\mathrm {angle} )=\left({\begin{bmatrix}e_{x}\\e_{y}\\e_{z}\end{bmatrix}},\theta \right)=\left({\begin{bmatrix}0\\0\\-1\end{bmatrix}},{\frac {-\pi }{2}}\right).}
1480:
896:{\displaystyle \mathbf {v} _{\mathrm {rot} }=(\cos \theta )\mathbf {v} +(\sin \theta )(\mathbf {e} \times \mathbf {v} )+(1-\cos \theta )(\mathbf {e} \cdot \mathbf {v} )\mathbf {e} \,.}
2298:
3040:
2340:
2689:
713:
2840:, where the off-axis terms do not give information about the rotation axis (which is still defined up to a sign ambiguity). In that case, we must reconsider the above formula.
1948:
1756:
2273:
2390:
2363:
2318:
2187:
152:, which dictates that any rotation or sequence of rotations of a rigid body in a three-dimensional space is equivalent to a pure rotation about a single fixed axis.
2845:
2247:
2227:
2207:
3030:{\displaystyle R=I+2\mathbf {K} ^{2}=I+2({\boldsymbol {\omega }}\otimes {\boldsymbol {\omega }}-I)=2{\boldsymbol {\omega }}\otimes {\boldsymbol {\omega }}-I}
978:
546:
3408:
175:. In this case, both the rotation axis and the angle are represented by a vector codirectional with the rotation axis whose length is the rotation angle
2607:
182:
3507:
1050:
2150:{\displaystyle {\boldsymbol {\omega }}={\frac {1}{2\sin \theta }}{\begin{bmatrix}R_{32}-R_{23}\\R_{13}-R_{31}\\R_{21}-R_{12}\end{bmatrix}}~,}
137:, the angle and axis determine a transformation that rotates three-dimensional vectors. The rotation occurs in the sense prescribed by the
1441:
3532:
This holds for the triplet representation of the rotation group, i.e., spin 1. For higher dimensional representations/spins, see
245:. Thus, there are at least a countable infinity of rotation vectors corresponding to any rotation. Furthermore, all rotations by
156:
3604:
3215:{\displaystyle \mathbf {q} =\left(\cos {\tfrac {\theta }{2}},{\boldsymbol {\omega }}\sin {\tfrac {\theta }{2}}\right)}
3130:
1127:
1852:
672:
667:
134:
2278:
2323:
1891:
968:
962:
630:
121:
105:
3515:, a representation of rigid-body motions and velocities using the concepts of twists, screws, and wrenches
1424:
682:
658:
is a convenient construction of the
Cartesian representation of the Rotation Matrix in three dimensions.
149:
1942:
2397:
1845:
1374:
3305:
3117:
and the signs (up to sign ambiguity) can be determined from the signs of the off-axis terms of
2435:
1902:
continue to denote the 3 × 3 matrix that effects the cross product with the rotation axis
227:
Many rotation vectors correspond to the same rotation. In particular, a rotation vector of length
3497:
68:
3098:{\displaystyle B:={\boldsymbol {\omega }}\otimes {\boldsymbol {\omega }}={\frac {1}{2}}(R+I)\,,}
958:
3239:
64:
2255:
2375:
2348:
2303:
221:
2774:{\displaystyle d_{g}(A,B):=\left\|\log \left(A^{\mathsf {T}}B\right)\right\|_{\mathrm {F} }}
2162:
279:
Say you are standing on the ground and you pick the direction of gravity to be the negative
3564:
1887:
84:
1047:
one derives a closed-form relation between these two representations. Given a unit vector
8:
655:
638:
611:
3568:
3580:
3554:
2232:
2212:
2192:
950:
217:
1373:
is skew-symmetric, and the sum of the squares of its above-diagonal entries is 1, the
954:
634:
92:
20:
3584:
1851:
This is a Lie-algebraic derivation, in contrast to the geometric one in the article
3572:
3534:
2404:
1044:
676:
619:
108:. Only two numbers, not three, are needed to define the direction of a unit vector
88:
2795:. In that case, the off-axis terms will actually provide better information about
2005:{\displaystyle \theta =\arccos \left({\frac {\operatorname {Tr} (R)-1}{2}}\right)}
1835:{\displaystyle R=I+(\sin \theta )\mathbf {K} +(1-\cos \theta )\mathbf {K} ^{2}\,,}
2587:
1938:
972:
924:
138:
71:
3609:
3479:
2601:
654:
and their associated three orthogonal axes in a
Cartesian representation into
519:
The above example can be represented as a rotation vector with a magnitude of
3598:
3542:
3538:
2914:{\displaystyle R=I+\mathbf {K} \sin \theta +\mathbf {K} ^{2}(1-\cos \theta )}
2788:
may be numerically imprecise as the derivative of arccos goes to infinity as
1858:
Due to the existence of the above-mentioned exponential map, the unit vector
905:
For the rotation of a single vector it may be more efficient than converting
1099:{\textstyle {\boldsymbol {\omega }}\in {\mathfrak {so}}(3)=\mathbb {R} ^{3}}
971:
effects a transformation from the axis-angle representation of rotations to
3576:
3512:
3502:
314:
2816:. (This is because these are the first two terms of the Taylor series for
618:, and also for converting between different representations of rigid body
741:
304:
75:
56:
1525:
This cyclic pattern continues indefinitely, and so all higher powers of
3140:
2590:. In this case, the log is not unique. However, even in the case where
2583:
626:
3545:(2014). "A compact formula for rotations as spin matrix polynomials".
