232:
216:
1984:
31:
1715:
904:
1019:
3857:
From a physical perspective, active transformations can be characterized as transformations of physical space, while passive transformations are characterized as redundancies in the description of physical space. This plays an important role in mathematical
1425:
411:
740:
1116:
2377:
1517:
1969:
3445:
1854:
Note the equivalence between the two kinds of transformations: the coordinates of the new point in the active transformation and the new coordinates of the point in the passive transformation are the same, namely
314:
485:
1827:
2195:
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1254:
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327:
4040:
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1215:
600:
3307:
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2703:
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may consist of a translation and a linear transformation. In the following, the translation will be omitted, and the linear transformation will be represented by a 3×3 matrix
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748:
3808:
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3155:
3113:
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2133:
2111:
1508:
913:
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1245:
3472:
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2540:
1024:
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3215:
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2650:
2428:
2408:
2215:
2089:
2069:
1849:
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533:
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1217:
is left unchanged, while the coordinate system and its basis vectors are transformed in the opposite direction, that is, with the inverse transformation
2005:
1858:
1722:
3355:
1710:{\displaystyle \mathbf {v} =(v_{x},v_{y},v_{z})=v_{X}\mathbf {e} _{X}+v_{Y}\mathbf {e} _{Y}+v_{Z}\mathbf {e} _{Z}=T^{-1}(v_{X},v_{Y},v_{Z}).}
1122:. It makes sense to write the new vector in the unprimed basis (as above) only when the transformation is from the space into itself.
250:
3884:
440:
2138:
4006:
3528:
2742:
1153:
538:
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which leave points fixed but change the frame of reference or coordinate system relative to which they are described (
4060:
3950:
3257:
2569:
2031:
165:
2013:
1118:
in the original basis. Note that active transformations make sense even as a linear transformation into a different
2655:
145:. On the other hand, passive transformations may be useful in human motion analysis to observe the motion of the
2831:
1420:{\displaystyle \mathbf {e} _{X}=T^{-1}(1,0,0),\ \mathbf {e} _{Y}=T^{-1}(0,1,0),\ \mathbf {e} _{Z}=T^{-1}(0,0,1)}
2948:
2009:
2433:
2044:
The distinction between active and passive transformations can be seen mathematically by considering abstract
406:{\displaystyle R={\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{pmatrix}},}
71:
after the active transformation relative to the original coordinate system are the same as the coordinates of
3220:
2882:
4065:
3732:
3651:
3576:
3312:
3054:
2220:
3494:. This then gives a sharp distinction between active and passive transformations: active transformations
1432:
899:{\displaystyle \{\mathbf {e} '_{x}=T(1,0,0),\ \mathbf {e} '_{y}=T(0,1,0),\ \mathbf {e} '_{z}=T(0,0,1)\}}
3820:
3695:
3160:
3123:
Often one restricts to the case where the maps are invertible, so that active transformations are the
3023:
2496:
3782:
3620:
3504:
3129:
3094:
2545:
1994:
79:
2116:
2094:
1491:
1014:{\displaystyle \mathbf {v} '=v_{x}\mathbf {e} '_{x}+v_{y}\mathbf {e} '_{y}+v_{z}\mathbf {e} '_{z}}
4055:
1998:
200:
907:
181:
177:
3994:
1111:{\displaystyle \mathbf {v} =v_{x}\mathbf {e} _{x}+v_{y}\mathbf {e} _{y}+v_{z}\mathbf {e} _{z}}
3934:
3914:
3863:
2470:
1220:
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422:
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53:
8:
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2413:
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2200:
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1133:
735:{\displaystyle \mathbf {v} '=(v'_{x},v'_{y},v'_{z})=T\mathbf {v} =T(v_{x},v_{y},v_{z})}
518:
490:
231:
204:
196:
173:
95:
141:
For instance, active transformations are useful to describe successive positions of a
4002:
3946:
3498:
on bases, while the passive transformations act from the right, due to the inverse.
