1098:
1792:
934:
3170:
1596:
560:; it instead means that the planes have no nonzero vectors in common, and that every vector in one plane is orthogonal to every vector in the other plane. This can only happen in four or more dimensions. In two dimensions there is only one plane, while in three dimensions all planes have at least one nonzero vector in common, along their
1637:. Most importantly the planes of rotation are not uniquely identified. There are instead an infinite number of pairs of orthogonal planes that can be treated as planes of rotation. For example any point can be taken, and the plane it rotates in together with the plane orthogonal to it can be used as two planes of rotation.
1359:
1152:
perpendicular to, and so is defined by and defines, an axis of rotation, so any description of a rotation in terms of a plane of rotation can be described in terms of an axis of rotation, and vice versa. But unlike the axis of rotation the plane generalises into other, in particular higher, dimensions.
1711:
A general rotation is not simple, and has the maximum number of planes of rotation as given above. In the general case the angles of rotations in these planes are distinct and the planes are uniquely defined. If any of the angles are the same then the planes are not unique, as in four dimensions with
1782:
The examples given above were chosen to be clear and simple examples of rotations, with planes generally parallel to the coordinate axes in three and four dimensions. But this is not generally the case: planes are not usually parallel to the axes, and the matrices cannot simply be written down. In
1377:
there are two planes of rotation, no fixed planes, and the only fixed point is the origin. The rotation can be said to take place in both planes of rotation, as points in them are rotated within the planes. These planes are orthogonal, that is they have no vectors in common so every vector in one
953:
there are an infinite number of planes of rotation, only one of which is involved in any given rotation. That is, for a general rotation there is precisely one plane which is associated with it or which the rotation takes place in. The only exception is the trivial rotation, corresponding to the
2835:
of the complex roots are the magnitudes of the bivectors associated with the planes of rotations. The form of the characteristic equation is related to the planes, making it possible to relate its algebraic properties like repeated roots to the bivectors, where repeated bivector magnitudes have
1151:
In any three dimensional rotation the plane of rotation is uniquely defined. Together with the angle of rotation it fully describes the rotation. Or in a continuously rotating object the rotational properties such as the rate of rotation can be described in terms of the plane of rotation. It is
2762:
Given a rotor the bivector associated with it can be recovered by taking the logarithm of the rotor, which can then be split into simple bivectors to determine the planes of rotation, although in practice for all but the simplest of cases this may be impractical. But given the simple bivectors
2355:
Every rotation plane in a rotation has a simple bivector associated with it. This is parallel to the plane and has magnitude equal to the angle of rotation in the plane. These bivectors are summed to produce a single, generally non-simple, bivector for the whole rotation. This can generate a
717:, so any pair of orthogonal planes generates the rotation. But for a general rotation it is at least theoretically possible to identify a unique set of orthogonal planes, in each of which points are rotated through an angle, so the set of planes and angles fully characterise the rotation.
1427:
1183:. In a simple rotation there is a fixed plane, and rotation can be said to take place about this plane, so points as they rotate do not change their distance from this plane. The plane of rotation is orthogonal to this plane, and the rotation can be said to take place in this plane.
1830:-dimensional subspace. To generate simple rotations only reflections that fix the origin are needed, so the vector does not have a position, just direction. It does also not matter which way it is facing: it can be replaced with its negative without changing the result. Similarly
1092:
2037:
708:
1364:
In two and three dimensions all rotations are simple, in that they have only one plane of rotation. Only in four and more dimensions are there rotations that are not simple rotations. In particular in four dimensions there are also double and isoclinic rotations.
1815:-dimensional subspace to reflect in, so a two-dimensional reflection is in a line, a three-dimensional reflection is in a plane, and so on. But this becomes increasingly difficult to apply in higher dimensions, so it is better to use vectors instead, as follows.
428:
to itself by the rotation. The plane is not fixed, but all vectors in the plane are mapped to other vectors in the same plane by the rotation. This transformation of the plane to itself is always a rotation about the origin, through an angle which is the
2529:
923:
2305:
Conversely all simple rotations can be generated this way, with two reflections, by two unit vectors in the plane of rotation separated by half the desired angle of rotation. These can be composed to produce more general rotations, using up to
1220:
1607:
61:, as in two dimensions there is only one plane (so, identifying the plane of rotation is trivial and rarely done), while in three dimensions the axis of rotation serves the same purpose and is the more established approach.
2673:
This is a simple bivector, associated with the simple rotation described. More general rotations in four or more dimensions are associated with sums of simple bivectors, one for each plane of rotation, calculated as above.
1170:
has only one fixed point, the origin. Therefore an axis of rotation cannot be used in four dimensions. But planes of rotation can be used, and each non-trivial rotation in four dimensions has one or two planes of rotation.
2155:
1405:, so in a sense they together determine the amount of rotation. For a general double rotation the planes of rotation and angles are unique, and given a general rotation they can be calculated. For example a rotation of
1591:{\displaystyle {\begin{pmatrix}\cos \alpha &-\sin \alpha &0&0\\\sin \alpha &\cos \alpha &0&0\\0&0&\cos \beta &-\sin \beta \\0&0&\sin \beta &\cos \beta \end{pmatrix}}.}
381:
957:
In any rotation in three dimensions there is always a fixed axis, the axis of rotation. The rotation can be described by giving this axis, with the angle through which the rotation turns about it; this is the
1783:
all dimensions the rotations are fully described by the planes of rotation and their associated angles, so it is useful to be able to determine them, or at least find ways to describe them mathematically.
987:
827:
2430:
2212:
581:
2822:
1688:
1378:
plane is at right angles to every vector in the other plane. The two rotation planes span four-dimensional space, so every point in the space can be specified by two points, one on each of the planes.
480:
2242:, which must be distinct otherwise the reflections are the same and no rotation takes place. As either vector can be replaced by its negative the angle between them can always be acute, or at most
188:
1897:
316:
2668:
2348:, which generalise the idea of vectors into two dimensions. As vectors are to lines, so are bivectors to planes. So every plane (in any dimension) can be associated with a bivector, and every
2590:
1935:
2441:
966:
of the plane. The rotation then rotates this plane through the same angle as it rotates around the axis, that is everything in the plane rotates by the same angle about the origin.
845:
1693:
so the complexity quickly increases with more than four dimensions and categorising rotations as above becomes too complex to be practical, but some observations can be made.