1034:{\displaystyle \exp \colon {\mathfrak {so}}(3)\to \mathrm {SO} (3)\,.}
596:{\displaystyle {\begin{bmatrix}0\\0\\{\frac {\pi }{2}}\end{bmatrix}}.}
101:
3461:{\displaystyle \theta =2\operatorname {atan2} (|\mathbf {v} |,r)\,,}
3401:
A more numerically stable expression of the rotation angle uses the
26:
3251:, the axis–angle coordinates can be extracted using the following:
615:
241:, encodes exactly the same rotation as a rotation vector of length
130:
suffice to locate it in any particular
Cartesian coordinate frame.
3559:
2781:
is the geodesic distance on the 3D manifold of rotation matrices.
2671:{\displaystyle \|\log(R)\|_{\mathrm {F} }={\sqrt {2}}|\theta |\,.}
2252:
The axis-angle representation is not unique since a rotation of
744:
rooted at the origin describing an axis of rotation about which
167:
The axis–angle representation is equivalent to the more concise
42:
define a rotation, concisely represented by the rotation vector
3136:
3135:
The following expression transforms axis–angle coordinates to
754:, Rodrigues' rotation formula to obtain the rotated vector is
610:
The axis–angle representation is convenient when dealing with
209:{\displaystyle {\boldsymbol {\theta }}=\theta \mathbf {e} \,.}
3402:
252:
are the same as no rotation at all, so, for a given integer
3381:
2560:
642:
16:
Parameterization of a rotation into a unit vector and angle
285:
direction. Then if you turn to your left, you will rotate
944:
918:
3533:
3324:
3196:
3170:
2053:
1053:
555:
450:
377:
3411:
3309:
3257:
3149:
3043:
2937:
2848:
2692:
2610:
2462:
2417:
2378:
2351:
2326:
2306:
2281:
2258:
2235:
2215:
2195:
2165:
2018:
1951:
1759:
1549:
1444:
1168:
981:
760:
685:
549:
323:
185:
1106:representing the unit rotation axis, and an angle,
3460:
3391:
3214:
3097:
3029:
2913:
2773:
2670:
2566:
2384:
2357:
2334:
2312:
2292:
2267:
2241:
2221:
2201:
2181:
2149:
2004:
1834:
1745:
1474:
1357:
1098:
1033:
895:
707:
622:, such as homogeneous transformations and twists.
595:
509:
313:axis. Viewing the axis-angle representation as an
208:
1846:Taylor series formula for trigonometric functions
1543:. Thus, from the above equation, it follows that
1475:{\displaystyle \mathbf {K} ^{3}=-\mathbf {K} \,.}
148:. The axis–angle representation is predicated on
3596:
2830:This formulation also has numerical problems at
1881:
2012:and then use that to find the normalized axis,
1937:To retrieve the axis–angle representation of a
719:without computing the full matrix exponential.
2784:For small rotations, the above computation of
1864:representing the rotation axis, and the angle
114:rooted at the origin because the magnitude of
915:into a rotation matrix to rotate the vector.
2630:
2611:
1941:, calculate the angle of rotation from the
963:Rotation group SO(3) § Exponential map
959:Lie algebra § Relation to Lie groups
144:The rotation axis is sometimes called the
100:describing the magnitude and sense (e.g.,
3558:
3454:
3091:
2664:
2189:is the component of the rotation matrix,
1828:
1739:
1468:
1086:
1027:
889:
202:
2293:{\displaystyle -{\boldsymbol {\omega }}}
25:
19:For broader coverage of this topic, see
3289:
3185:
3059:
3051:
3017:
3009:
2989:
2981:
2372:is symmetric. For the general case the
2335:{\displaystyle {\boldsymbol {\omega }}}
2328:
2286:
2020:
1055:
187:
157:rotation formalisms in three dimensions
3597:
2744:
2501:
2345:The above calculation of axis vector
945:Exponential map from 𝔰𝔬(3) to SO(3)
919:Relationship to other representations
648:Plugging the three eigenvalues 1 and
637:rotation axis and the rotation angle
2398:rotation matrix#Determining the axis
661:
614:. It is useful to both characterize
224:maps involving this representation.