2372:{\textstyle (v_{i})_{1\leq i\leq n}=(v_{1},\cdots ,v_{n})\mapsto \sum _{i}v_{i}e_{i}}
99:
3091:
are isomorphic, they are not canonically isomorphic. Nevertheless a choice of basis
3975:
3889:
91:
3879:
321:
4027:
3197:
The transformations can then be understood as acting on the space of bases for
426:
3979:
60:
stays fixed, while the coordinate system rotates counterclockwise by an angle
4049:
3692:
from the left by composition, while passive transformations, identified with
3573:. The space of bases is equivalently the space of such isomorphisms, denoted
2045:
127:
24:
215:
3939:
Robots and screw theory: applications of kinematics and statics to robotics
3859:
2387:
1119:
56:
of a fixed coordinate system. In the passive transformation (right), point
20:
3932:
4019:
3495:
3440:{\textstyle \{e_{i}\}\mapsto \left\{\sum _{j}(T^{-1})_{ji}e_{j}\right\}}
157:) coordinate system which moves together with the femur, rather than a (
316:, be a vector in the plane. A rotation of the vector through an angle
142:
131:
118:
1983:
432:
135:
1719:
From this equation one sees that the new coordinates are given by
3935:"§4.4.1 The active interpretation and the active transformation"
30:
3811:
3157:
of transformations while passive transformations are the group
16:
Distinction between meanings of
Euclidean space transformations
309:{\displaystyle \mathbf {v} =(v_{1},v_{2})\in \mathbb {R} ^{2}}
150:
146:
104:
46:
480:{\displaystyle T\colon \mathbb {R} ^{3}\to \mathbb {R} ^{3}}
19:
For the concept of "passive transformation" in grammar, see
1964:{\displaystyle (v_{X},v_{Y},v_{Z})=(v'_{x},v'_{y},v'_{z}).}
3866:
are described mathematically by transition maps which act
1822:{\displaystyle (v_{X},v_{Y},v_{Z})=T(v_{x},v_{y},v_{z}).}
2190:{\displaystyle {\mathcal {B}}=\{e_{i}\}_{1\leq i\leq n}}
3566:{\displaystyle \Phi _{\mathcal {B}}:V\rightarrow K^{n}}
3501:
This observation is made more natural by viewing bases
3450:
The inverse in the passive transformation ensures the
3358:
2572:
2262:
342:
108:
meaning "being somewhere else at the same time"); and
3823:
3785:
3735:
3698:
3654:
3623:
3579:
3531:
3507:
3480:
3460:
3315:
3260:
3223:
3203:
3163:
3132:
3097:
3057:
3026:
2951:
2885:
2834:
2814:
2745:
2718:
2658:
2638:
2548:
2528:
2499:
2473:
2436:
2416:
2396:
2223:
2203:
2141:
2119:
2097:
2077:
2057:
1861:
1837:
1725:
1520:
1494:
1435:
1257:
1223:
1156:
1136:
1027:
916:
751:
608:
541:
521:
493:
443:
330:
253:
172:, whether active or passive, can be represented as a
3999:
The theory of the Moiré phenomenon: Aperiodic layers
3933:Joseph K. Davidson, Kenneth Henderson Hunt (2004).
3846:
3802:
3765:
3721:
3684:
3640:
3609:
3565:
3517:
3486:
3466:
3439:
3344:
3301:
3246:
3209:
3186:
3149:
3107:
3083:
3043:
3009:
2937:
2871:
2820:
2800:
2731:
2697:
2644:
2624:
2558:
2534:
2514:
2485:
2459:
2422:
2402:
2371:
2248:
2209:
2189:
2127:
2105:
2083:
2063:
1963:
1851:transforms the old coordinates into the new ones.
1843:
1821:
1709:
1502:
1480:
1419:
1239:
1209:
1142:
1110:
1013:
898:
734:
594:
527:
499:
479:
405:
308:
161:) coordinate system which is fixed to the floor.
4047:
2801:{\displaystyle v_{i}\mapsto T_{ij}v_{j}=:v'_{i}}
2625:{\textstyle \tau e_{i}=\sum _{j}\tau _{ji}e_{j}}
1210:{\displaystyle \mathbf {v} =(v_{x},v_{y},v_{z})}
1150:as a passive transformation, the initial vector
595:{\displaystyle \mathbf {v} =(v_{x},v_{y},v_{z})}
433:Spatial transformations in the Euclidean space R
3302:{\displaystyle \{e_{i}\}\mapsto \{\tau e_{i}\}}
130:usually refer to active transformations, while
90:which change the physical position of a set of
320:in counterclockwise direction is given by the
3912:
3118:
34:In the active transformation (left), a point
4024:Lectures on Analytic and Projective Geometry
3372:
3359:
3296:
3280:
3274:
3261:
2866:
2850:
2698:{\displaystyle v_{i}\mapsto \tau _{ij}v_{j}}
2166:
2152:
1973:
893:
752:
122:meaning "going under a different name"). By
2012:. Unsourced material may be challenged and
3966:Bargmann, Valentine (1957). "Relativity".
3928:
3926:
3617:. Active transformations, identified with
2872:{\displaystyle {\mathcal {B}}'=\{e'_{i}\}}
1510:with respect to the new coordinate system
1021:in the new basis are the same as those of
75:relative to the rotated coordinate system.
3919:. Cambridge University Press. p. 22.
3010:{\displaystyle e'_{i}=(T^{-1})_{ji}e_{j}}
2121:
2099:
2032:Learn how and when to remove this message
1125:
910:, then the coordinates of the new vector
467:
452:
296:
3965:
3885:Covariance and contravariance of vectors
2460:{\displaystyle \tau \in {\text{End}}(V)}
2430:to itself. Taking such a transformation
510:
230:
214:
29:
3992:
3923:
3247:{\displaystyle \tau \in {\text{GL}}(V)}
3115:allows construction of an isomorphism.
