140:
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43:
634:
2194:
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403:
1078:
1410:
2270:, are in some ways the least intuitive representation of three-dimensional rotations. They are not the three-dimensional instance of a general approach. They are more compact than matrices and easier to work with than all other methods, so are often preferred in real-world applications.
1994:
1813:
1219:
938:
1686:{\displaystyle {\begin{aligned}e^{i\theta }z&=(\cos \theta +i\sin \theta )(x+iy)\\&=x\cos \theta +iy\cos \theta +ix\sin \theta -y\sin \theta \\&=(x\cos \theta -y\sin \theta )+i(x\sin \theta +y\cos \theta )\\&=x'+iy',\end{aligned}}}
1983:
2804:
If a rotation of
Minkowski space is in a space-like plane, then this rotation is the same as a spatial rotation in Euclidean space. By contrast, a rotation in a plane spanned by a space-like dimension and a time-like dimension is a
705:
for details. Instead the rotation has two mutually orthogonal planes of rotation, each of which is fixed in the sense that points in each plane stay within the planes. The rotation has two angles of rotation, one for each
686:
2189:{\displaystyle \mathbf {A} {\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}}{\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}x'\\y'\\z'\end{pmatrix}}}
1318:
2693:
2336:
1707:
1415:
1113:
1265:
2420:
1702:
1108:
2438:, is itself a rotation, but in four dimensions. Any four-dimensional rotation about the origin can be represented with two quaternion multiplications: one left and one right, by two
257:
about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed. These two types of rotation are called
2281:
of the quaternion is 1. This constraint limits the degrees of freedom of the quaternion to three, as required. Unlike matrices and complex numbers two multiplications are needed:
1073:{\displaystyle {\begin{bmatrix}x'\\y'\end{bmatrix}}={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}}
2633:
593:, the second rotates around the line of nodes and the third is an intrinsic rotation (a spin) around an axis fixed in the body that moves. Euler angles are typically denoted as
2472:. Rotations represented in other ways are often converted to matrices before being used. They can be extended to represent rotations and transformations at the same time using
3124:
3082:
2589:
3041:
3003:
1900:
406:
A plane rotation around a point followed by another rotation around a different point results in a total motion which is either a rotation (as in this picture), or a
1385:
644:(pictured at the right) specifies an angle with the axis about which the rotation takes place. It can be easily visualised. There are two variants to represent it:
554:, so the order in which rotations are applied is important even about the same point. Also, unlike the two-dimensional case, a three-dimensional direct motion, in
2468:
Matrices are often used for doing transformations, especially when a large number of points are being transformed, as they are a direct representation of the
3396:
3196:
899:
186:): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions:
702:
566:
for details), the same as the number of dimensions. A three-dimensional rotation can be specified in a number of ways. The most usual methods are:
1835:
563:
60:
859:, a (proper) rotation is different from an arbitrary fixed-point motion in its preservation of the orientation of the vector space. Thus, the
577:
of three rotations defined as the motion obtained by changing one of the Euler angles while leaving the other two constant. They constitute a
356:
of rotations is a particular formalism, either algebraic or geometric, used to parametrize a rotation map. This meaning is somehow inverse to
3785:
3758:
3343:
3801:
3405:
1841:
3525:
2809:, and if this plane contains the time axis of the reference frame, is called a "Lorentz boost". These transformations demonstrate the
905:
2450:
More generally, coordinate rotations in any dimension are represented by orthogonal matrices. The set of all orthogonal matrices in
746:. Rotations in four dimensions about a fixed point have six degrees of freedom. A four-dimensional direct motion in general position
3318:
2503:
The main disadvantage of matrices is that they are more expensive to calculate and do calculations with. Also in calculations where
258:
126:
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under the rotation. Unlike the axis, its points are not fixed themselves. The axis (where present) and the plane of a rotation are
107:
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2255:
79:
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2454:
dimensions which describe proper rotations (determinant = +1), together with the operation of matrix multiplication, forms the
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3449:
3139:
1823:
64:
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641:
86:
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that visualize (1 + 1)-dimensional pseudo-Euclidean geometry on planar drawings. The study of relativity is deals with the
3822:
3773:
3763:
3653:
3472:
3211:
1270:
773:
454:. Any direct Euclidean motion can be represented as a composition of a rotation about the fixed point and a translation.
3827:
3768:
3633:
3573:
3367:
2368:
2287:
1227:
1808:{\displaystyle {\begin{aligned}x'&=x\cos \theta -y\sin \theta \\y'&=x\sin \theta +y\cos \theta .\end{aligned}}}
1214:{\displaystyle {\begin{aligned}x'&=x\cos \theta -y\sin \theta \\y'&=x\sin \theta +y\cos \theta .\end{aligned}}}
2377:
93:
863:
of a rotation orthogonal matrix must be 1. The only other possibility for the determinant of an orthogonal matrix is
2246:
Another possibility to represent a rotation of three-dimensional
Euclidean vectors are quaternions described below.
3837:
3561:
3382:
3335:
800:. Alternatively, the vector description of rotations can be understood as a parametrization of geometric rotations
550:
differ from those in two dimensions in a number of important ways. Rotations in three dimensions are generally not
427:
341:
295:
278:
585:, rather than a single frame that is purely external or purely intrinsic. Specifically, the first angle moves the
53:
3832:
3357:"A review of useful theorems involving proper orthogonal matrices referenced to three-dimensional physical space"
2477:
777:
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points, in general, do not commute. Any two-dimensional direct motion is either a translation or a rotation; see
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between any two points unchanged after the transformation. But a (proper) rotation also has to preserve the
211:
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3730:
3673:
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3221:
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matrices. They are not rotation matrices, but a transformation that represents a
Euclidean rotation has a
2354:
793:
698:
407:
187:
3623:
3602:
3546:
3048:
2598:
2200:
832:
458:
3047:; and its subgroup representing proper rotations (those that preserve the orientation of space) is the
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transformations of the celestial sphere. It is a broader class of the sphere transformations known as
2560:
1978:{\displaystyle \mathbf {A} ={\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}}}
1822:, vector rotations in two dimensions are commutative, unlike in higher dimensions. They have only one
3506:
3492:
3436:
3015:
2977:
2204:
805:
769:
574:
498:
474:
357:
290:
226:
3005:, which represent rotations in complex space. The set of all unitary matrices in a given dimension
3638:
3566:
3454:
3216:
2932:
2913:
2806:
2764:
2635:. It can be conveniently described in terms of a Clifford algebra. Unit quaternions give the group
2520:
2504:
2278:
466:
222:
167:
253:
results in the body being at the same coordinates. For example, in two dimensions rotating a body
100:
3700:
3688:
3535:
3530:
3482:
2952:
2798:
2789:, spanned by three space dimensions and one of time. In special relativity, this space is called
2782:
2704:
2237:) as are its columns, making it simple to spot and check if a matrix is a valid rotation matrix.
