3548:
1235:
1738:
22:
3958:. Different authors have termed the plane-based GA part of PGA "Euclidean space" and "Antispace". This form of duality, combined with the fact that geometric objects are represented homogeneously (meaning that multiplication by scalars does not change them), is the reason that the system is known as "Projective" Geometric Algebra (even though it does not contain the full projective group, unlike Conformal Geometric Algebra, which contains the full conformal group).
3818:
4189:
3770:
3099:
160:
3692:), e.g. when they are applied to sets of objects, the relative distances between those objects does not change; nor does their handedness, which is to say that a right-handed glove will not turn into a left-handed glove. All transformations in 3D euclidean plane-based geometric algebra preserve distances, but reflections, rotoreflections, and transflections do not preserve handedness.
300:
1154:
3664:
3790:. Reflections in planes are a special case of inversions in spheres, because a plane is a sphere with infinite radius. Since plane-based geometric algebra is generated by composition of reflections, it is a special case of inversive geometry. Inversive geometry itself can be performed with the larger system known as
3961:
Alternatively, conformal geometric algebra can be used (since plane-based GA is a subalgebra of CGA), but defining the PGA regressive product within it is complicated by the fact that CGA has its own regressive product, which is a different product. Loosely because the join of three points in CGA is
303:
In Plane-based GA, grade-1 elements are planes and can be used to perform planar reflections; grade-2 elements are lines and can be used to perform "line reflections"; grade-3 elements are points and can be used to perform "point reflections". Rotations and translations are constructed out of these
4204:
In these systems, the points, planes, and lines have the same coordinates that they have in plane-based GA. But transformations like rotations and reflections will have very different effects on the geometry. In all cases below, the algebra is a double cover of the group of reflections, rotations,
3917:
There is variation across authors as to the precise definition given for dual that is used to define the regressive product in PGA. No matter which definition is given, the regressive product functions to give completely identical outputs; for this reason, precise discussion of the dual is usually
3555:
degree of freedom. The yellow cube is a reflection of the black cube; the green cube is a reflection of the yellow cube. But while the yellow cube changes as the planes change, the final green cube will be unchanged while the reflection planes have the same angle/distance and intersect in the same
1238:
Plane-based GA includes elements "at infinity". A star in the night sky is an intuitive example of a "point at infinity", in the sense that it defines some direction, but practically speaking it is impossible to reach. The milky way forms a hazy stripe of stars across the sky; it behaves, in some
4208:
All formulae from the euclidean case carry over to these other geometries â the meet still functions as a way of taking the intersection of objects; the geometric product still functions as a way of composing transformations; and in the hyperbolic case the inner product become able to measure
291:(generally) vectors in the sense that one could meaningfully take their cross product - so it is not informative to visualize them as arrows. Therefore to avoid conflict over different algebraic and visual connotations coming from the word 'vector', this article avoids use of the word.
1019:
1741:
The orange objects here are projected onto the green objects to get the dark grey objects, all using the unified projection formula (a·b)bâ»Âč. Since PGA includes points, lines, and planes, this involves projection of planes onto points, points onto planes, lines onto planes,
528:
would be the plane midway between the y- and z-plane. In general, combining two geometric objects in plane-based GA will always be as a weighted average of them â combining points will give a point between them, as will combining lines, and indeed rotations.
3931:
relates elements of plane-based geometric algebra to other elements of plane based geometric algebra (eg, other euclidean transformations); for example, the Hodge dual of a planar reflection is a point reflection. PGA was originally defined using the Hodge
210:, in which points, translations, rotation axes, and plane normals are all modelled as "vectors". However, use of vectors in advanced engineering problems often require subtle distinctions between different kinds of vector because of this, including
3106:
on the tetrahedron. In 3D plane-based GA, points 3-reflections. Algebraically this means they are grade-3 â but their geometric interpretation is very different from the usual geometric interpretation of a "trivector" as an "oriented volume
3090:. The information needed to specify that the intersection line is contained inside the transform composition of the two planes, because a reflection in a pair of planes will result in a rotation around their intersection line.