2938:{\displaystyle v_{i}e_{i}=v'_{i}e'_{i}}
4048:
3986:
3941:. Oxford University Press. p. 74
2051:Fix a finite-dimensional vector space
3776:This turns the space of bases into a
3766:{\displaystyle {\text{Iso}}(V,K^{n})}
3685:{\displaystyle {\text{Iso}}(V,K^{n})}
3610:{\displaystyle {\text{Iso}}(V,K^{n})}
3345:{\displaystyle T\in {\text{GL}}(n,K)}
3309:. Meanwhile a passive transformation
3084:{\displaystyle {\text{End}}({K^{n}})}
2739:. This is applied to the components:
2217:. This basis provides an isomorphism
1247:. This gives a new coordinate system
235:Translation and rotation as passive (
153:, that is, its motion relative to a (
82:can be distinguished into two types:
64:about its origin. The coordinates of
3913:Crampin, M.; Pirani, F.A.E. (1986).
2249:{\displaystyle C:K^{n}\rightarrow V}
2010:adding citations to reliable sources
1977:
437:In general a spatial transformation
3773:from the right by pre-composition.
1481:{\displaystyle (v_{X},v_{Y},v_{Z})}
13:
3538:
3533:
3510:
3100:
2838:
2551:
2144:
1130:On the other hand, when one views
219:Rotation considered as an active (
14:
4077:
4034:
413:which can be viewed either as an
195:were first introduced in 1957 by
166:three-dimensional Euclidean space
3916:Applicable Differential Geometry
3847:{\displaystyle {\text{GL}}(n,K)}
3722:{\displaystyle {\text{GL}}(n,K)}
3187:{\displaystyle {\text{GL}}(n,K)}
1982:
1636:
1611:
1586:
1522:
1496:
1370:
1315:
1260:
1158:
1098:
1073:
1048:
1029:
998:
970:
942:
919:
853:
805:
757:
680:
611:
543:
255:
3044:{\displaystyle {\text{End}}(V)}
2515:{\displaystyle v\mapsto \tau v}
3959:
3906:
3841:
3829:
3803:{\displaystyle {\text{GL}}(V)}
3797:
3791:
3760:
3741:
3716:
3704:
3679:
3660:
3641:{\displaystyle {\text{GL}}(V)}
3635:
3629:
3604:
3585:
3550:
3518:{\displaystyle {\mathcal {B}}}
3410:
3393:
3375:
3339:
3327:
3277:
3241:
3235:
3181:
3169:
3150:{\displaystyle {\text{GL}}(V)}
3144:
3138:
3108:{\displaystyle {\mathcal {B}}}
3078:
3063:
3038:
3032:
2985:
2968:
2756:
2712:is instead an endomorphism on
2669:
2559:{\displaystyle {\mathcal {B}}}
2503:
2454:
2448:
2336:
2333:
2301:
2277:
2263:
2240:
1955:
1907:
1901:
1862:
1813:
1774:
1765:
1726:
1701:
1662:
1568:
1529:
1475:
1436:
1414:
1396:
1359:
1341:
1304:
1286:
1204:
1165:
890:
872:
842:
824:
794:
776:
729:
690:
670:
622:
589:
550:
535:transforms the initial vector
462:
288:
262:
247:As an example, let the vector
1:
3900:
2879:is determined by asking that
2828:is invertible, the new basis
2566:are defined via the equation
2410:, that is, a linear map from
515:As an active transformation,
3454:transform identically under
2945:, from which the expression
2128:{\displaystyle \mathbb {C} }
2106:{\displaystyle \mathbb {R} }
1831:As a passive transformation
1503:{\displaystyle \mathbf {v} }
7:
3873:
3525:as a choice of isomorphism
3217:. An active transformation
170:proper rigid transformation
10:
4082:
3119:As left- and right-actions
2632:. Then, the components of
2542:with respect to the basis
210:
18:
4001:. Springer. p. 346.
3995:"Appendix D: Remark D.12"
3980:10.1103/RevModPhys.29.161
3968:Reviews of Modern Physics
1974:In abstract vector spaces
80:Geometric transformations
45:by rotating clockwise by
4061:Mathematical terminology
38:is transformed to point
3993:Amidror, Isaac (2007).
429:), as described below.