1354:{\displaystyle {\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&\cos \theta &-\sin \theta \\0&0&\sin \theta &\cos \theta \end{pmatrix}}}
977:-plane, so everything in that plane it kept in the plane by the rotation. This could be described by a matrix like the following, with the rotation being through an angle
321:
This is true in all dimensions, and can be taken as the definition on the plane. In particular, from the properties of the exterior product it is satisfied by both
1633:, and it differs from a general double rotation in a number of ways. For example in an isoclinic rotation, all non-zero points rotate through the same angle,
1381:
A double rotation has two angles of rotation, one for each plane of rotation. The rotation is specified by giving the two planes and two non-zero angles,
3101:
2865:
1161:
1770:
planes and angles of rotation, the same as the even dimension one lower. These do not span the space, but leave a line which does not rotate – like the
2081:
1742:
planes of rotation span the space, so a general rotation rotates all points except the origin which is the only fixed point. In odd dimensions (
3398:
3045:
339:
136:
of the space. It is completely specified by any two non-zero and non-parallel vectors that lie in the plane, that is by any two vectors
1087:{\displaystyle {\begin{pmatrix}\cos \theta &-\sin \theta &0\\\sin \theta &\cos \theta &0\\0&0&1\end{pmatrix}}.}
3204:
703:{\displaystyle {\begin{pmatrix}-1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}},}
3154:
3070:
3020:
2783:
has either one (in odd dimensions) or zero (in even dimensions) real roots. The other roots are in complex conjugate pairs, exactly
767:
2383:
2789:
1655:
3408:
450:
3094:
2367:
Bivectors are related to rotors through the exponential map (which applied to bivectors generates rotors and rotations using
154:
1696:
Simple rotations can be identified in all dimensions, as rotations with just one plane of rotation. A simple rotation in
3403:
2166:
1771:
713:
describes a rotation in four dimensions in which every plane through the origin is a plane of rotation through an angle
285:
81:
2625:
3189:
2832:
2032:{\displaystyle \mathbf {x} ''=-\mathbf {nx} '\mathbf {n} =-\mathbf {n} (-\mathbf {mxm} )\mathbf {n} =\mathbf {nmxmn} }
730:
2552:
1857:
3413:
3393:
3037:
2056:. It can be checked using geometric algebra that this is a rotation, and that it rotates all vectors as expected.
3087:
1774:
in three dimensions, except rotations do not take place about this line but in multiple planes orthogonal to it.
92:
they are related to other algebraic and geometric properties, which can then be generalised to other dimensions.
3124:
2524:{\displaystyle {R_{\mathbf {B} }}^{2}=e^{\frac {\mathbf {B} }{2}}e^{\frac {\mathbf {B} }{2}}=e^{\mathbf {B} },}
1800:
3332:
3327:
3307:
950:
918:{\displaystyle {\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{pmatrix}}.}
962:
representation of a rotation. The plane of rotation is the plane orthogonal to this axis, so the axis is a
3317:
3312:
3292:
2368:
1167:
556:
to each other, with only the origin in common. This is a stronger condition than to say the planes are at
421:
35:
28:
3322:
3302:
3297:
726:
567:
In more than three dimensions planes of rotation are not always unique. For example the negative of the
2046:
dimensions, through twice the angle between the subspaces, which is also the angle between the vectors
2759:
which are parallel to the two planes of rotation and have magnitudes equal to the angles of rotation.
2435:
This is a simple rotation if the bivector is simple, a more general rotation otherwise. When squared,
3062:
2860:
2361:
742:
572:
254:
is the angle between the vectors; hence the requirement that the vectors be nonzero and nonparallel.
118:
3199:
3194:
2357:
2066:
1133:
1129:
1113:
126:
64:
Mathematically such planes can be described in a number of ways. They can be described in terms of
3373:
3214:
3169:
2763:
geometric algebra is a useful tool for studying planes of rotation using algebra like the above.
1190:-plane: points in that plane and only in that plane are unchanged. The plane of rotation is the
758:
42:, where they can be used to break down the rotations into simpler parts. This can be done using
3209:
3066:
3041:
3016:
2677:
Examples include the two rotations in four dimensions given above. The simple rotation in the
2337:
1903:
1145:
489:
is the dimension. The maximum number of planes up to eight dimensions is shown in this table:
430:
202:
77:
69:
43:
39:
745:
fixed. It is specified completely by the signed angle of rotation, in the range for example −
3139:
2780:
2540:
is simple then this is the same rotation as is generated by two reflections, as the product
2345:
2341:
1106:
561:
437:
198:
113:
65:
1389:(if either angle is zero the rotation is simple). Points in the first plane rotate through
3184:
3129:
2850:
2845:
2352:
is associated with a plane. This makes them a good fit for describing planes of rotation.
836:
734:
568:
441:
133:
85:
58:
2595:
from which it follows that the bivector associated with the plane of rotation containing
2325:
is odd, by choosing pairs of reflections given by two vectors in each plane of rotation.
3266:
3251:
3002:
963:
408:
ranges over the whole plane, so this can be taken as another definition of the plane.
3387:
3256:
738:
206:
54:
2546:
gives a rotation through twice the angle between the vectors. These can be equated,
741:. Any rotation therefore is of the whole plane, i.e. of the space, keeping only the
3276:
3241:
3134:
1097:
3056:
3031:
3006:
3361:
3144:
2855:
2281:
1831:
557:
122:
2831:
of the matrix, which can be calculated using algebraic techniques. In addition
1618:
A special case of the double rotation is when the angles are equal, that is if
34:
The main use for planes of rotation is in describing more complex rotations in
3356:
3236:
2828:
2772:
2150:{\displaystyle (\mathbf {mn} )(\mathbf {nm} )=\mathbf {mnnm} =\mathbf {mm} =1}
1121:
1117:
1102:
969:
One example is shown in the diagram, where the rotation takes place about the
959:
553:
425:
106:
3337:
3246:
3159:
3110:
1611:
1137:
89:
1791:
933:
3261:
3224:
3149:
2349:
2333:
1141:
729:
there is only one plane of rotation, the plane of the space itself. In a
73:
47:
20:
3271:
1125:
1606:
16:
Geometric object used to describe rotation in any number of dimensions
3058:
Applications of geometric algebra in computer science and engineering
3012:
1845:-dimensional space is given by the unit vector perpendicular to it,
273:, then the condition that a point lies on the plane associated with
1700:
dimensions takes place about (that is at a fixed distance from) an
376:{\displaystyle \mathbf {c} =\lambda \mathbf {a} +\mu \mathbf {b} ,}
3228:
1795:
Two different reflections in two dimensions generating a rotation.