785:
510:
451:
423:
250:
179:
1401:
249:), because for any motion of a body there is an inverse transformation which if applied to the
3750:
3643:
3628:
3444:
3339:
3314:
2925:
2820:
2715:
2543:
2367:. The quaternion can be related to the rotation vector form of the axis angle rotation by the
2234:
2222:
1847:
883:
845:
819:
on vectors that preserves the same geometric structure but expressed in terms of vectors. For
781:
707:
431:
332:
300:
282:
246:
218:
163:
2797:, have a physical interpretation. These transformations preserve a quadratic form called the
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1819:
1355:
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171:
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Euler angles and axis–angle representations can be easily converted to a rotation matrix.
923:
816:
763:
514:
447:
415:
393:
317:
203:
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and equating real and imaginary parts gives the same result as a two-dimensional matrix:
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3298:
3135:
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2535:
2508:
1334:
828:
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their composition with translations. In other words, one vector rotation presents many
139:
3149:
2873:
3816:
3710:
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3514:
3010:
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for more physical aspects. Euclidean rotations and, more generally, Lorentz symmetry
2847:
2824:
849:
586:
542:
Euler rotations of the Earth. Intrinsic (green), precession (blue) and nutation (red)
522:
434:" term refers to isometries that reverse (flip) the orientation. In the language of
3715:
3678:
3663:
3658:
3578:
3519:
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to be rotated counterclockwise is written as a column vector, then multiplied by a
789:
663:
624:
570:
482:
435:
368:
364:
17:
3266:
Weisstein, Eric W. "Alias
Transformation." From MathWorld--A Wolfram Web Resource.
3255:
Weisstein, Eric W. "Alibi
Transformation." From MathWorld--A Wolfram Web Resource.
3683:
3551:
3487:
3363:
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2226:
860:
751:
648:
573:(pictured at the left). Any rotation about the origin can be represented as the
551:
155:
42:
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nature of the
Minkowski space. Hyperbolic rotations are sometimes described as
2229:
1. That it is an orthogonal matrix means that its rows are a set of orthogonal
701:
has only one fixed point, the centre of rotation, and no axis of rotation; see
2592:
2469:
2364:
2261:
876:
633:
559:
371:
are not always clearly distinguished. The former are sometimes referred to as
345:
175:
3306:
2786:
776:, important in pure mathematics, can be erased because there is a canonical
690:
506:
274:
254:
234:
2507:
is a concern matrices can be more prone to it, so calculations to restore
3541:
2944:
2909:
2708:
2539:
1988:
This is multiplied by a vector representing the point to give the result
629:. This presentation is convenient only for rotations about a fixed point.
419:
159:
31:
2553:
In the case of a positive-definite
Euclidean quadratic form, the double
538:
3590:
2936:
672:
238:
2511:, which are expensive to do for matrices, need to be done more often.
1826:, as such rotations are entirely determined by the angle of rotation.
581:
system because angles are measured with respect to a mix of different
402:
3720:
3477:
3236:
3201:
3089:
2839:
2785:, where it can be considered to operate on a four-dimensional space,
2534:
In general (even for vectors equipped with a non-Euclidean
Minkowski
2266:
910:
In two dimensions, to carry out a rotation using a matrix, the point
3374:
2928:
is an invariance with respect to all rotation about the fixed axis.
658:
obtained by multiplying the angle with this unit vector, called the
2199:
The set of all appropriate matrices together with the operation of
3740:
3426:
801:
728:
then all points not in the planes rotate through an angle between
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rotations, in physics and astronomy, correspond to rotations of
768:
When one considers motions of the
Euclidean space that preserve
600:
562:. Rotations about the origin have three degrees of freedom (see
3378:
2971:-valued matrices analogous to real orthogonal matrices are the
3144:
2868:
612:
36:
1852:
As in two dimensions, a matrix can be used to rotate a point
676:
2827:
generated by the space rotations and hyperbolic rotations.
27:
Motion of a certain space that preserves at least one point
3088:. These complex rotations are important in the context of
710:, through which points in the planes rotate. If these are
375:(although the term is misleading), whereas the latter are
517:, which implies that all two-dimensional rotations about
2931:
As was stated above, Euclidean rotations are applied to
2738:, most of these motions do not have fixed points on the
2688:{\displaystyle \mathrm {Spin} (3)\cong \mathrm {SU} (2)}
754:
Euclidean dimensions), but screw operations exist also.
3161:
2884:
2842:
in the
Euclidean 3-space, Lorentz transformations from
2538:) the rotation of a vector space can be expressed as a
143:
Rotation of an object in two dimensions around a point
2143:
2107:
2044:
2008:
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have the same magnitude and are separated by an angle
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1111:
941:
2277:) consists of four real numbers, constrained so the
1313:{\displaystyle {\begin{bmatrix}x'\\y'\end{bmatrix}}}
815:
A motion that preserves the origin is the same as a
689:
A perspective projection onto three-dimensions of a
3794:
3749:
3616:
3505:
3463:
3435:
3412:
67:. Unsourced material may be challenged and removed.
3118:
3076:
3035:
2997:
2757:geometries are not different from Euclidean ones.
2742:-sphere and, strictly speaking, are not rotations
2687:
2627:
2583:
2414:
2331:{\displaystyle \mathbf {x'} =\mathbf {qxq} ^{-1},}
2330:
2188:
1977:
1807:
1685:
1379:
1349:in the plane is represented by the complex number
1312:
1260:{\displaystyle {\begin{bmatrix}x\\y\end{bmatrix}}}
1259:
1213:
1072:
693:being rotated in four-dimensional Euclidean space.