201:Lorentz transformations
176:, the composition of a
3848:
3804:
3767:
3723:
3686:
3642:
3611:
3567:
3519:
3488:
3468:
3441:
3346:
3303:
3248:
3211:
3188:
3151:
3109:
3085:
3045:
3011:
2939:
2873:
2822:
2802:
2733:
2710:passive transformation
2699:
2646:
2626:
2560:
2536:
2516:
2487:
2486:{\displaystyle v\in V}
2461:
2424:
2404:
2373:
2256:via the component map
2250:
2211:
2191:
2129:
2107:
2085:
2065:
1965:
1845:
1823:
1711:
1504:
1482:
1421:
1241:
1240:{\displaystyle T^{-1}}
1211:
1144:
1126:Passive transformation
1112:
1015:
900:
736:
596:
529:
501:
481:
419:passive transformation
407:
310:
244:
228:
193:passive transformation
76:
3864:gauge transformations
3849:
3805:
3768:
3724:
3687:
3643:
3612:
3568:
3520:
3489:
3469:
3467:{\displaystyle \tau }
3442:
3347:
3304:
3249:
3212:
3189:
3152:
3110:
3086:
3046:
3012:
2940:
2874:
2823:
2803:
2734:
2732:{\displaystyle K^{n}}
2700:
2647:
2627:
2561:
2537:
2535:{\displaystyle \tau }
2517:
2488:
2462:
2425:
2405:
2384:active transformation
2374:
2251:
2212:
2192:
2130:
2108:
2086:
2066:
1966:
1846:
1824:
1712:
1505:
1483:
1422:
1242:
1212:
1145:
1113:
1016:
901:
737:
597:
530:
511:Active transformation
502:
482:
415:active transformation
408:
311:
234:
218:
189:active transformation
114:alias transformations
88:alibi transformations
33:
3821:
3783:
3733:
3696:
3652:
3621:
3577:
3529:
3505:
3478:
3458:
3356:
3313:
3258:
3221:
3201:
3161:
3130:
3125:general linear group
3095:
3055:
3024:
3020:Although the spaces
2949:
2883:
2832:
2812:
2743:
2716:
2656:
2636:
2570:
2546:
2526:
2522:. The components of
2497:
2471:
2434:
2414:
2394:
2260:
2221:
2201:
2139:
2117:
2095:
2075:
2055:
2006:improve this section
1859:
1835:
1723:
1518:
1492:
1433:
1429:The new coordinates
1255:
1251:with basis vectors:
1221:
1154:
1134:
1025:
914:
749:
606:
539:
519:
491:
441:
328:
251:
180:along an axis and a
94:relative to a fixed
4066:Concepts in physics
3895:Translation of axes
2964:
2934:
2921:
2865:
2797:
1954:
1938:
1922:
1010:
982:
954:
865:
817:
769:
669:
653:
637:
138:could mean either.
3844:
3800:
3763:
3719:
3682:
3638:
3607:
3563:
3515:
3484:
3464:
3437:
3392:
3342:
3299:
3244:
3207:
3184:
3147:
3105:
3081:
3041:
3007:
2952:
2935:
2922:
2909:
2869:
2853:
2818:
2798:
2785:
2729:
2695:
2642:
2622:
2598:
2556:
2532:
2512:
2483:
2457:
2420:
2400:
2369:
2348:
2246:
2207:
2187:
2125:
2103:
2081:
2061:
1961:
1942:
1926:
1910:
1841:
1819:
1707:
1500:
1478:
1417:
1237:
1207:
1140:
1108:
1011:
996:
968:
940:
896:
851:
803:
755:
732:
657:
641:
625:
602:into a new vector
592:
525:
497:
477:
403:
394:
306:
245:
229:
205:special relativity
197:Valentine Bargmann
174:screw displacement
96:frame of reference
77:
4008:978-1-4020-5457-0
3827:
3789:
3739:
3702:
3658:
3627:
3583:
3496:act from the left
3487:{\displaystyle T}
3383:
3325:
3233:
3210:{\displaystyle V}
3167:
3136:
3061:
3030:
2821:{\displaystyle T}
2645:{\displaystyle v}
2589:
2446:
2423:{\displaystyle V}
2403:{\displaystyle V}
2339:
2210:{\displaystyle V}
2084:{\displaystyle K}
2064:{\displaystyle V}
2042:
2041:
2034:
1844:{\displaystyle T}
1367:
1312:
1143:{\displaystyle T}
850:
802:
528:{\displaystyle T}
500:{\displaystyle T}
421:(where the above
243:) transformations
184:about that axis.
100:coordinate system
4073:
4013:
4012:
3990:
3984:
3983:
3963:
3957:
3956:
3930:
3921:
3920:
3910:
3890:Rotation of axes
3853:
3851:
3850:
3845:
3828:
3825:
3809:
3807:
3806:
3801:
3790:
3787:
3772:
3770:
3769:
3764:
3759:
3758:
3740:
3737:
3728:
3726:
3725:
3720:
3703:
3700:
3691:
3689:
3688:
3683:
3678:
3677:
3659:
3656:
3647:
3645:
3644:
3639:
3628:
3625:
3616:
3614:
3613:
3608:
3603:
3602:
3584:
3581:
3572:
3570:
3569:
3564:
3562:
3561:
3543:
3542:
3541:
3524:
3522:
3521:
3516:
3514:
3513:
3493:
3491:
3490:
3485:
3473:
3471:
3470:
3465:
3446:
3444:
3443:
3438:
3436:
3432:
3431:
3430:
3421:
3420:
3408:
3407:
3391:
3371:
3370:
3352:sends the basis
3351:
3349:
3348:
3343:
3326:
3323:
3308:
3306:
3305:
3300:
3295:
3294:
3273:
3272:
3254:sends the basis
3253:
3251:
3250:
3245:
3234:
3231:
3216:
3214:
3213:
3208:
3193:
3191:
3190:
3185:
3168:
3165:
3156:
3154:
3153:
3148:
3137:
3134:
3114:
3112:
3111:
3106:
3104:
3103:
3090:
3088:
3087:
3082:
3077:
3076:
3075:
3062:
3059:
3050:
3048:
3047:
3042:
3031:
3028:
3017:can be derived.