1790:
1630:
1605:
1374:
1180:
1096:
932:
937:
A three-dimensional rotation, with an axis of rotation along the
552:
When a rotation has multiple planes of rotation they are always
3083:
1136:
Hemispheres. Other examples include mechanical devices like a
822:{\displaystyle e^{i\theta }=\cos {\theta }+i\sin {\theta },\,}
2425:{\displaystyle R_{\mathbf {B} }=e^{\frac {\mathbf {B} }{2}}.}
1645:
As already noted the maximum number of planes of rotation in
3079:
2827:
such pairs. These correspond to the planes of rotation, the
2771:
The planes of rotations for a particular rotation using the
2268:
or a half-turn. The sense of the rotation is to rotate from
1708:-dimensional subspace orthogonal to the plane of rotation.
2817:{\displaystyle \left\lfloor {\frac {n}{2}}\right\rfloor ,}
2534:
it gives a rotor that rotates through twice the angle. If
1683:{\displaystyle \left\lfloor {\frac {n}{2}}\right\rfloor ,}
1194:-plane, points in this plane are rotated through an angle
1822:
dimensions is specified by a vector perpendicular to the
475:{\displaystyle \left\lfloor {\frac {n}{2}}\right\rfloor }
832:
while the rotation in a
Cartesian plane is given by the
1101:
The Earth showing its axis and plane of rotation, both
219:
is the bivector associated with the plane specified by
1436:
1229:
996:
854:
590:
183:{\displaystyle \mathbf {a} \wedge \mathbf {b} \neq 0,}
2792:
2628:
2555:
2444:
2386:
2169:
2084:
1938:
1860:
1658:
1430:
1223:
990:
848:
770:
584:
453:
342:
288:
157:
27:
is an abstract object used to describe or visualize
3349:
3285:
3223:
3177:
3117:
1397:. All other points rotate through an angle between
1124:and the plane of rotation is the plane through the
954:identity matrix, in which no rotation takes place.
493:
2816:
2662:
2584:
2523:
2424:
2207:{\displaystyle \mathbf {x} ''=R\mathbf {x} R^{-1}}
2206:
2149:
2031:
1891:
1682:
1590:
1393:, while points in the second plane rotate through
1353:
1086:
917:
821:
702:
474:
375:
311:{\displaystyle \mathbf {x} \wedge \mathbf {B} =0.}
310:
182:
2663:{\displaystyle \mathbf {B} =\log(\mathbf {mn} ).}
3055:Dorst, Leo; Doran, Chris; Lasenby, Joan (2002).
1902:where the product is the geometric product from
1179:A rotation with only one plane of rotation is a
444:) has at least one plane of rotation, and up to
2585:{\displaystyle \mathbf {mn} =e^{\mathbf {B} },}
1923:-dimensional space, described by a unit vector
1892:{\displaystyle \mathbf {x} '=-\mathbf {mxm} \,}
1116:. The axis of rotation is the line joining the
46:, with the planes of rotations associated with
2886:
2884:
2882:
2880:
2230:The plane of rotation is the plane containing
1799:Every simple rotation can be generated by two
757:the rotation in the complex plane is given by
3095:
1148:in mass usually along the plane of rotation.
8:
2697:, a simple bivector. The double rotation by
1186:For example the following matrix fixes the
1105:relative to the plane and perpendicular of
209:can be used). More precisely, the quantity
3102:
3088:
3080:
2866:Rotations in 4-dimensional Euclidean space
1834:can be used to simplify the calculations.
1162:Rotations in 4-dimensional Euclidean space
2797:
2791:
2646:
2629:
2627:
2572:
2571:
2556:
2554:
2511:
2510:
2492:
2490:
2475:
2473:
2460:
2452:
2451:
2446:
2443:
2408:
2406:
2392:
2391:
2385:
2364:, which can be used to rotate an object.
2290:is the inverse rotation, with sense from
2195:
2186:
2171:
2168:
2133:
2116:
2102:
2088:
2083:
2012:
2004:
1990:
1979:
1968:
1956:
1940:
1937:
1888:
1877:
1862:
1859:
1663:
1657:
1431:
1429:
1224:
1222:
991:
989:
849:
847:
818:
810:
793:
775:
769:
585:
583:
458:
452:
365:
354:
343:
341:
297:
289:
287:
166:
158:
156:
2989:Dorst, Doran, Lasenby (2002) pp. 145–154
53:Planes of rotation are not used much in
3008:New Foundations for Classical Mechanics
2876:
2980:Dorst, Doran, Lasenby (2002) pp. 79–89
2836:particular geometric interpretations.
1202:-plane, that is it rotates around the
1198:. A general point rotates only in the
2775:. Given a general rotation matrix in
2264:the angle between the vectors, up to
941:-axis and a plane of rotation in the
7:
2371:). In particular given any bivector
973:-axis. The plane of rotation is the
1929:perpendicular to it, the result is
1915:is reflected in another, distinct,
333:, and so by any vector of the form
2735:, the sum of two simple bivectors
1803:. Reflections can be specified in
981:(about the axis or in the plane):
14:
3168:
2650:
2647:
2630:
2573:
2560:
2557:
2512:
2493:
2476:
2453:
2409:
2393:
2377:the rotor associated with it is
2187:
2172:
2137:
2134:
2126:
2123:
2120:
2117:
2106:
2103:
2092:
2089:
2025:
2022:
2019:
2016:
2013:
2005:
1997:
1994:
1991:
1980:
1969:
1960:
1957:
1941:
1884:
1881:
1878:
1863:
366:
355:
344:
298:
290:
167:
159:
2280:: the geometric product is not
2160:So the rotation can be written
2654:
2643:
2110:
2099:
2096:
2085:
2001:
1984:
1421:-plane is given by the matrix
733:it is the Cartesian plane, in
436:Every rotation except for the
72:. They can be associated with
1:
3033:Clifford algebras and spinors
2310:reflections if the dimension
2042:This is a simple rotation in
402:range over all real numbers,
197:is the exterior product from
3399:Rotation in three dimensions
1206:-plane by changing only its
82:eigenvalues and eigenvectors
2890:Lounesto (2001) pp. 222–223
2767:Eigenvalues and eigenplanes
1614:with an isoclinic rotation.
