1461:
762:
3137:
987:
302:
1456:{\displaystyle {\begin{aligned}R_{z}(\phi )=\mathrm {Roll} _{1}(\phi )&={\begin{bmatrix}\cos \phi &-\sin \phi &0\\\sin \phi &\cos \phi &0\\0&0&1\end{bmatrix}}\\R_{y}(\theta )=\mathrm {Pitch} (\theta )&={\begin{bmatrix}\cos \theta &0&\sin \theta \\0&1&0\\-\sin \theta &0&\cos \theta \end{bmatrix}}\\R_{z}(\psi )=\mathrm {Roll} _{2}(\psi )&={\begin{bmatrix}\cos \psi &-\sin \psi &0\\\sin \psi &\cos \psi &0\\0&0&1\end{bmatrix}}.\end{aligned}}}
2643:
757:{\displaystyle {\begin{aligned}\\R_{x}(\phi )=\mathrm {Roll} (\phi )&={\begin{bmatrix}1&0&0\\0&\cos \phi &-\sin \phi \\0&\sin \phi &\cos \phi \end{bmatrix}}\\R_{y}(\theta )=\mathrm {Pitch} (\theta )&={\begin{bmatrix}\cos \theta &0&\sin \theta \\0&1&0\\-\sin \theta &0&\cos \theta \end{bmatrix}}\\R_{z}(\psi )=\mathrm {Yaw} (\psi )&={\begin{bmatrix}\cos \psi &-\sin \psi &0\\\sin \psi &\cos \psi &0\\0&0&1\end{bmatrix}}.\end{aligned}}}
256:
3132:{\displaystyle {\begin{aligned}Y'&={}^{0}\!R_{2\rightarrow 1}\\&={}^{0}\!R_{1\rightarrow 0}{}^{1}\!R_{2\rightarrow 1}{}^{0}\!R_{1\rightarrow 0}^{-1}\\&=XYX^{-1}\\Z''&={}^{0}\!R_{3\rightarrow 2}\\&={}^{0}\!R_{1\rightarrow 0}{}^{1}\!R_{3\rightarrow 2}{}^{0}\!R_{1\rightarrow 0}^{-1}\\&=X\left({}^{1}\!R_{2\rightarrow 1}{}^{2}\!R_{3\rightarrow 2}{}^{1}\!R_{2\rightarrow 1}^{-1}\right)X^{-1}\\&=XYZY^{-1}X^{-1}\end{aligned}}}
110:
959:
183:
3435:
158:
2220:
2196:
1507:
1483:
944:
970:
Anyway, Euler rotations can still be used when speaking about a vehicle, though they will have a weird behavior. As the vertical axis is the origin for the angles, it is named "inclination" instead of "elevation". As before, describing the attitude of a vehicle, there is an axis considered pointing
165:
A set of
Davenport rotations is said to be complete if it is enough to generate any rotation of the space by composition. Speaking in matrix terms, it is complete if it can generate any orthonormal matrix of the space, whose determinant is +1. Due to the non-commutativity of the matrix product, the
169:
Sometimes the order is imposed by the geometry of the underlying problem. For example, when used for vehicles, which have a special axis pointing to the "forward" direction, only one of the six possible combinations of rotations is useful. The interesting composition is the one able to control the
966:
Euler rotations appear as the special case in which the first and third axes of rotations are overlapping. These Euler rotations are related to the proper Euler angles, which were thought to study the movement of a rigid body such as a planet. The angle to define the direction of the roll axis is
949:
Of the six possible combinations of yaw, pitch and roll, this combination is the only one in which the heading (direction of the roll axis) is equal to one of the rotations (the yaw), and the elevation (angle of the roll axis with the horizontal plane) is equal to other of the rotations (to the
3148:
173:
In the adjacent drawing, the yaw, pitch and roll (YPR) composition allows adjustment of the direction of an aircraft with the two first angles. A different composition like YRP would allow establishing the direction of the wings axis, which is obviously not useful in most cases.