593:
3648:). In the plane-based geometric algebra notation, this rotoreflection can be thought of as a planar reflection "added to" a point reflection. The plane part of this rotoreflection is the plane that is orthogonal to the line
2994:
2906:
1159:
The geometric interpretation of the first three defining equations is that if we perform the same planar reflection twice we get back to where we started; e.g. any grade-1 element (plane) multiplied by itself results in the
3667:
A transformation in 2D that takes a blue triangle to a red triangle, simplified using "gauging". The full transformation was composed from four reflections. Two of the reflection lines, gauged so that they coincide, can be
3809:. It can be difficult to see the connection between PGA and CGA, since CGA is often "point based", although some authors take a plane-based approach to CGA which makes the notations for Plane-based GA and CGA identical.
1149:{\displaystyle {\boldsymbol {e}}_{1}{\boldsymbol {e}}_{1}=1\qquad {\boldsymbol {e}}_{2}{\boldsymbol {e}}_{2}=1\qquad {\boldsymbol {e}}_{3}{\boldsymbol {e}}_{3}=1\qquad {\boldsymbol {e}}_{0}{\boldsymbol {e}}_{0}=0}
3966:, whereas in PGA it is a plane. Another problem is that PGA "points" have a fundamentally different algebraic representation than CGA points; to compare the two algebras, PGA points must be recognized as CGA
3456:
3046:. But this may be more than is desired; if we wish to take only the intersection line of the two planes, we simply need to look at just the "grade-2 part" of this result, e.g. the part with two lower indices
3970:, where the pair has one point at infinity. To get around this problem, some authors define the projective dual described above, in CGA, as an exchange of two different PGA-isomorphic subalgebras within it.
3044:
3307:
180 degrees, do not have a single specific geometric object which is used to visualize them; nevertheless, they can always be thought of as being made up of reflections, and can always be represented as a
3088:
1395:
1229:
1524:, cannot act as axes for a "rotation". Instead, these are axes for translations, and instead of having an algebra resembling complex numbers or quaternions, their algebraic behaviour is the same as the
884:
526:
3660:. A similar procedure can be used to find the line orthogonal to a plane and passing through a point, or the intersection of a line and a plane, or the intersection line of a plane with another plane.
711:, which is a point reflection in the origin, because that is the transformation that results from a 180-degree rotation followed by a planar reflection in a plane orthogonal to the rotation's axis.
3242:
are line reflections - which in 3D are the same thing as 180-degree rotations. The identity transform is the unique object that is constructed out of zero reflections. All of these are elements of
2437:
3547:
2330:
3348:
222:. The latter of these two, in plane-based GA, map to the concepts of "rotation axis" and "point", with the distinction between them being made clear by the notation: rotation axes such as
3597:. This simple fact can be used to give a geometric interpretation for the general behaviour of the geometric product as a device that solves geometric problems by "cancelling mirrors".
3734:
dimensions. This group has two commonly-used representations that allow them to be used in algebra and computation, one being the 4Ă4 matrices of real numbers, and the other being the
3537:
3881:
a point to another point to obtain a line, and can join a point and a line to obtain a plane. It has the further convenience that if any two elements (points, lines, or planes) have
3207:
2220:
709:
278:
3485:
3377:
3240:
2157:
1723:
1694:
1665:
1613:
1584:
1555:
1522:
1456:
1424:
975:
917:
680:
249:
3591:
3406:
3178:
2935:
2847:
1493:
1348:
1308:
1275:
1004:
946:
625:
479:
424:
369:
102:
1321:
is an example of such a line. For an observer standing on a plane, all planes parallel to the plane they stand on meet one another at the horizon line. Algebraically, if we take
2594:
2246:
1832:
1901:
2503:
2042:
25:
Elements of 3D Plane-based GA, which includes planes, lines, and points. All elements are constructed from reflections in planes. Lines are a special case of rotations.
2795:
2697:
2546:
1972:
2069:
3301:
3269:
3142:
4609:
651:
450:
395:
340:
1636:
3903:
3636:
2814:
2756:
2736:
2716:
2658:
2638:
2393:
2370:
2350:
2286:
2266:
2184:
2113:
2093:
2016:
1996:
1945:
1925:
1872:
1852:
1800:
1780:
1182:
811:
791:
752:
154:
134:
4685:
3742:
of SE(3). Since the Dual
Quaternions are closed under multiplication and addition and are made from an even number of basis elements in, they are called the
2776:
2678:
2618:
2523:
2468:
831:
772:
732:
3946:
the case that both are reflections; instead, the projective dual switches between the space that plane-based geometric algebra operates in and a different,
4099:
542:
3644:
in some sense produces a 5-reflection; however, as in the picture below, two of these reflections cancel, leaving a 3-reflection (sometimes known as a
2940:
2852:
219:
3954:. For example, planes in plane-based geometric algebra, which perform planar reflections, are mapped to points in dual space which are involved in
1314:", or alternatively "ideal points", or "points at infinity". Parallel lines such as metal rails on a railway line meet one another at such points.