731:Cartesian coordinate system
121:, that is they contain the
3430:
3038:Cambridge University Press
2971:Hestenes (1999) p. 278–280
2944:Hestenes (1999) pp 280–284
2260:. The rotation is through
1159:
485:planes of rotation, where
440:rotation (with matrix the
104:
80:. They are related to the
3370:
3166:
3030:Lounesto, Pertti (2001).
2953:Lounesto (2001) pp. 83–89
205:(in three dimensions the
2962:Lounesto (2001) p. 57–58
2935:Lounesto (2001) pp.27–28
1807:dimensions by giving an
571:in four dimensions (the
2781:characteristic equation
1837:So the reflection in a
1778:Mathematical properties
1712:an isoclinic rotation.
1112:Another example is the
951:three-dimensional space
117:are planes through the
3409:Orientation (geometry)
2917:Lounesto (2001) p. 222
2818:
2664:
2586:
2525:
2426:
2208:
2151:
2033:
1893:
1796:
1684:
1615:
1592:
1355:
1168:four-dimensional space
1166:A general rotation in
1109:
1088:
946:
919:
823:
704:
476:
377:
312:
184:
111:For this article, all
36:four-dimensional space
2908:Hestenes (1999) p. 48
2899:Lounesto (2001) p. 38
2819:
2713:-planes has bivector
2665:
2587:
2526:
2427:
2209:
2152:
2034:
1894:
1794:
1685:
1609:
1593:
1356:
1100:
1089:
936:
920:
824:
753:. So if the angle is
727:two-dimensional space
705:
477:
378:
313:
185:
132:is a two-dimensional
3286:Dimensions by number
2926:Lounesto (2001) p.87
2861:Rotation group SO(3)
2790:
2626:
2553:
2442:
2384:
2336:are quantities from
2167:
2082:
1936:
1858:
1715:In even dimensions (
1656:
1629:. This is called an
1428:
1221:
988:
846:
768:
582:
562:line of intersection
451:
340:
286:
231:, and has magnitude
155:
88:. And in particular
3404:Rotational symmetry
2681:-plane by an angle
2369:De Moivre's formula
1602:Isoclinic rotations
424:is a plane that is
3215:Degrees of freedom
3118:Dimensional spaces
2814:
2660:
2582:
2521:
2422:
2204:
2147:
2075:is its inverse as
2029:
1889:
1797:
1722:) there are up to
1680:
1631:isoclinic rotation
1616:
1610:A projection of a
1588:
1579:
1351:
1345:
1110:
1084:
1075:
947:
915:
906:
819:
700:
691:
472:
373:
308:
180:
130:-dimensional space
125:. Such a plane in
70:angles of rotation
3414:Planes (geometry)
3394:Geometric algebra
3381:
3380:
3190:Lebesgue covering
3155:Algebraic variety
3047:978-0-521-00551-7
2805:
2500:
2483:
2416:
2338:geometric algebra
1904:geometric algebra
1671:
1641:Higher dimensions
1146:rotational energy
573:central inversion
548:
547:
523:Number of planes
466:
431:angle of rotation
420:for a particular
418:plane of rotation
412:Plane of rotation
394:real numbers. As
203:geometric algebra
78:geometric algebra
44:geometric algebra
40:higher dimensions
25:plane of rotation
3421:
3178:Other dimensions
3172:
3140:Projective space
3104:
3097:
3090:
3081:
3076:
3051:
3026:
3011:(2nd ed.).
2990:
2987:
2981:
2978:
2972:
2969:
2963:
2960:
2954:
2951:
2945:
2942:
2936:
2933:
2927:
2924:
2918:
2915:
2909:
2906:
2900:
2897:
2891:
2888:
2823:
2821:
2820:
2815:
2810:
2806:
2798:
2778:
2758:
2746:
2734:
2712:
2708:
2704:
2700:
2696:
2684:
2680:
2669:
2667:
2666:
2661:
2653:
2633:
2618:
2612:
2606:
2600:
2591:
2589:
2588:
2583:
2578:
2577:
2576:
2563:
2545:
2539:
2530:
2528:
2527:
2522:
2517:
2516:
2515:
2502:
2501:
2496:
2491:
2485:
2484:
2479:
2474:
2465:
2464:
2459:
2458:
2457:
2456:
2431:
2429:
2428:
2423:
2418:
2417:
2412:
2407:
2398:
2397:
2396:
2376:
2346:exterior algebra
2342:clifford algebra
2324:
2320:
2313:
2309:
2301:
2295:
2289:
2279:
2273:
2267:
2259:
2257:
2256:
2253:
2250:
2249:
2241:
2235:
2226:
2213:
2211:
2210:
2205:
2203:
2202:
2190:
2179:
2175:
2156:
2154:
2153:
2148:
2140:
2129:
2109:
2095:
2074:
2064:
2055:
2045:
2038:
2036:
2035:
2030:
2028:
2008:
2000:
1983:
1972:
1967:
1963:
1948:
1944:
1928:
1922:
1914:
1898:
1896:
1895:
1890:
1887:
1870:
1866:
1850:
1844:
1829:
1821:
1818:A reflection in
1814:
1806:
1772:axis of rotation
1769:
1768:
1766:
1765:
1762:
1759:
1748:
1741:
1740:
1738:
1737:
1734:
1731:
1721:
1707:
1699:
1689:
1687:
1686:
1681:
1676:
1672:
1664:
1648:
1636:
1628:
1597:
1595:
1594:
1589:
1584:
1583:
1420:
1416:
1412:
1408:
1404:
1400:
1396:
1392:
1388:
1384:
1369:Double rotations
1360:
1358:
1357:
1352:
1350:
1349:
1213:
1209:
1205:
1201:
1197:
1193:
1189:
1175:Simple rotations
1114:Earth's rotation
1093:
1091:
1090:
1085:
1080:
1079:
980:
976:
972:
944:
940:
929:Three dimensions
924:
922:
921:
916:
911:
910:
835:
828:
826:
825:
820:
814:
797:
783:
782:
756:
752:
748:
716:
709:
707:
706:
701:
696:
695:
494:
488:
481:
479:
478:
473:
471:
467:
459:
407:
401:
397:
393:
389:
382:
380:
379:
374:
369:
358:
347:
332:
326:
317:
315:
314:
309:
301:
293:
278:
272:
266:
257:If the bivector
253:
249:
244:
238:
230:
224:
218:
199:exterior algebra
196:
189:
187:
186:
181:
170:
162:
147:
141:
129:
59:three dimensions
50:in the algebra.