47:
and Tait–Bryan rotations are particular cases of the
Davenport general rotation decomposition. The angles of rotation are called Davenport angles because the general problem of decomposing a rotation in a sequence of three was studied first by Paul B. Davenport.
2624:
773:
1518:
Davenport rotations are usually studied as an intrinsic rotation composition, because of the importance of the axes fixed to a moving body, but they can be converted to an extrinsic rotation composition, in case it could be more intuitive.
148:
Therefore, decompositions in Euler chained rotations and Tait–Bryan chained rotations are particular cases of this. The Tait–Bryan case appears when axes 1 and 3 are perpendicular, and the Euler case appears when they are overlapping.
2648:
128:
The general problem consists of obtaining the matrix decomposition of a rotation given the three known axes. In some cases one of the axes is repeated. This problem is equivalent to a decomposition problem of matrices.
3430:{\displaystyle {\begin{aligned}R&=Z''Y'X\\&=\left(XYZY^{-1}X^{-1}\right)\left(XYX^{-1}\right)X\\&=XYZY^{-1}\left(X^{-1}X\right)Y\left(X^{-1}X\right)\\&=XYZ\left(Y^{-1}Y\right)\\&=XYZ\end{aligned}}}
2434:
Since a rotation matrix can be represented among these three frames, let's use the left shoulder index to denote the representation frame. The following notation means the rotation matrix that transforms the frame
144:
According to the
Davenport theorem, a unique decomposition is possible if and only if the second axis is perpendicular to the other two axes. Therefore, axes 1 and 3 must be in the plane orthogonal to axis 2.
132:
Davenport proved that any orientation can be achieved by composing three elemental rotations using non-orthogonal axes. The elemental rotations can either occur about the axes of the fixed coordinate system
1891:, but this notation may be ambiguous as it may be identical to that used for extrinsic rotations. In this case, it becomes necessary to separately specify whether the rotations are intrinsic or extrinsic.
1879:. Sets of rotation axes associated with both proper Euler angles and Tait-Bryan angles are commonly named using this notation (see above for details). Sometimes, the same sequence is simply called
974:
The combination depends on how the axis are taken and what the initial position of the plane is. Using the one in the drawing, and combining rotations in such a way that an axis is repeated, only
3153:
992:
778:
307:
2323:
2170:
1957:
1614:
3498:
2494:
2505:
2414:
2231:
Any extrinsic rotation is equivalent to an intrinsic rotation by the same angles but with inverted order of elemental rotations, and vice versa. For instance, the intrinsic rotations
1522:
Any extrinsic rotation is equivalent to an intrinsic rotation by the same angles but with inverted order of elemental rotations, and vice versa. For instance, the intrinsic rotations
2419:
In this process there are three frames related in the intrinsic rotation sequence. Let's denote the frame 0 as the initial frame, the frame 1 after the first rotation around the
2499:
An intrinsic element rotation matrix represented in that frame where the rotation happens has the same value as that of the corresponding extrinsic element rotation matrix:
939:{\displaystyle {\begin{aligned}M&=\mathrm {Yaw} (\psi )\,\mathrm {Pitch} (\theta )\,\mathrm {Roll} (\phi )\\&=R_{z}(\psi )R_{y}(\theta )R_{x}(\phi ).\end{aligned}}}
1704:
1646:
1675:
2104:(or 3-1-3). Sets of rotation axes associated with both proper Euler angles and Tait–Bryan angles are commonly named using this notation (see above for details).
2346:
1724:
137:) or about the axes of a rotating coordinate system, which is initially aligned with the fixed one and modifies its orientation after each elemental rotation (
101:
Most of the cases belong to the second group, given that the generalized Euler rotations are a degenerated case in which first and third axes are overlapping.
240:
in the beginning, after performing intrinsic rotations Y, P and R in the yaw, pitch and roll axes (in this order) we obtain something similar to image 4
293:. In these conditions, the Heading (angle on the horizontal plane) will be equal to the yaw applied, and the Elevation will be equal to the pitch.
3440:
Therefore, the rotation matrix of an intrinsic element rotation sequence is the same as that of the inverse extrinsic element rotation sequence:
967:
normally named "longitude of the revolution axis" or "longitude of the line of nodes" instead of "heading", which makes no sense for a planet.