3124:
3111:are the squares of the elements of
1067:
1064:
993:
990:
708:{\displaystyle {\mathfrak {so}}(3)}
691:
688:
13:
2765:
2635:
1214:
1014:
1011:
775:
772:
769:
357:
354:
351:
348:
345:
337:
334:
331:
328:
162:
14:
3621:
2392:may be found using null space of
256:, all rotation vectors of length
120:is constrained. For example, the
3436:
3312:
3151:
3131:Quaternions and spatial rotation
2955:
2880:
2862:
1818:
1788:
1729:
1632:
1464:
1447:
1335:
1296:
1267:
1229:
1188:
1116:, an equivalent rotation matrix
885:
877:
869:
834:
826:
800:
763:
198:
3526:
3451:
3441:
3431:
3427:
3088:
3076:
2999:
2977:
2908:
2890:
2759:
2723:
2715:
2703:
2660:
2652:
2626:
2620:
2554:
2539:
1983:
1977:
1813:
1795:
1784:
1772:
1340:
1328:
1301:
1289:
1234:
1222:
1192:
1181:
1078:
1072:
1024:
1018:
1007:
1004:
998:
881:
865:
862:
844:
838:
822:
819:
807:
796:
784:
702:
696:
361:
324:
1:
3519:
2300:is the same as a rotation of
1882:Log map from SO(3) to 𝔰𝔬(3)
1531:can be expressed in terms of
3605:Rotation in three dimensions
1943:trace of the rotation matrix
1892:Infinitesimal transformation
633:, its axis–angle data are a
7:
3491:
1853:Rodrigues' rotation formula
1120:is given as follows, where
673:Rodrigues' rotation formula
668:Rodrigues' rotation formula
135:Rodrigues' rotation formula
10:
3626:
3508:Rotations without a matrix
3128:
1885:
948:
922:
665:
274:
18:
3105:so the diagonal terms of
2799:since, for small angles,
2576:An exception occurs when
1870:are sometimes called the
1375:characteristic polynomial
61:axis–angle representation
2678:Given rotation matrices
2268:{\displaystyle -\theta }
1043:Essentially, by using a
150:Euler's rotation theorem
3498:Homogeneous coordinates
2407:of the rotation matrix
2385:{\displaystyle \omega }
2358:{\displaystyle \omega }
2313:{\displaystyle \theta }
1874:of the rotation matrix
1872:exponential coordinates
1438:= 0, this implies that
1425:Cayley–Hamilton theorem
750:is rotated by an angle
605:
106:rotation about the axis
3577:10.3842/SIGMA.2014.084
3462:
3393:
3216:
3099:
3031:
2915:
2775:
2672:
2568:
2386:
2359:
2336:
2314:
2294:
2269:
2243:
2223:
2203:
2183:
2182:{\displaystyle R_{ij}}
2151:
2006:
1836:
1747:
1476:
1359:
1218:
1100:
1035:
897:
709:
639:continuously dependent
597:
511:
210:
52:
3463:
3394:
3238:represented with its
3217:
3100:
3032:
2916:
2776:
2673:
2569:
2387:
2360:
2337:
2315:
2295:
2270:
2244:
2224:
2204:
2184:
2152:
2007:
1886:Further information:
1837:
1748:
1477:
1360:
1198:
1101:
1036:
949:Further information:
923:Further information:
898:
710:
598:
512:
211:
122:elevation and azimuth
74:by two quantities: a
36:and axis unit vector
29:
3409:
3255:
3147:
3041:
2935:
2846:
2690:
2608:
2415:
2376:
2349:
2324:
2304:
2279:
2256:
2233:
2213:
2193:
2163:
2016:
1949:
1888:Rotation group SO(3)
1757:
1547:
1442:
1166:
1128:cross product matrix
1051:
979:
758:
683:
547:
321:
183:
3569:2014SIGMA..10..084C
631:around a fixed axis
612:rigid-body dynamics
216:It is used for the
3458:
3389:
3387:
3380:
3337:
3333:
3212:
3205:
3179:
3095:
3027:
2911:
2771:
2668:
2564:
2559:
2482:
2382:
2355:
2332:
2310:
2290:
2265:
2239:
2219:
2199:
2179:
2147:
2135:
2002:
1832:
1743:
1472:
1355:
1096:
1031:
951:Matrix exponential
893:
705:
593:
584:
507:
475:
420:
237:, for any integer
206:
171:, also called the
155:It is one of many
53:
3373:
3347:
3336:
3332:
3204:
3178:
3074:
2649:
2531:
2517:
2481:
2446:
2242:{\displaystyle j}
2222:{\displaystyle i}
2202:{\displaystyle R}
2143:
2046:
1996:
1934:in what follows.