48:simple bivectors
3429:
3428:
3424:
3423:
3422:
3420:
3419:
3418:
3384:
3383:
3382:
3377:
3366:
3345:
3281:
3219:
3173:
3164:
3130:Euclidean space
3113:
3108:
3073:
3054:
3048:
3029:
3023:
3003:Hestenes, David
3001:
2998:
2993:
2988:
2984:
2979:
2975:
2970:
2966:
2961:
2957:
2952:
2948:
2943:
2939:
2934:
2930:
2925:
2921:
2916:
2912:
2907:
2903:
2898:
2894:
2889:
2878:
2874:
2851:Givens rotation
2846:Charts on SO(3)
2842:
2793:
2788:
2787:
2779:dimensions its
2776:
2769:
2754:
2748:
2742:
2736:
2730:
2720:
2714:
2710:
2706:
2702:
2698:
2692:
2686:
2682:
2678:
2624:
2623:
2614:
2608:
2602:
2596:
2567:
2551:
2550:
2541:
2535:
2506:
2486:
2469:
2447:
2445:
2440:
2439:
2402:
2387:
2382:
2381:
2372:
2362:exponential map
2350:simple bivector
2331:
2322:
2315:
2311:
2307:
2297:
2291:
2285:
2284:so the product
2275:
2269:
2265:
2254:
2251:
2247:
2246:
2245:
2243:
2237:
2231:
2218:
2191:
2170:
2165:
2164:
2080:
2079:
2070:
2060:
2051:
2043:
1955:
1939:
1934:
1933:
1924:
1916:
1910:
1861:
1856:
1855:
1846:
1838:
1823:
1819:
1808:
1804:
1789:
1780:
1763:
1760:
1754:
1753:
1751:
1750:
1743:
1735:
1732:
1727:
1726:
1724:
1723:
1716:
1701:
1697:
1659:
1654:
1653:
1646:
1643:
1634:
1619:
1604:
1578:
1577:
1566:
1555:
1550:
1544:
1543:
1529:
1518:
1513:
1507:
1506:
1501:
1496:
1485:
1473:
1472:
1467:
1462:
1448:
1432:
1426:
1425:
1418:
1414:
1410:
1406:
1402:
1398:
1394:
1390:
1386:
1382:
1375:double rotation
1371:
1344:
1343:
1332:
1321:
1316:
1310:
1309:
1295:
1284:
1279:
1273:
1272:
1267:
1262:
1257:
1251:
1250:
1245:
1240:
1235:
1225:
1219:
1218:
1211:
1207:
1203:
1199:
1195:
1191:
1187:
1181:simple rotation
1177:
1164:
1158:
1156:Four dimensions
1074:
1073:
1068:
1063:
1057:
1056:
1051:
1040:
1028:
1027:
1022:
1008:
992:
986:
985:
978:
974:
970:
942:
938:
931:
905:
904:
893:
881:
880:
866:
850:
844:
843:
837:rotation matrix
833:
771:
766:
765:
759:Euler's formula
754:
750:
746:
735:complex numbers
723:
714:
690:
689:
681:
676:
671:
665:
664:
659:
651:
646:
640:
639:
634:
629:
621:
615:
614:
609:
604:
599:
586:
580:
579:
569:identity matrix
486:
454:
449:
448:
442:identity matrix
433:for the plane.
414:
403:
399:
395:
391:
387:
338:
337:
328:
322:
284:
283:
274:
268:
258:
251:
240:
234:
232:
226:
220:
210:
194:
153:
152:
143:
137:
134:linear subspace
127:
109:
103:
98:
86:rotation matrix
17:
12:
11:
5:
3427:
3425:
3417:
3416:
3411:
3406:
3401:
3396:
3386:
3385:
3379:
3378:
3371:
3368:
3367:
3365:
3364:
3359:
3353:
3351:
3347:
3346:
3344:
3343:
3335:
3330:
3325:
3320:
3315:
3310:
3305:
3300:
3295:
3289:
3287:
3283:
3282:
3280:
3279:
3274:
3269:
3267:Cross-polytope
3264:
3259:
3254:
3252:Hyperrectangle
3249:
3244:
3239:
3233:
3231:
3221:
3220:
3218:
3217:
3212:
3207:
3202:
3197:
3192:
3187:
3181:
3179:
3175:
3174:
3167:
3165:
3163:
3162:
3157:
3152:
3147:
3142:
3137:
3132:
3127:
3121:
3119:
3115:
3114:
3109:
3107:
3106:
3099:
3092:
3084:
3078:
3077:
3071:
3052:
3046:
3027:
3021:
2997:
2994:
2992:
2991:
2982:
2973:
2964:
2955:
2946:
2937:
2928:
2919:
2910:
2901:
2892:
2875:
2873:
2870:
2869:
2868:
2863:
2858:
2853:
2848:
2841:
2838:
2825:
2824:
2813:
2809:
2804:
2801:
2796:
2768:
2765:
2752:
2740:
2728:
2718:
2690:
2671:
2670:
2659:
2656:
2652:
2649:
2645:
2642:
2639:
2636:
2632:
2593:
2592:
2581:
2575:
2570:
2566:
2562:
2559:
2532:
2531:
2520:
2514:
2509:
2505:
2499:
2495:
2489:
2482:
2478:
2472:
2468:
2463:
2455:
2450:
2433:
2432:
2421:
2415:
2411:
2405:
2401:
2395:
2390:
2330:
2327:
2227:is the rotor.