3016:
3014:
3013:
3008:
3006:
3005:
2996:
2995:
2983:
2982:
2960:
2944:
2942:
2941:
2936:
2930:
2917:
2905:
2904:
2895:
2894:
2878:
2876:
2875:
2870:
2861:
2846:
2842:
2841:
2827:
2825:
2824:
2819:
2808:. Provided that
2807:
2805:
2804:
2799:
2793:
2781:
2780:
2771:
2770:
2755:
2754:
2738:
2736:
2735:
2730:
2728:
2727:
2704:
2702:
2701:
2696:
2694:
2693:
2684:
2683:
2668:
2667:
2651:
2649:
2648:
2643:
2631:
2629:
2628:
2623:
2621:
2620:
2611:
2610:
2597:
2585:
2584:
2565:
2563:
2562:
2557:
2555:
2554:
2541:
2539:
2538:
2533:
2521:
2519:
2518:
2513:
2492:
2490:
2489:
2484:
2466:
2464:
2463:
2458:
2447:
2444:
2429:
2427:
2426:
2421:
2409:
2407:
2406:
2401:
2378:
2376:
2375:
2370:
2368:
2367:
2358:
2357:
2347:
2332:
2331:
2313:
2312:
2297:
2296:
2275:
2274:
2255:
2253:
2252:
2247:
2239:
2238:
2216:
2214:
2213:
2208:
2196:
2194:
2193:
2188:
2186:
2185:
2164:
2163:
2148:
2147:
2134:
2132:
2131:
2126:
2124:
2112:
2110:
2109:
2104:
2102:
2090:
2088:
2087:
2082:
2070:
2068:
2067:
2062:
2037:
2030:
2026:
2023:
2017:
1986:
1978:
1970:
1968:
1967:
1962:
1950:
1934:
1918:
1900:
1899:
1887:
1886:
1874:
1873:
1850:
1848:
1847:
1842:
1828:
1826:
1825:
1820:
1812:
1811:
1799:
1798:
1786:
1785:
1764:
1763:
1751:
1750:
1738:
1737:
1716:
1714:
1713:
1708:
1700:
1699:
1687:
1686:
1674:
1673:
1661:
1660:
1645:
1644:
1639:
1633:
1632:
1620:
1619:
1614:
1608:
1607:
1595:
1594:
1589:
1583:
1582:
1567:
1566:
1554:
1553:
1541:
1540:
1525:
1509:
1507:
1506:
1501:
1499:
1487:
1485:
1484:
1479:
1474:
1473:
1461:
1460:
1448:
1447:
1426:
1424:
1423:
1418:
1395:
1394:
1379:
1378:
1373:
1365:
1340:
1339:
1324:
1323:
1318:
1310:
1285:
1284:
1269:
1268:
1263:
1246:
1244:
1243:
1238:
1236:
1235:
1216:
1214:
1213:
1208:
1203:
1202:
1190:
1189:
1177:
1176:
1161:
1149:
1147:
1146:
1141:
1117:
1115:
1114:
1109:
1107:
1106:
1101:
1095:
1094:
1082:
1081:
1076:
1070:
1069:
1057:
1056:
1051:
1045:
1044:
1032:
1020:
1018:
1017:
1012:
1006:
1001:
995:
994:
978:
973:
967:
966:
950:
945:
939:
938:
926:
922:
905:
903:
902:
897:
861:
856:
848:
813:
808:
800:
765:
760:
741:
739:
738:
733:
728:
727:
715:
714:
702:
701:
683:
665:
649:
633:
618:
614:
601:
599:
598:
593:
588:
587:
575:
574:
562:
561:
546:
534:
532:
531:
526:
506:
504:
503:
498:
486:
484:
483:
478:
476:
475:
470:
461:
460:
455:
412:
410:
409:
404:
399:
398:
315:
313:
312:
307:
305:
304:
299:
287:
286:
274:
273:
258:
227:) transformation
149:relative to the
74:
70:
69:
63:
59:
51:
44:
43:
37:
4081:
4080:
4076:
4075:
4074:
4072:
4071:
4070:
4046:
4045:
4037:
4016:
4009:
3991:
3987:
3964:
3960:
3953:
3931:
3924:
3911:
3907:
3903:
3880:Change of basis
3876:
3824:
3822:
3819:
3818:
3786:
3784:
3781:
3780:
3754:
3750:
3736:
3734:
3731:
3730:
3699:
3697:
3694:
3693:
3673:
3669:
3655:
3653:
3650:
3649:
3624:
3622:
3619:
3618:
3598:
3594:
3580:
3578:
3575:
3574:
3557:
3553:
3537:
3536:
3532:
3530:
3527:
3526:
3509:
3508:
3506:
3503:
3502:
3479:
3476:
3475:
3459:
3456:
3455:
3426:
3422:
3413:
3409:
3400:
3396:
3387:
3382:
3378:
3366:
3362:
3357:
3354:
3353:
3322:
3314:
3311:
3310:
3290:
3286:
3268:
3264:
3259:
3256:
3255:
3230:
3222:
3219:
3218:
3202:
3199:
3198:
3164:
3162:
3159:
3158:
3133:
3131:
3128:
3127:
3121:
3099:
3098:
3096:
3093:
3092:
3071:
3067:
3066:
3058:
3056:
3053:
3052:
3027:
3025:
3022:
3021:
3001:
2997:
2988:
2984:
2975:
2971:
2956:
2950:
2947:
2946:
2926:
2913:
2900:
2896:
2890:
2886:
2884:
2881:
2880:
2857:
2837:
2836:
2835:
2833:
2830:
2829:
2813:
2810:
2809:
2789:
2776:
2772:
2763:
2759:
2750:
2746:
2744:
2741:
2740:
2723:
2719:
2717:
2714:
2713:
2689:
2685:
2676:
2672:
2663:
2659:
2657:
2654:
2653:
2637:
2634:
2633:
2616:
2612:
2603:
2599:
2593:
2580:
2576:
2571:
2568:
2567:
2550:
2549:
2547:
2544:
2543:
2527:
2524:
2523:
2498:
2495:
2494:
2472:
2469:
2468:
2443:
2435:
2432:
2431:
2415:
2412:
2411:
2395:
2392:
2391:
2363:
2359:
2353:
2349:
2343:
2327:
2323:
2308:
2304:
2280:
2276:
2270:
2266:
2261:
2258:
2257:
2234:
2230:
2222:
2219:
2218:
2202:
2199:
2198:
2169:
2165:
2159:
2155:
2143:
2142:
2140:
2137:
2136:
2135:), and a basis
2120:
2118:
2115:
2114:
2098:
2096:
2093:
2092:
2091:(thought of as
2076:
2073:
2072:
2056:
2053:
2052:
2038:
2027:
2021:
2018:
2003:
1987:
1976:
1946:
1930:
1914:
1895:
1891:
1882:
1878:
1869:
1865:
1860:
1857:
1856:
1836:
1833:
1832:
1807:
1803:
1794:
1790:
1781:
1777:
1759:
1755:
1746:
1742:
1733:
1729:
1724:
1721:
1720:
1695:
1691:
1682:
1678:
1669:
1665:
1653:
1649:
1640:
1635:
1634:
1628:
1624:
1615:
1610:
1609:
1603:
1599:
1590:
1585:
1584:
1578:
1574:
1562:
1558:
1549:
1545:
1536:
1532:
1521:
1519:
1516:
1515:
1514:are given by:
1495:
1493:
1490:
1489:
1469:
1465:
1456:
1452:
1443:
1439:
1434:
1431:
1430:
1387:
1383:
1374:
1369:
1368:
1332:
1328:
1319:
1314:
1313:
1277:
1273:
1264:
1259:
1258:
1256:
1253:
1252:
1228:
1224:
1222:
1219:
1218:
1198:
1194:
1185:
1181:
1172:
1168:
1157:
1155:
1152:
1151:
1135:
1132:
1131:
1128:
1102:
1097:
1096:
1090:
1086:
1077:
1072:
1071:
1065:
1061:
1052:
1047:
1046:
1040:
1036:
1028:
1026:
1023:
1022:
1002:
997:
990:
986:
974:
969:
962:
958:
946:
941:
934:
930:
918:
917:
915:
912:
911:
857:
852:
809:
804:
761:
756:
750:
747:
746:
723:
719:
710:
706:
697:
693:
679:
661:
645:
629:
610:
609:
607:
604:
603:
583:
579:
570:
566:
557:
553:
542:
540:
537:
536:
520:
517:
516:
513:
492:
489:
488:
471:
466:
465:
456:
451:
450:
442:
439:
438:
435:
393:
392:
381:
369:
368:
354:
338:
337:
329:
326:
325:
322:rotation matrix
300:
295:
294:
282:
278:
269:
265:
254:
252:
249:
248:
213:
199:for describing
72:
67:
65:
61:
57:
49:
41:
39:
35:
28:
17:
12:
11:
5:
4079:
4069:
4068:
4063:
4058:
4056:Systems theory
4044:
4043:
4036:
4035:External links
4033:
4032:
4031:
4028:Addison-Wesley
4015:
4014:
4007:
3985:
3974:(2): 161–174.