1714:
1689:
1664:
1618:
1593:
1326:
1287:
1252:
973:rotation matrices
955:Orthogonal matrix
662:Rotating a vector
580:
497:
93:angle of rotation
69:three-dimensional
21:3D rotation group
3617:
3589:
3588:
3562:
3535:Curtright, T. L.
3530:
3487:
3482:of the 3-vector
3477:
3475:
3467:
3465:
3464:
3459:
3444:
3439:
3434:
3398:
3396:
3395:
3390:
3388:
3384:
3383:
3374:
3371:
3348:
3345:
3338:
3335:
3334:
3325:
3315:
3310:
3292:
3250:
3244:
3237:
3221:
3219:
3218:
3213:
3211:
3207:
3206:
3197:
3188:
3180:
3171:
3154:
3125:Unit quaternions
3120:
3116:
3110:
3104:
3102:
3101:
3096:
3075:
3067:
3062:
3054:
3036:
3034:
3033:
3028:
3020:
3012:
2992:
2984:
2964:
2963:
2958:
2930:
2920:
2918:
2917:
2912:
2889:
2888:
2883:
2865:
2839:
2826:
2815:
2798:
2794:
2787:
2780:
2778:
2777:
2772:
2770:
2769:
2768:
2762:
2758:
2757:
2753:
2749:
2748:
2747:
2702:
2701:
2685:
2681:
2677:
2675:
2674:
2669:
2663:
2655:
2650:
2645:
2640:
2639:
2638:
2599:
2581:
2573:
2571:
2570:
2565:
2563:
2562:
2532:
2529:
2518:
2515:
2511:
2507:
2506:
2505:
2504:
2483:
2480:
2463:
2447:
2444:
2410:
2405:matrix logarithm
2395:
2391:
2389:
2388:
2383:
2371:
2364:
2362:
2361:
2356:
2341:
2339:
2338:
2333:
2331:
2319:
2317:
2316:
2311:
2299:
2297:
2296:
2291:
2289:
2274:
2272:
2271:
2266:
2248:
2246:
2245:
2240:
2228:
2226:
2225:
2220:
2208:
2206:
2205:
2200:
2188:
2186:
2185:
2180:
2178:
2177:
2156:
2154:
2153:
2148:
2141:
2140:
2139:
2132:
2131:
2119:
2118:
2105:
2104:
2092:
2091:
2078:
2077:
2065:
2064:
2047:
2045:
2028:
2023:
2011:
2009:
2008:
2003:
2001:
1997:
1992:
1969:
1933:
1928:for all vectors
1927:
1907:
1901:
1877:
1869:
1863:
1841:
1839:
1838:
1833:
1827:
1826:
1821:
1791:
1752:
1750:
1749:
1744:
1738:
1737:
1732:
1726:
1722:
1715:
1713:
1705:
1704:
1695:
1690:
1688:
1680:
1679:
1670:
1665:
1663:
1655:
1654:
1645:
1635:
1630:
1626:
1619:
1617:
1609:
1608:
1599:
1594:
1592:
1584:
1583:
1574:
1542:
1536:
1530:
1521:
1511:
1501:
1491:
1481:
1479:
1478:
1473:
1467:
1456:
1455:
1450:
1437:
1423:. Since, by the
1422:
1392:
1386:
1372:
1364:
1362:
1361:
1356:
1348:
1347:
1338:
1327:
1325:
1314:
1309:
1308:
1299:
1288:
1286:
1275:
1270:
1253:
1251:
1243:
1242:
1241:
1232:
1220:
1217:
1212:
1191:
1161:
1152:for all vectors
1151:
1135:
1125:
1119:
1115:
1105:
1103:
1102:
1097:
1095:
1094:
1089:
1071:
1070:
1058:
1045:Taylor expansion
1040:
1038:
1037:
1032:
1017:
997:
996:
940:
934:
914:
910:
902:
900:
899:
894:
888:
880:
872:
837:
829:
803:
780:
779:
778:
766:
753:
749:
739:
733:
727:
718:
714:
712:
711:
706:
695:
694:
677:Olinde Rodrigues
656:Mercer's theorem
653:
602:
600:
599:
594:
589:
588:
581:
573:
542:
537:pointing in the
536:
535:
533:
532:
529:
526:
516:
514:
513:
508:
503:
499:
498:
493:
485:
480:
479:
436:
432:
425:
424:
417:
416:
403:
402:
389:
388:
360:
340:
317:, this would be
312:
302:
301:
299:
298:
295:
292:
284:
262:
255:
251:
244:
240:
236:
215:
213:
212:
207:
201:
190:
178:
129:
119:
113:
99:
89:axis of rotation
82:
63:parameterizes a
50:
41:
35:
3625:
3624:
3620:
3619:
3618:
3616:
3615:
3614:
3595:
3594:
3593:
3592:
3531:
3527:
3522:
3494:
3483:
3471:
3469:
3440:
3435:
3430:
3410:
3407:
3406:
3386:
3385:
3379:
3378:
3370:
3368:
3359:
3358:
3344:
3342:
3323:
3316:
3311:
3308:
3301:
3300:
3293:
3288:
3285:
3284:
3265:
3258:
3256:
3253:
3252:
3246:
3242:
3225:
3224:Given a versor
3195:
3184:
3169:
3162:
3158:
3150:
3148:
3145:
3144:
3133:
3127:
3118:
3112:
3106:
3066:
3058:
3050:
3042:
3039:
3038:
3016:
3008:
2988:
2980:
2959:
2954:
2953:
2936:
2933:
2932:
2922:
2884:
2879:
2878:
2861:
2847:
2844:
2843:
2831:
2817:
2800:
2796:
2789:
2785:
2764:
2763:
2743:
2742:
2738:
2737:
2733:
2726:
2722:
2721:
2697:
2693:
2691:
2688:
2687:
2683:
2679:
2659:
2651:
2644:
2634:
2633:
2629:
2609:
2606:
2605:
2591:
2577:
2558:
2557:
2530: and
2528:
2514:
2512:
2500:
2499:
2495:
2488:
2484:
2467:
2461:
2458:
2457:
2443:
2441:
2431:
2430:
2416:
2413:
2412:
2408:
2393:
2377:
2374:
2373:
2369:
2350:
2347:
2346:
2327:
2325:
2322:
2321:
2305:
2302:
2301:
2285:
2280:
2277:
2276:
2257:
2254:
2253:
2234:
2231:
2230:
2214:
2211:
2210:
2194:
2191:
2190:
2170:
2166:
2164:
2161:
2160:
2134:
2133:
2127:
2123:
2114:
2110:
2107:
2106:
2100:
2096:
2087:
2083:
2080:
2079:
2073:
2069:
2060:
2056:
2049:
2048:
2032:
2027:
2019:
2017:
2014:
2013:
1970:
1968:
1964:
1950:
1947:
1946:
1939:rotation matrix
1929:
1909:
1903:
1897:
1894:
1884:
1875:
1865:
1859:
1822:
1817:
1816:
1787:
1758:
1755:
1754:
1733:
1728:
1727:
1706:
1700:
1696:
1694:
1681:
1675:
1671:
1669:
1656:
1650:
1646:
1644:
1643:
1639:
1631:
1610:
1604:
1600:
1598:
1585:
1579:
1575:
1573:
1566:
1562:
1548:
1545:
1544:
1538:
1532:
1526:
1513:
1503:
1493:
1483:
1463:
1451:
1446:
1445:
1443:
1440:
1439:
1428:
1394:
1388:
1377:
1368:
1343:
1339:
1334:
1318:
1313:
1304:
1300:
1295:
1279:
1274:
1266:
1244:
1237:
1233:
1228:
1221:
1219:
1213:
1202:
1187:
1167:
1164:
1163:
1153:
1137:
1131:
1121:
1117:
1107:
1090:
1085:
1084:
1063:
1062:
1054:
1052:
1049:
1048:
1010:
989:
988:
980:
977:
976:
969:exponential map
965:
947:
936:
930:
927:
925:Charts on SO(3)
921:
912:
906:
884:
876:
868:
833:
825:
799:
768:
767:
762:
761:
759:
756:
755:
751:
745:
735:
729:
728:is a vector in
723:
716:
687:
686:
684:
681:
680:
670:
664:
649:
608:
583:
582:
572:
569:
568:
562:
561:
551:
550:
548:
545:
544:
538:
530:
527:
524:
523:
521:
520:
486:
484:
474:
473:
464:
463:
457:
456:
446:
445:
444:
440:
419:
418:
412:
408:
405:
404:
398:
394:
391:
390:
384:
380:
373:
372:
371:
367:
344:
327:
322:
319:
318:
308:
296:
293:
290:
289:
287:
286:
280:
277:
257:
253:
246:
242:
238:
228:
197:
186:
184:
181:
180:
176:
169:rotation vector
165:
163:Rotation vector
139:right-hand rule
125:
115:
109:
95:
83:indicating the
78:
72:Euclidean space
43:
37:
31:
24:
17:
12:
11:
5:
3623:
3613:
3612:
3607:
3591:
3590:
3539:Fairlie, D. B.