2215:
2214:
2201:
2198:
2194:
2189:
2185:
2182:
2178:
2174:
2158:
2157:
2146:
2143:
2139:
2136:
2132:
2128:
2125:
2122:
2119:
2115:
2112:
2108:
2105:
2101:
2098:
2094:
2091:
2087:
2040:
2039:
2027:
2024:
2021:
2018:
2015:
2011:
2007:
2003:
1999:
1996:
1993:
1989:
1986:
1982:
1978:
1975:
1971:
1966:
1962:
1959:
1954:
1951:
1947:
1943:
1900:
1899:
1886:
1883:
1880:
1876:
1873:
1869:
1865:
1788:
1785:
1779:
1776:
1747:= 3, 5, 7, ...
1691:
1690:
1679:
1675:
1670:
1667:
1662:
1649:dimensions is
1642:
1639:
1603:
1600:
1599:
1598:
1587:
1582:
1576:
1573:
1570:
1567:
1565:
1562:
1559:
1556:
1554:
1551:
1549:
1546:
1545:
1542:
1539:
1536:
1533:
1530:
1528:
1525:
1522:
1519:
1517:
1514:
1512:
1509:
1508:
1505:
1502:
1500:
1497:
1495:
1492:
1489:
1486:
1484:
1481:
1478:
1475:
1474:
1471:
1468:
1466:
1463:
1461:
1458:
1455:
1452:
1449:
1447:
1444:
1441:
1438:
1437:
1435:
1370:
1367:
1362:
1361:
1348:
1342:
1339:
1336:
1333:
1331:
1328:
1325:
1322:
1320:
1317:
1315:
1312:
1311:
1308:
1305:
1302:
1299:
1296:
1294:
1291:
1288:
1285:
1283:
1280:
1278:
1275:
1274:
1271:
1268:
1266:
1263:
1261:
1258:
1256:
1253:
1252:
1249:
1246:
1244:
1241:
1239:
1236:
1234:
1231:
1230:
1228:
1176:
1173:
1160:Main article:
1157:
1154:
1095:
1094:
1083:
1078:
1072:
1069:
1067:
1064:
1062:
1059:
1058:
1055:
1052:
1050:
1047:
1044:
1041:
1039:
1036:
1033:
1030:
1029:
1026:
1023:
1021:
1018:
1015:
1012:
1009:
1007:
1004:
1001:
998:
997:
995:
964:surface normal
930:
927:
926:
925:
914:
909:
903:
900:
897:
894:
892:
889:
886:
883:
882:
879:
876:
873:
870:
867:
865:
862:
859:
856:
855:
853:
830:
829:
817:
813:
809:
806:
803:
800:
796:
792:
789:
786:
781:
778:
774:
722:
721:Two dimensions
719:
711:
710:
699:
694:
688:
685:
682:
680:
677:
675:
672:
670:
667:
666:
663:
660:
658:
655:
652:
650:
647:
645:
642:
641:
638:
635:
633:
630:
628:
625:
622:
620:
617:
616:
613:
610:
608:
605:
603:
600:
598:
595:
592:
591:
589:
550:
549:
546:
545:
542:
539:
536:
533:
530:
527:
524:
520:
519:
516:
513:
510:
507:
504:
501:
498:
483:
482:
470:
465:
462:
457:
413:
410:
384:
383:
372:
368:
364:
361:
357:
353:
350:
346:
319:
318:
307:
304:
300:
296:
292:
191:
190:
179:
176:
173:
169:
165:
161:
102:
99:
97:
94:
15:
13:
10:
9:
6:
4:
3:
2:
3426:
3415:
3412:
3410:
3407:
3405:
3402:
3400:
3397:
3395:
3392:
3391:
3389:
3376:
3375:
3369:
3363:
3360:
3358:
3355:
3354:
3352:
3348:
3342:
3340:
3336:
3334:
3331:
3329:
3326:
3324:
3321:
3319:
3316:
3314:
3311:
3309:
3306:
3304:
3301:
3299:
3296:
3294:
3291:
3290:
3288:
3284:
3278:
3275:
3273:
3270:
3268:
3265:
3263:
3260:
3258:
3257:Demihypercube
3255:
3253:
3250:
3248:
3245:
3243:
3240:
3238:
3235:
3234:
3232:
3230:
3226:
3222:
3216:
3213:
3211:
3208:
3206:
3203:
3201:
3198:
3196:
3193:
3191:
3188:
3186:
3183:
3182:
3180:
3176:
3171:
3161:
3158:
3156:
3153:
3151:
3148:
3146:
3143:
3141:
3138:
3136:
3133:
3131:
3128:
3126:
3123:
3122:
3120:
3116:
3112:
3105:
3100:
3098:
3093:
3091:
3086:
3085:
3082:
3074:
3072:0-8176-4267-6
3068:
3064:
3060:
3059:
3053:
3049:
3043:
3039:
3036:. Cambridge:
3035:
3034:
3028:
3024:
3022:0-7923-5302-1
3018:
3014:
3010:
3009:
3004:
3000:
2999:
2995:
2986:
2983:
2977:
2974:
2968:
2965:
2959:
2956:
2950:
2947:
2941:
2938:
2932:
2929:
2923:
2920:
2914:
2911:
2905:
2902:
2896:
2893:
2887:
2885:
2883:
2881:
2877:
2871:
2867:
2864:
2862:
2859:
2857:
2854:
2852:
2849:
2847:
2844:
2843:
2839:
2837:
2834:
2830:
2811:
2807:
2802:
2799:
2794:
2786:
2785:
2784:
2782:
2774:
2766:
2764:
2760:
2757:
2751:
2745:
2739:
2733:
2727:
2723:
2717:
2695:
2689:
2685:has bivector
2675:
2657:
2640:
2637:
2634:
2622:
2621:
2620:
2617:
2611:
2607:that rotates
2605:
2599:
2579:
2568:
2564:
2549:
2548:
2547:
2544:
2538:
2518:
2507:
2503:
2497:
2487:
2480:
2470:
2466:
2461:
2448:
2438:
2437:
2436:
2419:
2413:
2403:
2399:
2388:
2380:
2379:
2378:
2375:
2370:
2365:
2363:
2359:
2353:
2351:
2347:
2343:
2339:
2335:
2328:
2326:
2318:
2303:
2300:
2294:
2288:
2283:
2278:
2272:
2263:
2240:
2234:
2228:
2225:
2221:
2199:
2196:
2192:
2183:
2180:
2176:
2163:
2162:
2161:
2144:
2141:
2130:
2113:
2078:
2077:
2076:
2073:
2068:
2063:
2059:The quantity
2057:
2054:
2049:
2009:
1987:
1976:
1973:
1964:
1952:
1949:
1945:
1932:
1931:
1930:
1927:
1920:
1913:
1907:
1905:
1874:
1871:
1867:
1854:
1853:
1852:
1849:
1842:
1835:
1833:
1827:
1816:
1812:
1802:
1793:
1786:
1784:
1777:
1775:
1773:
1757:
1746:
1730:
1719:
1713:
1709:
1705:
1694:
1677:
1673:
1668:
1665:
1660:
1652:
1651:
1650:
1640:
1638:
1632:
1626:
1622:
1613:
1608:
1601:
1585:
1580:
1574:
1571:
1568:
1563:
1560:
1557:
1552:
1547:
1540:
1537:
1534:
1531:
1526:
1523:
1520:
1515:
1510:
1503:
1498:
1493:
1490:
1487:
1482:
1479:
1476:
1469:
1464:
1459:
1456:
1453:
1450:
1445:
1442:
1439:
1433:
1424:
1423:
1422:
1379:
1376:
1368:
1366:
1346:
1340:
1337:
1334:
1329:
1326:
1323:
1318:
1313:
1306:
1303:
1300:
1297:
1292:
1289:
1286:
1281:
1276:
1269:
1264:
1259:
1254:
1247:
1242:
1237:
1232:
1226:
1217:
1216:
1215:
1214:coordinates.