3547:
M. Shuster and L. Markley, Generalization of Euler angles, Journal of the
Astronautical Sciences, Vol. 51, No. 2, April–June 2003, pp. 123–123
59:
coordinate system. Given that rotation axes are solidary with the moving body, the generalized rotations can be divided into two groups (here
3574:
3579:
55:
rotating coordinate system may be imagined to be rigidly attached to a rigid body. In this case, it is sometimes called a
2269:
2116:
1903:
1560:
2619:{\displaystyle {}^{0}\!R_{1\rightarrow 0}=X,\quad {}^{1}\!R_{2\rightarrow 1}=Y,\quad {}^{2}\!R_{3\rightarrow 2}=Z.}
2018:) are the amplitudes of these elemental rotations. For instance, the target orientation can be reached as follows:
1793:) are the amplitudes of these elemental rotations. For instance, the target orientation can be reached as follows:
194:
Tait–Bryan rotations are a special case in which the first and third axes are perpendicular among them. Assuming a
2453:
221:
187:
121:
into three composed movements about intrinsic axes was studied by P. Davenport, under the name "generalized
3589:
1757:
Intrinsic rotations are elemental rotations that occur about the axes of the rotating coordinate system
3509:
1982:
Extrinsic rotations are elemental rotations that occur about the axes of the fixed coordinate system
1863:
allows us to summarize this as follows: the three elemental rotations of the XYZ-system occur about
3557:
3446:
1706:
are the elementary rotation matrices of the corresponding angles. The product of these matrices,
195:
3556:
J. Wittenburg, L. Lilov, Decomposition of a finite rotation in three rotations about given axes
1680:
1622:
3533:
2374:
1651:
255:
971:
forward, and therefore only one out of the possible combinations of rotations will be useful.
244:
228:
2002:, a composition of three extrinsic rotations can be used to reach any target orientation for
1777:, a composition of three intrinsic rotations can be used to reach any target orientation for
1470:
imposes the "heading", Pitch is the "inclination" (complementary of the elevation), and Roll
214:
3514:
2368:
can be obtained by the sequential intrinsic element rotations from the right to the left:
2353:
2184:
2107:
1971:
1894:
1747:
1731:
259:
Image 4: Heading, elevation and bank angles after yaw, pitch and roll rotations (Z-Y’-X’’)
3584:
2331:
1709:
109:
125:", but later these angles were named "Davenport angles" by M. Shuster and L. Markley.
3568:
2349:
2180:
1967:
1727:
52:
958:
17:
122:
44:
182:
978:
will allow controlling the longitude and the inclination with one rotation each.
32:
2223:
Image 9: The same rotation represented by (γ, β, α) = (45°, 30°, −60°), using
1735:
1510:
Image 7: The same rotation represented by (γ, β, α) = (45°, 30°, −60°), using
170:
heading and the elevation of the aircraft with one independent rotation each.