3524:
3523:
3521:
3518:
3517:
3516:
3510:
3505:
3500:
3493:
3490:
3480:Euclidean norm
3457:
3453:
3450:
3447:
3443:
3438:
3433:
3429:
3426:
3423:
3420:
3417:
3414:
3382:
3377:
3369:
3367:
3364:
3361:
3360:
3357:
3354:
3351:
3343:
3341:
3331:
3328:
3322:
3319:
3314:
3307:
3306:
3304:
3299:
3296:
3294:
3291:
3287:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3266:
3264:
3261:
3260:
3210:
3203:
3200:
3194:
3191:
3187:
3183:
3177:
3174:
3168:
3165:
3161:
3157:
3153:
3129:Main article:
3126:
3123:
3094:
3090:
3087:
3084:
3081:
3078:
3073:
3070:
3065:
3061:
3057:
3053:
3049:
3046:
3026:
3023:
3019:
3015:
3011:
3007:
3004:
3001:
2998:
2995:
2991:
2987:
2983:
2979:
2976:
2973:
2970:
2967:
2962:
2957:
2952:
2949:
2946:
2943:
2940:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2887:
2882:
2877:
2874:
2871:
2868:
2864:
2860:
2857:
2854:
2851:
2767:
2761:
2756:
2752:
2746:
2741:
2736:
2732:
2729:
2725:
2720:
2717:
2714:
2711:
2708:
2705:
2700:
2696:
2667:
2662:
2658:
2654:
2648:
2643:
2637:
2632:
2628:
2625:
2622:
2619:
2616:
2613:
2604:of the log is
2602:Frobenius norm
2561:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2535:
2527:
2524:
2521:
2513:
2510:
2503:
2498:
2494:
2491:
2487:
2479:
2476:
2473:
2470:
2466:
2460:
2459:
2456:
2453:
2450:
2442:
2440:
2437:
2436:
2434:
2429:
2426:
2423:
2420:
2381:
2354:
2330:
2309:
2288:
2284:
2264:
2261:
2238:
2218:
2198:
2176:
2173:
2169:
2146:
2138:
2130:
2126:
2122:
2117:
2113:
2109:
2108:
2103:
2099:
2095:
2090:
2086:
2082:
2081:
2076:
2072:
2068:
2063:
2059:
2055:
2054:
2052:
2044:
2041:
2038:
2035:
2031:
2026:
2022:
2000:
1995:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1967:
1963:
1960:
1957:
1954:
1883:
1880:
1831:
1825:
1820:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1790:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1742:
1736:
1731:
1725:
1721:
1718:
1712:
1709:
1703:
1699:
1693:
1687:
1684:
1678:
1674:
1668:
1662:
1659:
1653:
1649:
1642:
1638:
1634:
1629:
1625:
1622:
1616:
1613:
1607:
1603:
1597:
1591:
1588:
1582:
1578:
1572:
1569:
1565:
1561:
1558:
1555:
1552:
1471:
1466:
1462:
1459:
1454:
1449:
1354:
1351:
1346:
1342:
1337:
1333:
1330:
1324:
1321:
1317:
1312:
1307:
1303:
1298:
1294:
1291:
1285:
1282:
1278:
1273:
1269:
1265:
1262:
1259:
1256:
1250:
1247:
1240:
1236:
1231:
1227:
1224:
1216:
1211:
1208:
1205:
1201:
1197:
1194:
1190:
1186:
1183:
1180:
1177:
1174:
1171:
1093:
1088:
1083:
1080:
1077:
1074:
1069:
1066:
1061:
1057:
1030:
1026:
1023:
1020:
1016:
1013:
1009:
1006:
1003:
1000:
995:
992:
987:
984:
946:
943:
920:
917:
892:
887:
883:
879:
875:
871:
867:
864:
861:
858:
855:
852:
849:
846:
843:
840:
836:
832:
828:
824:
821:
818:
815:
812:
809:
806:
802:
798:
795:
792:
789:
786:
783:
777:
774:
771:
765:
704:
701:
698:
693:
690:
675:, named after
666:Main article:
663:
660:
607:
604:
592:
587:
579:
576:
571:
570:
567:
564:
563:
560:
557:
556:
554:
506:
502:
496:
492:
489:
483:
478:
472:
469:
466:
465:
462:
459:
458:
455:
452:
451:
449:
443:
439:
435:
431:
428:
423:
415:
411:
407:
406:
401:
397:
393:
392:
387:
383:
379:
378:
376:
370:
366:
363:
359:
356:
353:
350:
347:
343:
339:
336:
333:
330:
326:
276:
273:
205:
200:
196:
193:
189:
164:
161:
15:
9:
6:
4:
3:
2:
3622:
3611:
3608:
3606:
3603:
3602:
3600:
3586:
3582:
3578:
3574:
3570:
3566:
3561:
3556:
3552:
3548:
3544:
3543:Zachos, C. K.