394:Euclidean space § Rotations and reflections
3134:), as well as respective transformations of the
2753:. Rotations about a fixed point in elliptic and
2431:is the rotation vector treated as a quaternion.
2415:{\displaystyle \mathbf {q} =e^{\mathbf {v} /2},}
2363:is the vector treated as a quaternion with zero
1084:The coordinates of the point after rotation are
867:, and this result means the transformation is a
174:. It can describe, for example, the motion of a
2726:-dimensional Euclidean space about the origin (
299:in a broader group of (orientation-preserving)
2935:. Moreover, most of mathematical formalism in
489:). The rotation is acting to rotate an object
3390:
2793:, and the four-dimensional rotations, called
2519:As was demonstrated above, there exist three
784:. The same is true for geometries other than
241:, this concept is frequently understood as a
8:
2746:; such motions are sometimes referred to as
886:. Matrices of all proper rotations form the
3197:Rotations and reflections in two dimensions
900:Rotations and reflections in two dimensions
671:Matrices, versors (quaternions), and other
221:. All rotations about a fixed point form a
3460:
3397:
3383:
3375:
2781:A generalization of a rotation applies in
2527:, for two dimensions, and two others with
2500:rotation matrix in the upper left corner.
856:
750:a rotation about certain point (as in all
703:rotations in 4-dimensional Euclidean space
473:is needed to specify a rotation about the
178:around a fixed point. Rotation can have a
3099:
3097:
3057:
3055:
3019:
3017:
2981:
2979:
2959:is not a precise symmetry law of nature.
2668:
2642:
2640:
2602:
2600:
2564:
2562:
2515:More alternatives to the matrix formalism
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2394:
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2381:
2379:
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2305:
2291:
2289:
2138:
2102:
2039:
2003:
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1996:
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1112:
1110:
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981:
942:
940:
127:Learn how and when to remove this message
2767:have not a distinct notion of rotation.
758:Linear and multilinear algebra formalism
678:Linear and Multilinear Algebra Formalism
647:as a pair consisting of the angle and a
320:of its fixed points. They exist only in
30:For broader coverage of this topic, see
3303:New Foundations for Classical Mechanics
3247:
1836:Rotation formalisms in three dimensions
564:rotation formalisms in three dimensions
3130:-dimensional Euclidean rotations (see
2908:Rotations define important classes of
774:distinction between points and vectors
662:(although, strictly speaking, it is a
505:for details. Composition of rotations
2434:A single multiplication by a versor,
2213:is a member of the three-dimensional
1390:This can be rotated through an angle
379:. See the article below for details.
245:(importantly, a transformation of an
7:
3802:List of computer graphics algorithms
2524:
65:adding citations to reliable sources
1842:Three-dimensional rotation operator
1400:, then expanding the product using
502:
289:and is usually identified with the
265:Related definitions and terminology
3103:
3100:
3061:
3058:
3020:
2982:
2948:
2672:
2669:
2652:
2649:
2646:
2643:
2628:{\displaystyle \mathrm {Spin} (n)}
2612:
2609:
2606:
2603:
2568:
2565:
906:Rotation of axes in two dimensions
835:, such operator is expressed with
797:
259:active and passive transformations
190:, which have no fixed points, and
25:
2531:, for three and four dimensions.
2240:
481:that specifies an element of the
3148:
3131:
3119:{\displaystyle \mathrm {SU} (2)}
3077:{\displaystyle \mathrm {SU} (n)}
2872:
2584:{\displaystyle \mathrm {SO} (n)}
2528:
2395:
2382:
2312:
2309:
2306:
2293:
2256:Quaternions and spatial rotation
1999:
1905:
450:, where the former comprise the
438:the distinction is expressed as
233:(of a particular space). But in
217:Mathematically, a rotation is a
194:, each of them having an entire
41:
3036:{\displaystyle \mathrm {U} (n)}
2998:{\displaystyle \mathrm {U} (n)}
2718:) is the same as a rotation of
1333:plane can be also presented as
383:Definitions and representations
358:the meaning in the group theory
281:. This (common) fixed point or
52:needs additional citations for
3140:representation theory of SU(2)
3113:
3107:
3071:
3065:
3030:
3024:
2992:
2986:
2682:
2676:
2662:
2656:
2622:
2616:
2578:
2572:
2550:representation of Lie groups.
2523:rotation formalisms: one with
1641:
1611:
1602:
1572:
1486:
1471:
1468:
1441:
1:
3759:3D computer graphics software
3332:Clifford algebras and spinors
3212:Infinitesimal rotation matrix
2943:) is rotation-invariant; see
1818:Since complex numbers form a
3574:Hidden-surface determination
3368:Sandia National Laboratories
3355:Brannon, Rebecca M. (2002).
2546:and, more generally, in the
2542:. This formalism is used in
170:that preserves at least one
158:is a concept originating in
3285:Hestenes 1999, pp. 580–588.
3227:Rodrigues' rotation formula
2699:In non-Euclidean geometries
823:, this expression is their
306:For a particular rotation:
76:"Rotation" mathematics
3854:
3336:Cambridge University Press
3311:Kluwer Academic Publishers
2861:
2774:
2478:Projective transformations
2253:
1845:
1839:
1833:
926:calculated from the angle
903:
897:
761:
558:, is not a rotation but a
391:
293:. The rotation group is a
29:
3330:Lounesto, Pertti (2001).
2819:and frequently appear on
2707:, a direct motion of the
778:one-to-one correspondence
642:Axis–angle representation
589:around the external axis
365:(affine) spaces of points
243:coordinate transformation
3126:are used to parametrize
2525:U(1), or complex numbers
2456:special orthogonal group
2273:A versor (also called a
2215:special orthogonal group
888:special orthogonal group
788:, but whose space is an
675:things: see the section
531:Euclidean plane isometry
398:Special orthogonal group
237:and, more generally, in
192:(hyperplane) reflections
3786:Vector graphics editors
3781:Raster graphics editors
3187:Aircraft principal axes
2953:symmetry laws of nature
2795:Lorentz transformations
2529:versors, or quaternions
2474:homogeneous coordinates
1884:. The matrix used is a
1094:, and the formulae for
548:three-dimensional space
3669:Checkerboard rendering
3222:Orientation (geometry)
3120:
3078:
3037:
2999:
2881:This section is empty.