3958:
3951:
3922:
3904:
3902:
3899:
3898:
3897:
3892:
3887:
3882:
3875:
3872:
3868:from the right
3843:
3840:
3837:
3834:
3831:
3799:
3796:
3793:
3762:
3757:
3753:
3749:
3746:
3743:
3718:
3715:
3712:
3709:
3706:
3681:
3676:
3672:
3668:
3665:
3662:
3637:
3634:
3631:
3606:
3601:
3597:
3593:
3590:
3587:
3560:
3556:
3552:
3549:
3546:
3540:
3535:
3512:
3483:
3463:
3435:
3429:
3425:
3419:
3416:
3412:
3406:
3403:
3399:
3395:
3390:
3386:
3381:
3377:
3374:
3369:
3365:
3361:
3341:
3338:
3335:
3332:
3329:
3321:
3318:
3298:
3293:
3289:
3285:
3282:
3279:
3276:
3271:
3267:
3263:
3243:
3240:
3237:
3229:
3226:
3206:
3183:
3180:
3177:
3174:
3171:
3146:
3143:
3140:
3120:
3117:
3102:
3080:
3074:
3070:
3065:
3040:
3037:
3034:
3004:
3000:
2994:
2991:
2987:
2981:
2978:
2974:
2970:
2967:
2963:
2959:
2955:
2933:
2929:
2925:
2920:
2916:
2912:
2908:
2903:
2899:
2893:
2889:
2868:
2864:
2860:
2856:
2852:
2849:
2845:
2840:
2817:
2796:
2792:
2788:
2784:
2779:
2775:
2769:
2766:
2762:
2758:
2753:
2749:
2726:
2722:
2692:
2688:
2682:
2679:
2675:
2671:
2666:
2662:
2641:
2619:
2615:
2609:
2606:
2602:
2596:
2592:
2588:
2583:
2579:
2575:
2553:
2531:
2511:
2508:
2505:
2502:
2493:transforms as
2482:
2479:
2476:
2456:
2453:
2450:
2442:
2439:
2419:
2399:
2366:
2362:
2356:
2352:
2346:
2342:
2338:
2335:
2330:
2326:
2322:
2319:
2316:
2311:
2307:
2303:
2300:
2295:
2292:
2289:
2286:
2283:
2279:
2273:
2269:
2265:
2245:
2242:
2237:
2233:
2229:
2226:
2206:
2184:
2181:
2178:
2175:
2172:
2168:
2162:
2158:
2154:
2151:
2146:
2123:
2101:
2080:
2060:
2040:
2039:
2022:September 2023
1990:
1988:
1981:
1975:
1972:
1960:
1957:
1953:
1949:
1945:
1941:
1937:
1933:
1929:
1925:
1921:
1917:
1913:
1909:
1906:
1903:
1898:
1894:
1890:
1885:
1881:
1877:
1872:
1868:
1864:
1840:
1818:
1815:
1810:
1806:
1802:
1797:
1793:
1789:
1784:
1780:
1776:
1773:
1770:
1767:
1762:
1758:
1754:
1749:
1745:
1741:
1736:
1732:
1728:
1706:
1703:
1698:
1694:
1690:
1685:
1681:
1677:
1672:
1668:
1664:
1659:
1656:
1652:
1648:
1643:
1638:
1631:
1627:
1623:
1618:
1613:
1606:
1602:
1598:
1593:
1588:
1581:
1577:
1573:
1570:
1565:
1561:
1557:
1552:
1548:
1544:
1539:
1535:
1531:
1528:
1524:
1498:
1477:
1472:
1468:
1464:
1459:
1455:
1451:
1446:
1442:
1438:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1393:
1390:
1386:
1382:
1377:
1372:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1338:
1335:
1331:
1327:
1322:
1317:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1283:
1280:
1276:
1272:
1267:
1262:
1234:
1231:
1227:
1206:
1201:
1197:
1193:
1188:
1184:
1180:
1175:
1171:
1167:
1164:
1160:
1139:
1127:
1124:
1105:
1100:
1093:
1089:
1085:
1080:
1075:
1068:
1064:
1060:
1055:
1050:
1043:
1039:
1035:
1031:
1009:
1005:
1000:
993:
989:
985:
981:
977:
972:
965:
961:
957:
953:
949:
944:
937:
933:
929:
925:
921:
895:
892:
889:
886:
883:
880:
877:
874:
871:
868:
864:
860:
855:
847:
844:
841:
838:
835:
832:
829:
826:
823:
820:
816:
812:
807:
799:
796:
793:
790:
787:
784:
781:
778:
775:
772:
768:
764:
759:
754:
731:
726:
722:
718:
713:
709:
705:
700:
696:
692:
689:
686:
682:
678:
675:
672:
668:
664:
660:
656:
652:
648:
644:
640:
636:
632:
628:
624:
621:
617:
613:
591:
586:
582:
578:
573:
569:
565:
560:
556:
552:
549:
545:
524:
512:
509:
496:
474:
469:
464:
459:
454:
449:
446:
434:
431:
402:
397:
391:
388:
385:
382:
380:
377:
374:
371:
370:
367:
364:
361:
358:
355:
353:
350:
347:
344:
343:
341:
336:
333:
303:
298:
293:
290:
285:
281:
277:
272:
268:
264:
261:
257:
223:) or passive (
212:
209:
128:mathematicians
124:transformation
15:
9:
6:
4:
3:
2:
4078:
4067:
4064:
4062:
4059:
4057:
4054:
4053:
4051:
4042:
4039:
4038:
4029:
4025:
4021:
4018:
4017:
4010:
4004:
4000:
3996:
3989:
3981:
3977:
3973:
3969:
3962:
3954:
3952:0-19-856245-4
3948:
3944:
3940:
3936:
3929:
3927:
3918:
3917:
3909:
3905:
3896:
3893:
3891:
3888:
3886:
3883:
3881:
3878:
3877:
3871:
3869:
3865:
3861:
3855:
3838:
3835:
3832:
3817:
3813:
3794:
3779:
3774:
3755:
3751:
3747:
3744:
3713:
3710:
3707:
3674:
3670:
3666:
3663:
3632:
3599:
3595:
3591:
3588:
3558:
3554:
3547:
3544:
3499:
3497:
3481:
3461:
3453:
3448:
3433:
3427:
3423:
3417:
3414:
3404:
3401:
3397:
3388:
3384:
3379:
3367:
3363:
3336:
3333:
3330:
3319:
3316:
3291:
3287:
3283:
3269:
3265:
3238:
3227:
3224:
3204:
3195:
3178:
3175:
3172:
3141:
3126:
3116:
3072:
3068:
3035:
3018:
3002:
2998:
2992:
2989:
2979:
2976:
2972:
2965:
2961:
2957:
2953:
2931:
2927:
2923:
2918:
2914:
2910:
2906:
2901:
2897:
2891:
2887:
2862:
2858:
2854:
2847:
2843:
2815:
2794:
2790:
2786:
2782:
2777:
2773:
2767:
2764:
2760:
2751:
2747:
2724:
2720:
2711:
2706:
2690:
2686:
2680:
2677:
2673:
2664:
2660:
2652:transform as
2639:
2617:
2613:
2607:
2604:
2600:
2594:
2590:
2586:
2581:
2577:
2573:
2529:
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2071:over a field
2058:
2049:
2047:
2046:vector spaces
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2015:
2011:
2007:
2001:
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1996:
1991:This section
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745:If one views
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55:
48:
32:
26:
25:passive voice
22:
4041:UI ambiguity
4023:
3998:
3988:
3971:
3967:
3961:
3942:
3938:
3915:
3908:
3867:
3860:gauge theory
3856:
3815:
3777:
3775:
3500:
3451:
3449:
3196:
3122:
3019:
2709:
2707:
2388:endomorphism
2383:
2381:
2050:
2043:
2028:
2019:
2004:Please help
1992:
1853:
1830:
1718:
1511:
1428:
1248:
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1120:vector space
744:
514:
436:
418:
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87:
83:
78:
21:active voice
4026:, page 84,
4020:Dirk Struik
3870:on fibers.
2467:, a vector
2386:is then an
178:translation
4050:Categories
3901:References
3452:components
187:The terms
143:rigid body
132:physicists
52:about the
3854:-torsor.
3648:, act on
3551:→
3534:Φ
3462:τ
3402:−
3385:∑
3376:↦
3320:∈
3284:τ
3278:↦
3228:∈
3225:τ
2977:−
2757:↦
2674:τ
2670:↦
2601:τ
2591:∑
2574:τ
2530:τ
2507:τ
2504:↦
2478:∈
2441:∈
2438:τ
2341:∑
2337:↦
2318:⋯
2291:≤
2285:≤
2241:→
2180:≤
2174:≤
1993:does not
1655:−
1389:−
1334:−
1279:−
1230:−
906:as a new
463:→
448::
390:θ
387:
379:θ
376:
366:θ
363:
357:−
352:θ
349:
292:∈
136:engineers
3874:See also
3862:, where
3729:acts on
2962:′
2932:′
2919:′
2863:′
2844:′
2795:′
1952:′
1936:′
1920:′
1008:′
980:′
952:′
924:′
863:′
815:′
767:′
667:′
651:′
635:′
616:′
427:inverted
425:will be
182:rotation
4022:(1953)
2014:removed
1999:sources
211:Example
110:passive
4005:
3949:
3814:and a
3812:torsor
1366:
1311:
849:
801:
423:matrix
168:, any
159:global
92:points
84:active
54:origin
3816:right
908:basis
417:or a
241:alibi
237:alias
225:alias
221:alibi
155:local
151:femur
147:tibia
119:alias
105:alibi
47:angle
4003:ISBN
3947:ISBN
3778:left
3474:and
3051:and
1997:any
1995:cite
191:and
134:and
23:and
3976:doi
3738:Iso
3657:Iso
3582:Iso
3447:.
3060:End
3029:End
2445:End
2390:on
2382:An
2197:of
2113:or
2048:.
2008:by
1512:XYZ
1488:of
1249:XYZ
384:cos
373:sin
360:sin
346:cos
203:in
164:In
112:or
98:or
86:or
4052::
3997:.
3972:29
3970:.
3945:.
3943:ff
3937:.
3925:^
3826:GL
3788:GL
3701:GL
3626:GL
3324:GL
3232:GL
3194:.
3166:GL
3135:GL
2783:=:
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2705:.
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126:,
4030:.
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