3540:
3536:
3529:
3525:
3514:
3511:
3509:
3506:
3504:
3501:
3499:
3496:
3495:
3489:
3486:
3481:
3474:
3455:
3448:
3445:
3424:
3421:
3418:
3415:
3412:
3404:
3399:
3375:
3365:
3362:
3355:
3352:
3349:
3339:
3329:
3326:
3320:
3317:
3302:
3297:
3295:
3281:
3278:
3275:
3272:
3269:
3267:
3262:
3249:
3241:
3236:
3232:
3228:
3222:
3208:
3201:
3198:
3192:
3189:
3181:
3175:
3172:
3166:
3163:
3159:
3155:
3142:
3138:
3132:
3122:
3115:
3109:
3092:
3085:
3082:
3079:
3071:
3068:
3063:
3055:
3047:
3044:
3024:
3021:
3013:
3005:
3002:
2996:
2993:
2985:
2974:
2971:
2968:
2965:
2960:
2950:
2947:
2944:
2941:
2938:
2929:
2925:
2905:
2902:
2899:
2896:
2893:
2885:
2875:
2872:
2869:
2866:
2858:
2855:
2852:
2849:
2841:
2838:
2834:
2828:
2824:
2821:
2814:
2811:
2807:
2803:
2792:
2782:
2754:
2750:
2739:
2734:
2730:
2727:
2718:
2712:
2709:
2706:
2698:
2694:
2665:
2656:
2646:
2641:
2623:
2617:
2614:
2603:
2598:
2594:
2589:
2585:
2580:
2574:
2551:
2548:
2545:
2542:
2536:
2533:
2525:
2522:
2519:
2508:
2496:
2492:
2489:
2485:
2477:
2474:
2471:
2468:
2464:
2454:
2451:
2448:
2438:
2432:
2427:
2424:
2421:
2418:
2406:
2401:
2399:
2379:
2367:
2366:does not work
2352:
2343:
2307:
2282:
2262:
2259:
2250:
2236:
2216:
2196:
2174:
2171:
2167:
2157:
2144:
2136:
2128:
2124:
2120:
2115:
2111:
2101:
2097:
2093:
2088:
2084:
2074:
2070:
2066:
2061:
2057:
2050:
2042:
2039:
2036:
2033:
2029:
2024:
1998:
1993:
1989:
1986:
1980:
1974:
1971:
1965:
1961:
1958:
1955:
1952:
1944:
1940:
1935:
1932:
1926:
1922:
1921:
1916:
1912:
1906:
1900:
1893:
1889:
1879:
1873:
1868:
1862:
1856:
1854:
1849:
1847:
1842:
1829:
1823:
1810:
1807:
1804:
1801:
1798:
1792:
1781:
1778:
1775:
1769:
1766:
1763:
1760:
1740:
1734:
1723:
1719:
1716:
1710:
1707:
1701:
1697:
1691:
1685:
1682:
1676:
1672:
1666:
1660:
1657:
1651:
1647:
1640:
1636:
1627:
1623:
1620:
1614:
1611:
1605:
1601:
1595:
1589:
1586:
1580:
1576:
1570:
1567:
1563:
1559:
1556:
1553:
1550:
1541:
1535:
1529:
1523:
1520:
1516:
1510:
1506:
1500:
1496:
1490:
1486:
1482:As a result,
1469:
1460:
1457:
1452:
1435:
1431:
1426:
1420:
1416:
1412:
1409:
1405:
1401:
1397:
1391:
1384:
1380:
1376:
1371:
1365:
1352:
1349:
1344:
1331:
1322:
1319:
1315:
1310:
1305:
1292:
1283:
1280:
1276:
1271:
1263:
1260:
1257:
1254:
1248:
1245:
1238:
1225:
1209:
1206:
1203:
1199:
1195:
1184:
1178:
1175:
1172:
1169:
1160:
1156:
1150:
1146:
1145:
1140:
1134:
1129:
1124:
1114:
1110:
1091:
1081:
1075:
1059:
1046:
1041:
1028:
1021:
1001:
985:
982:
974:
970:
964:
960:
956:
952:
942:
939:
933:
926:
916:
909:
903:
890:
873:
859:
856:
853:
850:
847:
841:
830:
816:
813:
810:
804:
793:
790:
787:
781:
748:
743:
738:
732:
726:
720:
699:
678:
674:
669:
659:
657:
652:
646:
644:
640:
636:
632:
628:
623:
621:
617:
613:
603:
590:
585:
577:
574:
565:
558:
552:
541:
517:
504:
500:
494:
490:
487:
481:
476:
470:
467:
460:
453:
447:
441:
437:
433:
429:
426:
421:
413:
409:
399:
395:
385:
381:
374:
368:
364:
341:
316:
311:
306:
283:
272:
270:
266:
261:
250:
235:
231:
225:
223:
219:
203:
194:
191:
174:
170:
160:
158:
153:
151:
147:
142:
140:
136:
131:
128:
123:
118:
112:
107:
103:
98:
94:
90:
86:
81:
77:
73:
70:
66:
62:
58:
49:
46:
40:
34:
28:
22:
3550:
3546:
3528:
3513:Screw theory
3503:Pseudovector
3484:
3472:
3400:
3247:
3234:
3230:
3226:
3223:
3134:
3113:
3107:
2927:
2923:
2842:
2836:
2832:
2829:
2822:
2819:
2812:
2809:
2805:
2801:
2790:
2783:
2596:
2592:
2578:
2575:
2402:
2365:
2344:
2251:
2249:-th column.