2852:Möbius transformations
2777:Lorentz transformation
2689:
2629:
2585:
2557:of the isometry group
2416:
2371:over the quaternions,
2332:
2190:
1979:
1809:
1687:
1394:by multiplying it by
1381:
1380:{\displaystyle z=x+iy}
1314:
1261:
1215:
1074:
882:), or another kind of
848:that is multiplied to
697:A general rotation in
694:
637:
579:mixed axes of rotation
543:
411:
148:
3624:Affine transformation
3603:Surface triangulation
3547:Anisotropic filtering
3276:Lounesto 2001, p. 30.
3121:
3079:
3049:special unitary group
3038:
3000:
2957:reflectional symmetry
2690:
2630:
2586:
2505:numerical instability
2417:
2333:
2201:matrix multiplication
2191:
1980:
1846:Further information:
1810:
1688:
1382:
1315:
1262:
1216:
1075:
869:hyperplane reflection
812:points in the space.
792:with a supplementary
688:
636:
541:
459:one-dimensional space
428:orientation structure
405:
392:Further information:
388:In Euclidean geometry
277:of rotations about a
206:of fixed points in a
142:
3823:Euclidean symmetries
3096:
3054:
3016:
2978:
2639:
2599:
2561:
2436:either left or right
2378:
2288:
2205:rotation group SO(3)
1995:
1901:
1703:
1411:
1356:
1271:
1228:
1109:
939:
162:. Any rotation is a
61:improve this article
3828:Rotational symmetry
3639:Collision detection
3567:Global illumination
3217:Irrational rotation
2955:. In contrast, the
2933:rigid body dynamics
2922:particular rotation
2914:rotational symmetry
2807:hyperbolic rotation
2765:projective geometry
2714:(an example of the
2521:multilinear algebra
2480:are represented by
2275:rotation quaternion
2221:, that is it is an
780:between points and
418:is the same as its
18:Coordinate rotation
3689:Scanline rendering
3483:Parallax scrolling
3473:Isometric graphics
3160:. You can help by
3116:
3092:. The elements of
3074:
3033:
2995:
2951:are thought to be
2920:with respect to a
2858:Discrete rotations
2821:Minkowski diagrams
2799:spacetime interval
2783:special relativity
2705:spherical geometry
2685:
2625:
2581:
2442:unit quaternions.
2412:
2328:
2186:
2180:
2129:
2096:
2030:
1975:
1969:
1805:
1803:
1683:
1681:
1377:
1310:
1304:
1257:
1251:
1211:
1209:
1070:
1064:
1038:
972:
857:was already stated
695:
638:
544:
525:. Rotations about
452:identity component
446:isometries in the
412:
367:and of respective
287:center of rotation
251:frame of reference
149:
3838:Unitary operators
3810:
3809:
3751:Graphics software
3644:Planar projection
3629:Back-face culling
3501:
3500:
3445:Alpha compositing
3406:Computer graphics
3345:978-0-521-00551-7
3178:
3177:
2926:circular symmetry
2901:
2900:
2716:elliptic geometry
2544:geometric algebra
2235:orthonormal basis
2223:orthogonal matrix
1848:3D rotation group
1824:degree of freedom
884:improper rotation
846:orthogonal matrix
821:Euclidean vectors
708:plane of rotation
493:through an angle
479:angle of rotation
461:, there are only
432:improper rotation
333:plane of rotation
247:orthonormal basis
137:
136:
129:
111:
16:(Redirected from
3845:
3833:Linear operators
3736:Volume rendering
3608:Wire-frame model
3461:
3399:
3392:
3385:
3376:
3371:
3361:
3349:
3324:
3286:
3283:
3277:
3274:
3268:
3263:
3257:
3252:
3232:Rotation of axes
3173:
3170:
3152:
3145:
3125:
3123:
3122:
3117:
3106:
3087:
3083:
3081:
3080:
3075:
3064:
3046:
3042:
3040:
3039:
3034:
3023:
3008:
3004:
3002:
3001:
2996:
2985:
2973:unitary matrices
2896:
2893:
2883:You can help by
2876:
2869:
2845:
2836:celestial sphere
2833:
2816:squeeze mappings
2811:pseudo-Euclidean
2741:
2737:
2733:
2725:
2711:
2694:
2692:
2691:
2686:
2675:
2655:
2634:
2632:
2631:
2626:
2615:
2591:is known as the
2590:
2588:
2587:
2582:
2571:
2548:Clifford algebra
2499:
2498:
2495:
2489:
2488:
2485:
2464:
2453:
2430:
2421:
2419:
2418:
2413:
2408:
2407:
2403:
2398:
2385:
2362:
2352:
2346:
2337:
2335:
2334:
2329:
2324:
2323:
2315:
2300:
2299:
2233:(so they are an
2220:
2212:
2195:
2193:
2192:
2187:
2185:
2184:
2177:
2165:
2153:
2134:
2133:
2101:
2100:
2035:
2034:
2002:
1984:
1982:
1981:
1976:
1974:
1973:
1908:
1893:
1892:
1889:
1883:
1867:
1830:Three dimensions
1820:commutative ring
1814:
1812:
1811:
1806:
1804:
1763:
1717:
1692:
1690:
1689:
1684:
1682:
1675:
1661:
1647:
1565:
1492:
1430:
1429:
1399:
1393:
1386:
1384:
1383:
1378:
1348:
1332:
1323:
1319:
1317:
1316:
1311:
1309:
1308:
1301:
1289:
1266:
1264:
1263:
1258:
1256:
1255:
1220:
1218:
1217:
1212:
1210:
1169:
1123:
1101:
1097:
1093:
1079:
1077:
1076:
1071:
1069:
1068:
1043:
1042:
977:
976:
969:
957:
931:
921:
881:
873:point reflection
844:
808:rotations about
798:an example below
782:position vectors
745:
736:
727:
718:
656:Euclidean vector
651:for the axis, or
583:reference frames
556:general position
496:
491:counterclockwise
488:
469:, only a single
377:vector rotations
373:affine rotations