1184:
1182:
1174:
1172:
1169:
1163:
1155:
1153:
1149:
1147:
1143:
1139:
1135:
1131:
1127:
1123:
1119:
1115:
1108:
1107:Earth's orbit
1104:
1099:
1081:
1076:
1070:
1065:
1060:
1053:
1048:
1045:
1042:
1037:
1034:
1031:
1024:
1019:
1016:
1013:
1010:
1005:
1002:
999:
993:
984:
983:
982:
967:
965:
961:
955:
952:
935:
928:
912:
907:
901:
898:
895:
890:
887:
884:
877:
874:
871:
868:
863:
860:
857:
851:
842:
841:
840:
838:
815:
811:
807:
804:
801:
798:
794:
790:
787:
784:
779:
776:
772:
764:
763:
762:
760:
744:
740:
739:complex plane
736:
732:
728:
720:
718:
697:
692:
686:
683:
678:
673:
668:
661:
656:
653:
648:
643:
636:
631:
626:
623:
618:
611:
606:
601:
596:
593:
587:
578:
577:
576:
574:
570:
565:
563:
559:
555:
543:
540:
537:
534:
531:
528:
525:
522:
521:
517:
514:
511:
508:
505:
502:
499:
496:
495:
492:
491:
490:
468:
463:
460:
455:
447:
446:
445:
443:
439:
434:
432:
427:
423:
419:
411:
409:
406:
370:
362:
359:
351:
348:
336:
335:
334:
331:
325:
305:
302:
294:
282:
281:
280:
277:
271:
265:
261:
255:
248:
243:
239:| |
237:
229:
223:
217:
213:
208:
207:cross product
204:
200:
177:
174:
171:
163:
151:
150:
149:
146:
140:
135:
131:
124:
120:
116:
115:
108:
100:
95:
93:
91:
87:
83:
79:
75:
71:
67:
62:
60:
56:
51:
49:
45:
41:
37:
32:
30:
26:
22:
3372:
3338:
3277:Hyperpyramid
3242:Hypersurface
3135:Affine space
3125:Vector space
3057:
3032:
3007:
2985:
2976:
2967:
2958:
2949:
2940:
2931:
2922:
2913:
2904:
2895:
2826:
2770:
2761:
2755:
2749:
2743:
2737:
2731:
2725:
2721:
2715:
2693:
2687:
2676:
2672:
2615:
2609:
2603:
2597:
2594:
2542:
2536:
2533:
2434:
2373:
2366:
2360:through the
2354:
2332:
2316:
2304:
2298:
2292:
2286:
2276:
2270:
2261:
2238:
2232:
2229:
2223:
2219:
2216:
2159:
2071:
2061:
2058:
2052:
2047:
2041:
1925:
1918:
1911:
1908:
1901:
1847:
1840:
1836:
1832:unit vectors
1825:
1817:
1810:
1798:
1781:
1755:
1749:) there are
1744:
1728:
1720:= 2, 4, 6...
1717:
1714:
1710:
1703:
1695:
1692:
1644:
1624:
1620:
1617:
1380:
1372:
1363:
1185:
1178:
1165:
1150:
1144:which store
1128:between the
1111:
968:
956:
948:
831:
724:
712:
566:
558:right angles
551:
484:
435:
417:
415:
404:
385:
329:
323:
320:
275:
269:
263:
259:
256:
246:
241:
235:
227:
221:
215:
211:
192:
148:, such that
144:
138:
112:
110:
63:
52:
33:
24:
18:
3362:Codimension
3341:-dimensions
3262:Hypersphere
3145:Free module
2856:Quaternions
2829:eigenplanes
2773:eigenvalues
2709:-plane and
2282:commutative
1801:reflections
1787:Reflections
1413:-plane and
267:is written
245:| sin
123:zero vector
96:Definitions
31:in space.
3388:Categories
3357:Hyperspace
3237:Hyperplane
3063:Birkhäuser
2996:References
1122:South Pole
1118:North Pole
960:axis angle
737:it is the
554:orthogonal
497:Dimension
279:is simply
107:Hyperplane
105:See also:
90:dimensions
3247:Hypercube
3225:Polytopes
3205:Minkowski
3200:Hausdorff
3195:Inductive
3160:Spacetime
3111:Dimension
2833:arguments
2641:
2334:Bivectors
2329:Bivectors
2314:is even,
2197:−
1988:−
1977:−
1953:−
1875:−
1612:tesseract
1575:β
1572:
1564:β
1561:
1541:β
1538:
1532:−
1527:β
1524:
1494:α
1491:
1483:α
1480:
1460:α
1457:
1451:−
1446:α
1443:
1341:θ
1338:
1330:θ
1327:
1307:θ
1304:
1298:−
1293:θ
1290:
1138:gyroscope
1049:θ
1046:
1038:θ
1035:
1020:θ
1017:
1011:−
1006:θ
1003:
902:θ
899:
891:θ
888:
878:θ
875:
869:−
864:θ
861:
812:θ
808:
795:θ
791:
780:θ
684:−
654:−
624:−
594:−
363:μ
352:λ
295:∧
172:≠
164:∧
74:bivectors
29:rotations
3374:Category
3350:See also
3150:Manifold
3005:(1999).