40:
2110:
can be used to represent a sequence of extrinsic rotations. For instance,
1897:
can be used to represent a sequence of intrinsic rotations. For instance,
296:
Matrix expressions for the three Tait–Bryan rotations in 3 dimensions are:
118:
157:
113:
Image 1: Davenport possible axes for steps 1 and 3 given Z as the step 2
28:
962:
Image 5:Starting position of an aircraft to apply proper Euler angles
1761:, which changes its orientation after each elemental rotation. The
2219:
2195:
1506:
1482:
957:
254:
181:
156:
108:
3543:
3541:
3142:
The two equations above are substituted to the first equation:
2637:
represented in the frame 0 can be expressed as other forms:
2175:
represents a composition of extrinsic rotations about axes
1962:
represents a composition of intrinsic rotations about axes
2364:
The rotation matrix of the intrinsic rotation sequence
1361:
1208:
1059:
662:
519:
370:
3449:
3151:
2646:
2508:
2456:
2377:
2334:
2272:
2119:
1906:
1712:
1683:
1654:
1625:
1563:
990:
776:
305:
2360:
The proof of the conversion in the pre-multiply case
2191:
Conversion between intrinsic and extrinsic rotations
245:
229:
3534:
P. B. Davenport, Rotations about nonorthogonal axes
2427:axis, and the frame 3 as the third rotation around
2423:, the frame 2 after the second rotation around the
3492:
3429:
3131:
2618:
2488:
2408:
2354:ambiguities in the definition of rotation matrices
2352:. This is standard practice, but take note of the
2340:
2317:
2185:ambiguities in the definition of rotation matrices
2183:. This is standard practice, but take note of the
2164:
2088:In sum, the three elemental rotations occur about
1972:ambiguities in the definition of rotation matrices
1970:. This is standard practice, but take note of the
1951:
1732:ambiguities in the definition of rotation matrices
1718:
1698:
1669:
1640:
1608:
1455:
938:
756:
3037:
3011:
2985:
2933:
2907:
2881:
2845:
2763:
2737:
2711:
2675:
2590:
2554:
2518:
2466:
2199:Image 8: A rotation represented by Euler angles (
1486:Image 6: A rotation represented by Euler angles (
2318:{\displaystyle R=X(\alpha )Y(\beta )Z(\gamma )}
2165:{\displaystyle R=Z(\gamma )Y(\beta )X(\alpha )}
1952:{\displaystyle R=X(\alpha )Y(\beta )Z(\gamma )}
1609:{\displaystyle R=X(\alpha )Y(\beta )Z(\gamma )}
1836:axis is now in its final orientation, and the
8:
2489:{\displaystyle {}^{c}\!R_{a\rightarrow b}.}
2065:axis is now at angle β with respect to the
1860:
71:refer to the non-orthogonal moving frame):
2247:are equivalent to the extrinsic rotations
1875:″. Indeed, this sequence is often denoted
1847:system rotates a third time about the new
1726:, should then be pre-multiplied against a
1538:are equivalent to the extrinsic rotations
43:rotations about body-fixed specific axes.
3448:
3387:
3342:
3313:
3295:
3255:
3226:
3213:
3152:
3150:
3116:
3103:
3071:
3053:
3042:
3031:
3029:
3016:
3005:
3003:
2990:
2979:
2977:
2949:
2938:
2927:
2925:
2912:
2901:
2899:
2886:
2875:
2873:
2850:
2839:
2837:
2808:
2779:
2768:
2757:
2755:
2742:
2731:
2729:
2716:
2705:
2703:
2680:
2669:
2667:
2647:
2645:
2595:
2584:
2582:
2559:
2548:
2546:
2523:
2512:
2510:
2507:
2471:
2460:
2458:
2455:
2376:
2333:
2271:
2118:
2100:. Indeed, this sequence is often denoted
1905:
1711:
1682:
1653:
1624:
1562:
1356:
1334:
1320:
1301:
1203:
1170:
1152:
1054:
1032:
1018:
999:
991:
989:
914:
895:
876:
839:
838:
812:
811:
791:
777:
775:
657:
630:
612:
514:
481:
463:
365:
335:
317:
306:
304:
2218:
2194:
1505:
1481:
767:The matrix of the composed rotations is
138:
134:
93:x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z
81:z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y
3526:
2629:The intrinsic element rotation matrix
2076:system rotates a third time about the
2443:and that is represented in the frame
1824:system rotates about the now rotated
117:The general problem of decomposing a
7:
981:The three matrices to multiply are:
161:Image 2: Airplane resting on a plane
2263:. Both are represented by a matrix
1817:axis now lies on the line of nodes.
1554:. Both are represented by a matrix
289:The rotations are applied in order
2006:. The Euler or Tait-Bryan angles (
1840:axis remains on the line of nodes.
1781:. The Euler or Tait-Bryan angles (
1742:Relationship with physical motions
1330:
1327:
1324:
1321:
1183:
1180:
1177:
1174:
1171:
1028:
1025:
1022:
1019:
849:
846:
843:
840:
825:
822:
819:
816:
813:
798:
795:
792:
637:
634:
631:
494:
491:
488:
485:
482:
345:
342:
339:
336:
25:
1734:, since some definitions may use
1478:Conversion to extrinsic rotations
166:rotation system must be ordered.