326:
313:axis of rotation
296:point stabilizer
209:
201:
184:sign of an angle
146:
132:
125:
121:
118:
112:
110:
69:
45:
37:
21:
3853:
3852:
3848:
3847:
3846:
3844:
3843:
3842:
3813:
3812:
3811:
3806:
3790:
3745:
3612:
3557:Fluid animation
3497:
3459:
3431:
3422:Diffusion curve
3414:Vector graphics
3408:
3403:
3359:
3354:
3346:
3329:
3321:
3299:Hestenes, David
3297:
3294:
3289:
3284:
3280:
3275:
3271:
3264:
3260:
3253:
3249:
3245:
3207:Squeeze mapping
3192:Charts on SO(3)
3183:
3174:
3168:
3165:
3158:needs expansion
3094:
3093:
3085:
3052:
3051:
3044:
3014:
3013:
3006:
2976:
2975:
2965:
2963:Generalizations
2949:described above
2941:vector calculus
2906:
2897:
2891:
2888:
2867:
2860:
2843:
2831:
2791:Minkowski space
2779:
2773:
2761:Affine geometry
2739:
2735:
2727:
2719:
2709:
2701:
2637:
2636:
2597:
2596:
2559:
2558:
2517:
2496:
2493:
2491:
2486:
2483:
2481:
2470:linear operator
2458:
2451:
2448:
2426:
2389:
2376:
2375:
2369:exponential map
2358:
2348:
2347:is the versor,
2342:
2304:
2292:
2286:
2285:
2258:
2252:
2241:Above-mentioned
2218:
2208:
2179:
2178:
2170:
2167:
2166:
2158:
2155:
2154:
2146:
2139:
2128:
2127:
2121:
2120:
2114:
2113:
2103:
2095:
2094:
2089:
2084:
2078:
2077:
2072:
2067:
2061:
2060:
2055:
2050:
2040:
2029:
2028:
2022:
2021:
2015:
2014:
2004:
1993:
1992:
1968:
1967:
1962:
1957:
1951:
1950:
1945:
1940:
1934:
1933:
1928:
1923:
1913:
1899:
1898:
1890:
1887:
1885:
1869:
1853:
1850:
1844:
1838:
1832:
1802:
1801:
1764:
1756:
1753:
1752:
1718:
1710:
1701:
1700:
1680:
1679:
1668:
1654:
1645:
1644:
1563:
1562:
1490:
1489:
1434:
1418:
1409:
1408:
1402:Euler's formula
1395:
1391:
1354:
1353:
1338:
1335:complex numbers
1328:
1321:
1303:
1302:
1294:
1291:
1290:
1282:
1275:
1269:
1268:
1250:
1249:
1243:
1242:
1232:
1226:
1225:
1208:
1207:
1170:
1162:
1159:
1158:
1124:
1116:
1107:
1106:
1099:
1095:
1085:
1063:
1062:
1056:
1055:
1045:
1037:
1036:
1025:
1013:
1012:
998:
982:
971:
970:
962:
959:
958:
950:
943:
937:
936:
927:
924:rotation matrix
911:
908:
902:
896:
879:
836:
817:linear operator
766:
764:Rotation matrix
760:
744:
738:
735:
729:
726:
720:
717:
711:
699:four dimensions
660:rotation vector
560:screw operation
494:
486:
485:(also known as
448:Euclidean group
416:Euclidean space
400:
390:
385:
321:
267:
207:
195:
144:
133:
122:
116:
113:
70:
68:
58:
46:
35:
28:
23:
22:
15:
12:
11:
5:
3851:
3849:
3841:
3840:
3835:
3830:
3825:
3815:
3814:
3808:
3807:
3805:
3804:
3798:
3796:
3792:
3791:
3789:
3788:
3783:
3778:
3777:
3776:
3771:
3766:
3755:
3753:
3747:
3746:
3744:
3743:
3738:
3733:
3728:
3723:
3718:
3713:
3708:
3706:Shadow mapping
3703:
3698:
3693:
3692:
3691:
3686:
3681:
3676:
3671:
3666:
3661:
3651:
3646:
3641:
3636:
3631:
3626:
3620:
3618:
3614:
3613:
3611:
3610:
3605:
3600:
3599:
3598:
3588:
3581:
3576:
3571:
3570:
3569:
3559:
3554:
3549:
3544:
3539:
3533:
3528:
3522:
3517:
3511:
3509:
3503:
3502:
3499:
3498:
3496:
3495:
3490:
3485:
3480:
3475:
3469:
3467:
3458:
3457:
3452:
3447:
3441:
3439:
3433:
3432:
3430:
3429:
3424:
3418:
3416:
3410:
3409:
3404:
3402:
3401:
3394:
3387:
3379:
3373:
3372:
3351:
3350:
3344:
3326:
3325:
3319:
3293:
3290:
3288:
3287:
3278:
3269:
3258:
3246:
3244:
3241:
3240:
3239:
3234:
3229:
3224:
3219:
3214:
3209:
3204:
3199:
3194:
3189:
3182:
3179:
3176:
3175:
3155:
3153:
3115:
3112:
3109:
3105:
3102:
3073:
3070:
3067:
3063:
3060:
3032:
3029:
3026:
3022:
2994:
2991:
2988:
2984:
2964:
2961:
2905:
2902:
2899:
2898:
2879:
2877:
2859:
2856:
2775:Main article:
2772:
2769:
2700:
2697:
2684:
2681:
2678:
2674:
2671:
2667:
2664:
2661:
2658:
2654:
2651:
2648:
2645:
2624:
2621:
2618:
2614:
2611:
2608:
2605:
2580:
2577:
2574:
2570:
2567:
2555:covering group
2536:quadratic form
2516:
2513:
2509:orthonormality
2447:
2444:
2423:
2422:
2411:
2406:
2402:
2397:
2392:
2388:
2384:
2339:
2338:
2327:
2322:
2319:
2314:
2311:
2308:
2303:
2298:
2295:
2254:Main article:
2251:
2248:
2197:
2196:
2183:
2176:
2173:
2169:
2168:
2164:
2161:
2157:
2156:
2152:
2149:
2145:
2144:
2142:
2137:
2132:
2126:
2123:
2122:
2119:
2116:
2115:
2112:
2109:
2108:
2106:
2099:
2093:
2090:
2088:
2085:
2083:
2080:
2079:
2076:
2073:
2071:
2068:
2066:
2063:
2062:
2059:
2056:
2054:
2051:
2049:
2046:
2045:
2043:
2038:
2033:
2027:
2024:
2023:
2020:
2017:
2016:
2013:
2010:
2009:
2007:
2001:
1986:
1985:
1972:
1966:
1963:
1961:
1958:
1956:
1953:
1952:
1949:
1946:
1944:
1941:
1939:
1936:
1935:
1932:
1929:
1927:
1924:
1922:
1919:
1918:
1916:
1911:
1907:
1834:Main article:
1831:
1828:
1816:
1815:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1765:
1762:
1759:
1755:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1719:
1716:
1713:
1709:
1708:
1694:
1693:
1678:
1674:
1671:
1667:
1664:
1660:
1657:
1653:
1650:
1648:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1566:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1493:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1435:
1433:
1428:
1425:
1421:
1417:
1416:
1388:
1387:
1376:
1373:
1370:
1367:
1364:
1361:
1327:Points on the
1307:
1300:
1297:
1293:
1292:
1288:
1285:
1281:
1280:
1278:
1254:
1248:
1245:
1244:
1241:
1238:
1237:
1235:
1222:
1221:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1171:
1168:
1165:
1161:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1125:
1122:
1119:
1115:
1114:
1082:
1081:
1067:
1061:
1058:
1057:
1054:
1051:
1050:
1048:
1041:
1035:
1032:
1029:
1026:
1024:
1021:
1018:
1015:
1014:
1011:
1008:
1005:
1002:
999:
997:
994:
991:
988:
987:
985:
980:
975:
968:
965:
961:
960:
956:
953:
949:
948:
946:
898:Main article:
895:
894:Two dimensions
892:
850:column vectors
829:Euclidean norm
762:Main article:
759:
756:
742:
733:
724:
715:
683:
682:
669:
668:
667:
652:
631:
630:
467:two dimensions
465:rotations. In
414:A motion of a
389:
386:
384:
381:
354:representation
350:
349:
328:
285:is called the
271:rotation group
266:
263:
231:rotation group
135:
134:
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3850:
3839:
3836:
3834:
3831:
3829:
3826:
3824:
3821:
3820:
3818:
3803:
3800:
3799:
3797:
3793:
3787:
3784:
3782:
3779:
3775:
3772:
3770:
3767:
3765:
3762:
3761:
3760:
3757:
3756:
3754:
3752:
3748:
3742:
3739:
3737:
3734:
3732:
3729:
3727:
3724:
3722:
3719:
3717:
3714:
3712:
3711:Shadow volume
3709:
3707:
3704:
3702:
3699:
3697:
3694:
3690:
3687:
3685:
3682:
3680:
3677:
3675:
3672:
3670:
3667:
3665:
3662:
3660:
3657:
3656:
3655:
3652:
3650:
3647:
3645:
3642:
3640:
3637:
3635:
3632:
3630:
3627:
3625:
3622:
3621:
3619:
3615:
3609:
3606:
3604:
3601:
3597:
3594:
3593:
3592:
3589:
3586:
3585:Triangle mesh
3582:
3580:
3577:
3575:
3572:
3568:
3565:
3564:
3563:
3560:
3558:
3555:
3553:
3550:
3548:
3545:
3543:
3540:
3537:
3534:
3532:
3529:
3527:
3523:
3521:
3518:
3516:
3515:3D projection
3513:
3512:
3510:
3508:
3504:
3494:
3491:
3489:
3486:
3484:
3481:
3479:
3476:
3474:
3471:
3470:
3468:
3466:
3462:
3456:
3455:Text-to-image
3453:
3451:
3448:
3446:
3443:
3442:
3440:
3438:
3434:
3428:
3425:
3423:
3420:
3419:
3417:
3415:
3411:
3407:
3400:
3395:
3393:
3388:
3386:
3381:
3380:
3377:
3369:
3365:
3358:
3353:
3352:
3347:
3341:
3337:
3334:. Cambridge:
3333:
3328:
3327:
3322:
3320:0-7923-5514-8
3316:
3312:
3308:
3304:
3300:
3296:
3295:
3291:
3282:
3279:
3273:
3270:
3267:
3262:
3259:
3256:
3251:
3248:
3242:
3238:
3235:
3233:
3230:
3228:
3225:
3223:
3220:
3218:
3215:
3213:
3210:
3208:
3205:
3203:
3200:
3198:
3195:
3193:
3190:
3188:
3185:
3184:
3180:
3172:
3169:February 2014
3163:
3159:
3156:This section
3154:
3151:
3147:
3146:
3143:
3141:
3137:
3133:
3129:
3110:
3091:
3068:
3050:
3027:
3012:
3011:unitary group
2989:
2974:
2970:
2962:
2960:
2958:
2954:
2950:
2946:
2942:
2939:(such as the
2938:
2934:
2929:
2927:
2923:
2919:
2915:
2911:
2903:
2895:
2892:February 2014
2886:
2882:
2878:
2875:
2871:
2870:
2865:
2857:
2855:
2853:
2849:
2841:
2837:
2828:
2826:
2825:Lorentz group
2822:
2818:
2817:
2812:
2808:
2802:
2800:
2796:
2792:
2788:
2784:
2778:
2771:In relativity
2770:
2768:
2766:
2762:
2758:
2756:
2752:
2750:
2745:
2744:of the sphere
2731:
2723:
2717:
2713:
2706:
2698:
2696:
2679:
2665:
2659:
2619:
2594:
2575:
2556:
2551:
2549:
2545:
2541:
2537:
2532:
2530:
2526:
2522:
2514:
2512:
2510:
2506:
2501:
2479:
2475:
2471:
2466:
2462:
2457:
2446:Further notes
2445:
2443:
2441:
2437:
2432:
2429:
2409:
2404:
2400:
2390:
2386:
2374:
2373:
2372:
2370:
2366:
2361:
2356:
2351:
2345:
2325:
2320:
2317:
2301:
2296:
2284:
2283:
2282:
2280:
2276:
2271:
2269:
2268:
2263:
2257:
2249:
2247:
2244:
2242:
2238:
2236:
2232:
2228:
2224:
2216:
2211:
2207:. The matrix
2206:
2202:
2181:
2174:
2171:
2162:
2159:
2150:
2147:
2140:
2135:
2130:
2124:
2117:
2110:
2104:
2097:
2091:
2086:
2081:
2074:
2069:
2064:
2057:
2052:
2047:
2041:
2036:
2031:
2025:
2018:
2011:
2005:
1991:
1990:
1989:
1970:
1964:
1959:
1954:
1947:
1942:
1937:
1930:
1925:
1920:
1914:
1909:
1897:
1896:
1895:
1881:
1877:
1873:
1865:
1861:
1857:
1849:
1843:
1837:
1829:
1827:
1825:
1821:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1766:
1760:
1757:
1749:
1746:
1743:
1740:
1737:
1734:
1731:
1728:
1725:
1722:
1720:
1714:
1711:
1699:
1698:
1697:
1676:
1672:
1669:
1665:
1662:
1658:
1655:
1651:
1649:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1608:
1605:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1569:
1567:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1494:
1483:
1480:
1477:
1474:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1438:
1436:
1431:
1426:
1423:
1419:
1407:
1406:
1405:
1403:
1398:
1374:
1371:
1368:
1365:
1362:
1359:
1352:
1351:
1350:
1346:
1342:
1336:
1331:
1325:
1324:as expected.