2840:See also
2808:⌋
2795:⌊
2344:and the
2274:towards
2177:″
1965:′
1946:″
1868:′
1851:, thus:
1674:⌋
1661:⌊
1142:flywheel
1134:Southern
1130:Northern
1103:inclined
469:⌋
456:⌊
438:identity
422:rotation
250:, where
21:geometry
3272:Simplex
3210:Fractal
2705:in the
2258:
2244:
1767:
1752:
1739:
1725:
1417:in the
1409:in the
1126:equator
3229:shapes
3069:
3044:
3019:
3013:Kluwer
2217:where
2069:, and
945:-plane
743:origin
426:mapped
233:|
193:where
119:origin
114:planes
66:planes
3333:Eight
3328:Seven
3308:Three
3185:Krull
2872:Notes
2358:rotor
2262:twice
2067:rotor
2065:is a
1373:In a
834:2 × 2
386:with
101:Plane
84:of a
76:from
3318:Five
3313:Four
3293:Zero
3227:and
3067:ISBN
3042:ISBN
3017:ISBN
2747:and
2701:and
2601:and
2236:and
2050:and
1921:− 1)
1843:− 1)
1828:− 1)
1813:− 1)
1706:− 2)
1401:and
1385:and
1210:and
1132:and
1120:and
398:and
390:and
327:and
225:and
142:and
68:and
57:and
38:and
23:, a
3323:Six
3303:Two
3298:One
2638:log
2619:is
2613:to
2321:if
2319:− 2
2296:to
1909:If
1758:− 1
1627:≠ 0
1569:cos
1558:sin
1535:sin
1521:cos
1488:cos
1477:sin
1454:sin
1440:cos
1335:cos
1324:sin
1301:sin
1287:cos
1140:or
1043:cos
1032:sin
1014:sin
1000:cos
949:In
896:cos
885:sin
872:sin
858:cos
805:sin
788:cos
749:to
725:In
575:),
201:or
55:two
19:In
3390::
3065:.
3061:.
3040:.
3015:.
2879:^
2753:34
2741:12
2729:34
2724:+
2719:12
2711:zw
2707:xy
2691:34
2679:zw
2543:mn
2340:,
2302:.
2287:nm
2224:mn
2222:=
2072:nm
2062:mn
1912:x′
1906:.
1623:=
1419:zw
1411:xy
1204:xy
1200:zw
1192:zw
1188:xy
975:xy
943:xy
839::
761::
564:.
544:4
518:8
416:A
306:0.
262:∧
214:∧
3339:n
3103:e
3096:t
3089:v
3075:.
3050:.
3025:.
2812:,
2803:2
2800:n
2777:n
2756:β
2750:e
2744:α
2738:e
2732:β
2726:e
2722:α
2716:e
2703:β
2699:α
2694:θ
2688:e
2683:θ
2658:.
2655:)
2651:n
2648:m
2644:(
2635:=
2631:B
2616:n
2610:m
2604:n
2598:m
2580:,
2574:B
2569:e
2565:=
2561:n
2558:m
2537:B
2519:,
2513:B
2508:e
2504:=
2498:2
2494:B
2488:e
2481:2
2477:B
2471:e
2467:=
2462:2
2454:B
2449:R
2420:.
2414:2
2410:B
2404:e
2400:=
2394:B
2389:R
2374:B
2323:n
2317:n
2312:n
2308:n
2299:m
2293:n
2277:n
2271:m
2266:π
2255:2
2252:/
2248:π
2239:n
2233:m
2220:R
2200:1
2193:R
2188:x
2184:R
2181:=
2173:x
2145:1
2142:=
2138:m
2135:m
2131:=
2127:m
2124:n
2121:n
2118:m
2114:=
2111:)
2107:m
2104:n
2100:(
2097:)
2093:n
2090:m
2086:(
2053:n
2048:m
2044:n
2026:n
2023:m
2020:x
2017:m
2014:n
2010:=
2006:n
2002:)
1998:m
1995:x
1992:m
1985:(
1981:n
1974:=
1970:n
1961:x
1958:n
1950:=
1942:x
1926:n
1919:n
1917:(
1885:m
1882:x
1879:m
1872:=
1864:x
1848:m
1841:n
1839:(
1826:n
1824:(
1820:n
1811:n
1809:(
1805:n
1764:2
1761:/
1756:n
1745:n
1736:2
1733:/
1729:n
1718:n
1704:n
1702:(
1698:n
1678:,
1669:2
1666:n
1647:n
1635:α
1625:β
1621:α
1586:.
1581:)
1553:0
1548:0
1516:0
1511:0
1504:0
1499:0
1470:0
1465:0
1434:(
1415:β
1407:α
1403:β
1399:α
1395:β
1391:α
1387:β
1383:α
1347:)
1319:0
1314:0
1282:0
1277:0
1270:0
1265:0
1260:1
1255:0
1248:0
1243:0
1238:0
1233:1
1227:(
1212:w
1208:z
1196:θ
1082:.
1077:)
1071:1
1066:0
1061:0
1054:0
1025:0
994:(
979:θ
971:z
939:z
913:.
908:)
852:(
816:,
802:i
799:+
785:=
777:i
773:e
755:θ
751:π
747:π
715:π
698:,
693:)
687:1
679:0
674:0
669:0
662:0
657:1
649:0
644:0
637:0
632:0
627:1
619:0
612:0
607:0
602:0
597:1
588:(
541:3
538:3
535:2
532:2
529:1
526:1
515:7
512:6
509:5
506:4
503:3
500:2
487:n
464:2
461:n
405:c
400:μ
396:λ
392:μ
388:λ
371:,
367:b
360:+
356:a
349:=
345:c
330:b
324:a
303:=
299:B
291:x
276:B
270:B
264:b
260:a
252:φ
247:φ
242:b
236:a
228:b
222:a
216:b
212:a
195:∧
178:,
175:0
168:b
160:a
145:b
139:a
128:n
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