274:the plane pitch axis is on axis
236:, lying horizontal on the plane
219:⟨yaw, pitch, roll⟩
217:as in image 2, and a plane with
87:Generalized Tait–Bryan rotations
2580:
2544:
2053:system rotates again about the
1809:axis (which coincides with the
267:the plane roll axis is on axis
3046:
3020:
2994:
2942:
2916:
2890:
2854:
2772:
2746:
2720:
2684:
2599:
2563:
2527:
2475:
2312:
2306:
2300:
2294:
2288:
2282:
2159:
2153:
2147:
2141:
2135:
2129:
1946:
1940:
1934:
1928:
1922:
1916:
1693:
1687:
1664:
1658:
1635:
1629:
1603:
1597:
1591:
1585:
1579:
1573:
1346:
1340:
1313:
1307:
1193:
1187:
1164:
1158:
1044:
1038:
1011:
1005:
926:
920:
907:
901:
888:
882:
859:
853:
835:
829:
808:
802:
647:
641:
624:
618:
504:
498:
475:
469:
355:
349:
329:
323:
281:the plane yaw axis is on axis
1:
3493:{\displaystyle R=Z''Y'X=XYZ.}
3575:Rotation in three dimensions
2211:) = (−60°, 30°, 45°), using
1498:) = (−60°, 30°, 45°), using
178:Tait–Bryan chained rotations
153:Complete system of rotations
75:Generalized Euler rotations
37:Davenport chained rotations
3606:
2179:, if used to pre-multiply
1966:, if used to pre-multiply
1745:
1699:{\displaystyle Z(\gamma )}
1641:{\displaystyle X(\alpha )}
105:Davenport rotation theorem
2409:{\displaystyle R=Z''Y'X.}
2026:system rotates about the
1670:{\displaystyle Y(\beta )}
2348:is used to pre-multiply
1994:is fixed. Starting with
1769:is fixed. Starting with
263:In the beginning :
1466:In this convention Roll
954:Euler chained rotations
3494:
3431:
3133:
2620:
2490:
2410:
2342:
2319:
2228:
2216:
2166:
1990:system rotates, while
1953:
1765:system rotates, while
1720:
1700:
1671:
1642:
1610:
1515:
1503:
1457:
963:
940:
758:
285:of the reference frame
278:of the reference frame
271:of the reference frame
260:
191:
162:
114:
3495:
3432:
3134:
2621:
2491:
2411:
2343:
2320:
2222:
2198:
2167:
2038:axis is now at angle
1954:
1721:
1701:
1672:
1643:
1611:
1509:
1485:
1458:
961:
941:
759:
258:
185:
160:
112:
3580:Euclidean symmetries
3510:Matrix decomposition
3447:
3149:
2644:
2506:
2454:
2375:
2332:
2270:
2117:
2042:with respect to the
1904:
1859:The above-mentioned
1710:
1681:
1652:
1623:
1561:
1474:imposes the "tilt".