2229:-th row and
2158:
1936:
1930:
1924:
1919:
1918:
1914:
1910:
1904:
1898:
1895:
1871:
1866:
1860:
1857:
1850:
1843:
1539:
1533:
1527:
1524:
1518:
1514:
1508:
1504:
1498:
1494:
1488:
1484:
1433:
1429:
1418:
1414:
1410:
1407:
1403:
1399:
1395:
1389:
1382:
1378:
1369:
1366:
1158:
1154:
1148:
1143:
1142:
1138:
1132:
1122:
1112:
1108:
1042:
966:
937:
931:
928:
907:
904:
746:
736:
730:
724:
721:
671:
650:
647:
624:
609:
539:
518:
315:ordered pair
309:
307:) about the
303:radians (or
281:
278:
268:
264:
259:
248:
233:
229:
226:
173:Euler vector
172:
168:
166:
154:
145:
143:
132:
126:
116:
110:
96:
79:
60:
54:
47:
44:
38:
32:
3245:and vector
3141:quaternions
3037:and so let
2584:eigenvalues
1136:, that is,
935:instead of
742:unit vector
543:direction,
218:exponential
76:unit vector
57:mathematics
3599:Categories
3520:References
3405:function:
2931:, we have
627:rigid body
269:one-to-one
146:Euler axis
124:angles of
30:The angle
3560:1402.3541
3425:
3413:θ
3372:otherwise
3353:≠
3350:θ
3327:θ
3321:
3290:ω
3279:
3263:θ
3199:θ
3193:
3186:ω
3173:θ
3167:
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3056:⊗
3052:ω
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3014:⊗
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2990:ω
2986:⊗
2982:ω
2906:θ
2903:
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2873:θ
2870:
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2657:θ
2631:‖
2618:
2612:‖
2586:equal to
2552:π
2546:π
2543:−
2537:∈
2534:θ
2523:≠
2520:θ
2493:−
2478:θ
2475:
2465:θ
2449:θ
2422:
2380:ω
2353:ω
2329:ω
2308:θ
2287:ω
2283:−
2263:θ
2260:−
2209:, in the
2121:−
2094:−
2067:−
2043:θ
2040:
2021:ω
1987:−
1975:
1962:
1953:θ
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1808:
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1779:
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1720:⋯
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1698:θ
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1667:−
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1624:⋯
1621:−
1602:θ
1577:θ
1571:−
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1461:−
1353:⋯
1332:θ
1293:θ
1264:θ
1226:θ
1215:∞
1200:∑
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1179:
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1008:→
986::
874:⋅
860:θ
857:
851:−
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817:θ
814:
794:θ
791:
616:rotations
575:π
491:π
488:−
468:−
430:θ
222:logarithm
195:θ
188:θ
104:) of the
102:clockwise
91:, and an
85:direction
3585:18776942
3492:See also
3346:if
2760:‖
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2516:if
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1367:Because
635:constant
629:rotates
267:but not
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1844:by the
1126:is the
625:When a
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300:
288:
275:Example
3583:
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3468:where
3276:arccos
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3139:(unit
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2320:about
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961:, and
620:motion
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59:, the
3610:Angle
3581:S2CID
3555:arXiv
3547:SIGMA
3422:atan2
3403:atan2
740:is a
717:SO(3)
67:in a
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2682:and
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2403:The
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1896:Let
1890:and
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967:The
911:and
734:and
643:time
606:Uses
305:-90°
265:onto
220:and
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