1305:
1298:
1295:
1286:
1283:
1276:
1252:
1246:
1239:
1233:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1172:
1166:
1163:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1126:
1120:
1117:
1105:
1104:
1103:
1092:
1088:
1065:
1059:
1052:
1046:
1039:
1033:
1030:
1027:
1022:
1019:
1016:
1009:
1006:
1003:
1000:
995:
992:
989:
983:
978:
973:
966:
963:
954:
951:
944:
935:
934:
933:
930:
925:
919:
915:
907:
901:
893:
891:
889:
885:
878:
874:
870:
866:
862:
858:
853:
851:
847:
843:
839:
834:
830:
826:
822:
818:
813:
811:
807:
803:
799:
795:
791:
787:
783:
779:
775:
771:
765:
757:
755:
753:
749:
741:
732:
723:
714:
709:
704:
700:
692:
687:
680:
679:
674:
670:
665:
661:
657:
653:
650:
646:
645:
643:
640:
639:
635:
628:
627:
622:
621:
616:
615:
610:
609:
604:
603:
598:
597:
592:
588:
587:line of nodes
584:
580:
576:
572:
569:
568:
567:
565:
561:
557:
553:
549:
546:Rotations in
540:
536:
534:
533:for details.
532:
528:
524:
520:
516:
512:
509:their angles
508:
504:
500:
492:
484:
480:
476:
472:
468:
464:
460:
455:
453:
449:
445:
441:
437:
433:
429:
425:
421:
417:
409:
404:
399:
395:
387:
382:
380:
378:
374:
370:
369:vector spaces
366:
363:Rotations of
361:
359:
355:
347:
343:
339:
335:
334:
329:
324:
319:
315:
314:
309:
308:
307:
304:
302:
298:
297:
292:
288:
284:
280:
276:
272:
264:
262:
260:
256:
252:
248:
244:
240:
236:
232:
228:
224:
220:
215:
213:
205:
202:-dimensional
199:
193:
189:
185:
181:
177:
173:
169:
166:of a certain
165:
161:
157:
153:
141:
131:
128:
120:
117:February 2014
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: –
77:
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
33:
19:
3716:Shear matrix
3695:
3679:Path tracing
3664:Cone tracing
3659:Beam tracing
3579:Polygon mesh
3520:3D rendering
3331:
3302:
3281:
3272:
3261:
3250:
3166:
3162:adding to it
3157:
3127:
2966:
2930:
2921:
2907:
2889:
2885:adding to it
2880:
2829:
2814:
2803:
2780:
2759:
2751:translations
2747:
2743:
2729:
2721:
2702:
2552:
2533:
2518:
2502:
2467:
2460:
2449:
2439:
2433:
2427:
2424:
2359:
2349:
2343:
2340:
2274:
2272:
2265:
2259:
2245:
2239:
2231:unit vectors
2209:
2198:
1987:
1879:
1875:
1871:
1863:
1859:
1855:
1851:
1817:
1695:
1404:as follows:
1396:
1389:
1344:
1340:
1337:: the point
1329:
1326:
1224:The vectors
1223:
1090:
1086:
1083:
928:
917:
913:
909:
854:
841:
837:
824:
814:
809:
790:affine space
767:
747:
739:
730:
721:
712:
696:
681:for details.
677:
664:pseudovector
659:
625:
619:
613:
607:
601:
595:
590:
578:
571:Euler angles
545:
535:
526:
518:
483:circle group
478:
456:
443:
439:
436:group theory
424:the distance
422:: it leaves
413:
376:
372:
362:
353:
351:
331:
322:
311:
305:
294:
286:
270:
268:
230:
216:
197:
188:translations
151:
150:
123:
114:
104:
97:
90:
83:
71:
59:Please help
54:verification
51:
3731:Translation
3684:Ray casting
3674:Ray tracing
3552:Cel shading
3526:Image-based
3507:3D graphics
3488:Ray casting
3437:2D graphics
3364:Albuquerque
2864:point group
2734:). For odd
2365:scalar part
2262:quaternions
2250:Quaternions
2227:determinant
1868:to a point
861:determinant
649:unit vector
575:composition
552:commutative
408:translation
279:fixed point
229:called the
227:composition
212:dimensional
182:(as in the
156:mathematics
3817:Categories
3795:Algorithms
3649:Reflection
3292:References
3084:of degree
3043:of degree
2918:invariance
2904:Importance
2755:hyperbolic
2593:Spin group
1840:See also:
904:See also:
833:components
806:equivalent
770:the origin
497:about the
346:orthogonal
176:rigid body
87:newspapers
3774:rendering
3764:animation
3654:Rendering
3307:Dordrecht
3243:Footnotes
2862:See:
2848:conformal
2787:spacetime
2666:≅
2440:different
2318:−
1796:θ
1793:
1781:θ
1778:
1750:θ
1747:
1738:−
1735:θ
1732:
1639:θ
1636:
1624:θ
1621:
1600:θ
1597:
1588:−
1585:θ
1582:
1560:θ
1557:
1548:−
1545:θ
1542:
1527:θ
1524:
1509:θ
1506:
1466:θ
1463:
1451:θ
1448:
1427:θ
1202:θ
1199:
1187:θ
1184:
1156:θ
1153:
1144:−
1141:θ
1138:
1034:θ
1031:
1023:θ
1020:
1010:θ
1007:
1001:−
996:θ
993:
825:magnitude
794:structure
786:Euclidean
691:tesseract
673:algebraic
527:different
342:invariant
275:Lie group
255:clockwise
235:mechanics
3769:modeling
3696:Rotation
3634:Clipping
3617:Concepts
3596:Deferred
3562:Lighting
3542:Aliasing
3536:Unbiased
3531:Spectral
3301:(1999).