988:
774:
303:
238:⟨x, y⟩
18:Tait-Bryan rotations
3061:
2957:
2787:
2227:extrinsic rotations
2215:intrinsic rotations
1978:Extrinsic rotations
1801:system rotates by
1753:Intrinsic rotations
1730:. Take note of the
1514:extrinsic rotations
1502:intrinsic rotations
291:yaw, pitch and roll
224:like in the image 3
139:intrinsic rotations
135:extrinsic rotations
3490:
3427:
3425:
3129:
3127:
3038:
2934:
2764:
2616:
2486:
2406:
2338:
2315:
2229:
2217:
2162:
1949:
1716:
1696:
1667:
1638:
1606:
1516:
1504:
1453:
1451:
1440:
1287:
1138:
964:
936:
934:
754:
752:
741:
598:
449:
261:
192:
163:
115:
39:are three chained
2341:{\displaystyle R}
2108:Rotation matrices
1895:Rotation matrices
1719:{\displaystyle R}
16:(Redirected from
3597:
3559:
3554:
3548:
3545:
3536:
3531:
3499:
3497:
3496:
3491:
3471:
3463:
3436:
3434:
3433:
3428:
3426:
3407:
3403:
3399:
3395:
3394:
3362:
3358:
3354:
3350:
3349:
3329:
3325:
3321:
3320:
3303:
3302:
3275:
3268:
3264:
3263:
3262:
3239:
3235:
3234:
3233:
3221:
3220:
3188:
3181:
3173:
3138:
3136:
3135:
3130:
3128:
3124:
3123:
3111:
3110:
3083:
3079:
3078:
3066:
3062:
3060:
3052:
3036:
3035:
3030:
3027:
3026:
3010:
3009:
3004:
3001:
3000:
2984:
2983:
2978:
2961:
2956:
2948:
2932:
2931:
2926:
2923:
2922:
2906:
2905:
2900:
2897:
2896:
2880:
2879:
2874:
2865:
2861:
2860:
2844:
2843:
2838:
2828:
2816:
2815:
2791:
2786:
2778:
2762:
2761:
2756:
2753:
2752:
2736:
2735:
2730:
2727:
2726:
2710:
2709:
2704:
2695:
2691:
2690:
2674:
2673:
2668:
2658:
2625:
2623:
2622:
2617:
2606:
2605:
2589:
2588:
2583:
2570:
2569:
2553:
2552:
2547:
2534:
2533:
2517:
2516:
2511:
2495:
2493:
2492:
2487:
2482:
2481:
2465:
2464:
2459:
2415:
2413:
2412:
2407:
2399:
2391:
2347:
2345:
2344:
2339:
2324:
2322:
2321:
2316:
2171:
2169:
2168:
2163:
1958:
1956:
1955:
1950:
1748:Givens rotations
1725:
1723:
1722:
1717:
1705:
1703:
1702:
1697:
1676:
1674:
1673:
1668:
1647:
1645:
1644:
1639:
1615:
1613:
1612:
1607:
1462:
1460:
1459:
1454:
1452:
1445:
1444:
1339:
1338:
1333:
1306:
1305:
1292:
1291:
1186:
1157:
1156:
1143:
1142:
1037:
1036:
1031:
1004:
1003:
945:
943:
942:
937:
935:
919:
918:
900:
899:
881:
880:
865:
852:
828:
801:
763:
761:
760:
755:
753:
746:
745:
640:
617:
616:
603:
602:
497:
468:
467:
454:
453:
348:
322:
321:
309:
251:
247:
239:
235:
231:
220:
212:
96:
84:
21:
3605:
3604:
3600:
3599:
3598:
3596:
3595:
3594:
3565:
3564:
3563:
3562:
3555:
3551:
3546:
3539:
3532:
3528:
3523:
3515:Givens rotation
3506:
3464:
3456:
3445:
3444:
3424:
3423:
3405:
3404:
3383:
3382:
3378:
3360:
3359:
3338:
3337:
3333:
3309:
3308:
3304:
3291:
3273:
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1980:
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1103:
1091:
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1085:
1071:
1055:
1047:
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995:
986:
985:
976:roll–pitch–roll
956:
933:
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891:
872:
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784:
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241:
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198:
196:reference frame
180:
155:
107:
90:
78:
45:Euler rotations
23:
22:
15:
12:
11:
5:
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316:
312:
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308:
287:
286:
279:
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190:of an aircraft
188:principal axes
179:
176:
154:
151:
106:
103:
99:
98:
88:
85:
76:
24:
14:
13:
10:
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2439:to the frame
2438:
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2013:
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186:Image 3: The
184:
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3590:Aerodynamics
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2327:
2260:
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2015:
2011:
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1991:
1987:
1983:
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1963:
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1872:
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246:data missing
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33:engineering
3569:Categories
3521:References
2251:by angles
2235:by angles
1805:about the
1746:See also:
1542:by angles
1526:by angles
215:convention
53:orthogonal
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1371:ψ
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