3181:See also
3009:forms a
2945:rotation
2910:symmetry
2840:2-sphere
2830:Whereas
2749:Clifford
2540:bivector
2297:′
2175:′
2163:′
2151:′
1894:matrix,
1761:′
1715:′
1673:′
1659:′
1299:′
1287:′
1167:′
1121:′
967:′
955:′
519:the same
444:indirect
420:isometry
340:that is
160:geometry
152:Rotation
32:Rotation
3701:Scaling
3591:Shading
3090:spinors
2969:complex
2937:physics
2846:induce
2844:SO(3;1)
2712:-sphere
2355:inverse
2353:is its
2267:versors
2203:is the
523:commute
463:trivial
430:. The "
301:motions
239:physics
214:space.
101:scholar
3721:Shader
3493:Skybox
3478:Mode 7
3450:Layers
3342:
3317:
3237:Vortex
3202:CORDIC
2924:. The
2916:is an
2425:where
2357:, and
2341:where
855:As it
831:). In
796:; see
772:, the
626:ψ
620:θ
614:φ
608:γ
602:β
596:α
521:point
511:modulo
501:; see
499:origin
477:– the
475:origin
440:direct
396:, and
325:> 2
291:origin
283:center
225:under
164:motion
103:
96:
89:
82:
74:
3741:Voxel
3726:Texel
3427:Pixel
3360:(PDF)
3138:(see
3132:above
3128:three
2838:as a
2832:SO(3)
2264:, or
2260:Unit
2225:with
2219:SO(3)
875:(for
802:up to
654:as a
611:, or
503:below
471:angle
338:plane
336:is a
316:is a
273:is a
223:group
172:point
168:space
108:JSTOR
94:books
3465:2.5D
3340:ISBN
3315:ISBN
3136:spin
2967:The
2763:and
2732:+ 1)
2724:+ 1)
2279:norm
1267:and
1102:are
1098:and
871:, a
752:even
737:and
719:and
515:turn
507:sums
487:U(1)
330:The
318:line
310:The
269:The
204:flat
200:− 1)
180:sign
80:news
3164:.
3142:).
2887:.
2728:SO(
2703:In
2459:SO(
1790:cos
1775:sin
1744:sin
1729:cos
1633:cos
1618:sin
1594:sin
1579:cos
1554:sin
1539:sin
1521:cos
1503:cos
1460:sin
1445:cos
1196:cos
1181:sin
1150:sin
1135:cos
1028:cos
1017:sin
1004:sin
990:cos
877:odd
810:all
457:In
442:vs
219:map
154:in
63:by
3819::
3366::
3362:.
3338:.
3313:.
3309::
3305:.
2912::
2854:.
2801:.
2695:.
2595:,
2476:.
2465:.
2217:,
1880:z′
1878:,
1876:y′
1874:,
1872:x′
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1858:,
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1100:y′
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932::
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890:.
865:−1
852:.
840:×
748:is
666:).
623:,
617:,
605:,
599:,
513:1
360:.
352:A
303:.
261:.
3587:)
3583:(
3538:)
3524:(
3398:e
3391:t
3384:v
3370:.
3348:.
3323:.
3171:)
3167:(
3114:)
3111:2
3108:(
3104:U
3101:S
3086:n
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3069:n
3066:(
3062:U
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3045:n
3031:)
3028:n
3025:(
3021:U
3007:n
2993:)
2990:n
2987:(
2983:U
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2866:.
2740:n
2736:n
2730:n
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2710:n
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2680:2
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2673:U
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2660:3
2657:(
2653:n
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2613:n
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2607:p
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2569:O
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2497:3
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2492:3
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2484:×
2482:4
2463:)
2461:n
2452:n
2428:v
2410:,
2405:2
2401:/
2396:v
2391:e
2387:=
2383:q
2360:x
2350:q
2344:q
2326:,
2321:1
2313:q
2310:x
2307:q
2302:=
2294:x
2210:A
2182:)
2172:z
2160:y
2148:x
2141:(
2136:=
2131:)
2125:z
2118:y
2111:x
2105:(
2098:)
2092:i
2087:h
2082:g
2075:f
2070:e
2065:d
2058:c
2053:b
2048:a
2042:(
2037:=
2032:)
2026:z
2019:y
2012:x
2006:(
2000:A
1971:)
1965:i
1960:h
1955:g
1948:f
1943:e
1938:d
1931:c
1926:b
1921:a
1915:(
1910:=
1906:A
1891:3
1888:×
1886:3
1882:)
1870:(
1866:)
1864:z
1860:y
1856:x
1854:(
1799:.
1787:y
1784:+
1772:x
1769:=
1758:y
1741:y
1726:x
1723:=
1712:x
1677:,
1670:y
1666:i
1663:+
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1652:=
1642:)
1630:y
1627:+
1615:x
1612:(
1609:i
1606:+
1603:)
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1570:=
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1500:x
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1296:y
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1234:[
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1190:+
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1175:=
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1080:.
1066:]
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918:y
914:x
912:(
880:n
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740:ω
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495:θ
410:.
348:.
327:.
323:n
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208:n
198:n
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147:.
145:O
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124:(
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115:(
105:·
98:·
91:·
84:·
57:.
34:.
20:)
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