71:
as basic elements, and constructs all other transformations and geometric objects out of them. Formally: it identifies planar reflections with the
4783:
Aperçu historique sur l'origine et le développement des méthodes en géométrie, particuliÚrement de celles qui se rapportent à la géométrie moderne
3995:, since rigid transformations can be modelled in this algebra. However, it is possible to model other spaces by slightly varying the algebra.
4724:
4585:
4547:
4520:
4482:
4418:
4285:
3510:, which is a 4-reflection. For this reason, when considering screw motions, it is necessary to use the grade-4 element of 3D plane-based GA,
3506:
can also always be written as compositions of 3 planar reflections and so are called 3-reflections. The upper limit of this for 3D is a
1947:; if a transformation by the exact distance or angle is required, it can be obtained with the dual quaternion exponential and logarithm.
1397:
will be a plane parallel to the ground (displaced 5 meters from it). These two parallel planes meet one another at the line-at-infinity
3918:
not included in introductory material on projective geometric algebra. There are significant conceptual and philosophical differences:
3411:
3865:
Plane-based geometric algebra is able to represent all
Euclidean transformations, but in practice it is almost always combined with a
2999:
4816:
3490:
In fact, any rotation can be written as a composition of two planar reflections that pass through its axis; thus it can be called a
4192:
Plane-based GA usually handles the (3D version of) the middle case here. But we instead choose to have a basis element squaring to
3869:
operation of some kind to create the larger system known as "Projective
Geometric Algebra", PGA. Duality, as in other Clifford and
3049:
1353:
1187:
3711:, which are set in the context of PGA above). If the planes were parallel, composing their reflections would give a translation.
1755:
848:
487:
164:
3905:, the norm of their regressive product is equal to the distance between them. The join of several points is also known as their
3145:
3111:
The algebra of all distance-preserving transformations (essentially, rigid transformations and reflections) in 3D is called the
4302:
3718:, e.g. a rotation around a line in space followed by a translation directed along the same line. This group is usually called
176:
168:
4352:
4515:. The Morgan Kaufmann series in computer graphics (2nd corrected printing ed.). Amsterdam: Morgan Kaufmann/Elsevier.
3936:
3703:
of reflections. A rotations can thought of as a reflection in a plane followed by a reflection in another plane which is
1281:. It behaves differently from any other plane â intuitively, it can be "approached but never reached". In 3 dimensions,
1234:
2398:
3791:
2072:
45:
2291:
5065:
4862:
3315:
4469:, Lecture Notes in Computer Science, vol. 12221, Cham: Springer International Publishing, pp. 459â471,
4179:
21:
4060:
Most important for engineering applications, since transformations are rigid; also most "intuitive" for humans
1737:
5013:
3513:
56:
in mind. It has since been applied to machine learning, rigid body dynamics, and computer science, especially
3183:
2189:
685:
254:
3689:
3564:
plane results in no change. The algebraic interpretation for this geometry is that grade-1 elements such as
3461:
3353:
3216:
2118:
1699:
1670:
1641:
1589:
1560:
1531:
1498:
1432:
1400:
951:
893:
656:
225:
196:
3600:
To give an example of the usefulness of this, suppose we wish to find a plane orthogonal to a certain line
3567:
3382:
3154:
3151:
In plane-based GA, essentially all geometric objects can be thought of as a transformation. Planes such as
2911:
2823:
1469:
1324:
1284:
1251:
980:
922:
601:
455:
400:
345:
78:
4171:
3947:
3806:
1726:
188:
3877:. This is extremely useful for engineering applications - in plane-based GA, the regressive product can
3699:
preserve handedness, which in 3D Plane-based GA implies that they can be written as a composition of an
175:
Plane-based GA subsumes a large number of algebraic constructions applied in engineering, including the
2551:
1746:
There are several useful products that can be extracted from the geometric product, similar to how the
5025:
4314:
4217:
4054:
2447:
2225:
37:
75:
elements of a
Clifford Algebra, that is, elements that are written with a single subscript such as "
4122:
4118:
3928:
3924:
1805:
313:
1877:
4994:
4968:
4911:
4667:
4641:
4603:
4488:
4380:
4154:
4089:
3975:
Projective geometric algebra of non-euclidean geometries and
Classical Lie Groups in 3 dimensions
3787:
3774:
3309:
2373:
40:. Generally this is with the goal of solving applied problems involving these elements and their
2473:
308:
Plane-based geometric algebra starts with planes and then constructs lines and points by taking
2021:
5041:
4986:
4903:
4812:
4763:
4720:
4659:
4591:
4581:
4553:
4543:
4516:
4478:
4414:
4372:
4330:
4281:
3874:
3495:
1904:
1278:
1161:
57:
2780:
2682:
2531:
1957:
1907:(essentially the inverse). The transformation will be by twice the angle or distance between
5033:
4978:
4937:
4895:
4804:
4753:
4712:
4651:
4470:
4406:
4364:
4322:
4273:
4210:
4044:
3870:
3801:
CGA is also usually applied to 3D space, and is able to model general spheres, circles, and
3503:
3499:
3210:
3103:
2054:
192:
33:
3817:
4188:
3277:
3245:
3118:
3112:
2470:
to any object, including points, lines, planes and indeed other rigid transformations, is
1311:
184:
3979:
To a first approximation, the physical world is euclidean, i.e. most transformations are
630:
429:
374:
319:
5029:
4318:
3769:
1618:
4434:
3888:
3861:
of
Projective Geometric Algebra, a system which subsumes Plane-based Geometric Algebra.
3758:
3754:
3645:
3621:
3551:
When viewed as a composition of reflections, rotations and translations, both have one
2799:
2741:
2721:
2701:
2643:
2623:
2378:
2355:
2335:
2271:
2251:
2169:
2098:
2078:
2001:
1981:
1930:
1910:
1857:
1837:
1785:
1765:
1167:
796:
776:
737:
206:
The plane-based approach to geometry may be contrasted with the approach that uses the
139:
119:
283:
All objects considered below are still "vectors" in the technical sense that they are
48:, and their angle from one another in 3D space. Originally growing out of research on
5059:
4998:
4671:
4492:
4164:
3802:
1751:
304:
elements; line reflections in particular are the same things as 180-degree rotations.
207:
41:
4915:
4384:
4200:
instead of 0, euclidean geometry can be changed to spherical or hyperbolic geometry.
3955:
3507:
3098:
2761:
2663:
2603:
2508:
2453:
1975:
816:
757:
717:
588:{\displaystyle {\boldsymbol {e}}_{1}{\boldsymbol {e}}_{23}={\boldsymbol {e}}_{123}}
215:
211:
200:
4796:
4462:
3805:(angle-preserving) transformations, which include the transformations seen on the
4808:
4474:
3786:
Inversive geometry is the study of geometric objects and behaviours generated by
2989:{\displaystyle {\boldsymbol {e}}_{1}+{\boldsymbol {e}}_{2}+{\boldsymbol {e}}_{0}}
2901:{\displaystyle {\boldsymbol {e}}_{1}+{\boldsymbol {e}}_{2}+{\boldsymbol {e}}_{0}}
4956:
4629:
4221:
3906:
3738:. The Dual Quaternion representation (like the usual quaternions) is actually a
3379:
axis, and it can be written as a geometric product (a transform composition) of
2597:
2440:
1747:
1525:
1463:
1013:
This geometric interpretation is usually combined with the following assertion:
484:
Other planes may be obtained as weighted sums of the basis planes. for example,
159:
4982:
4883:
4655:
4539:
Geometric algebra for computer science: an object-oriented approach to geometry
4513:
Geometric algebra for computer science: an object-oriented approach to geometry
682:, which is a 180-degree rotation around the x-axis. Their geometric product is
4941:
4899:
4706:
4368:
4267:
4245:
3951:
3795:
3753:
Describing rigid transformations using planes was a major goal in the work of
3708:
2908:. Their geometric product is their "reflection composition" â a reflection in
180:
49:
5045:
4990:
4957:"Projective Geometric Algebra as a Subalgebra of Conformal Geometric algebra"
4955:
Hrdina, Jaroslav; NĂĄvrat, AleĆĄ; VaĆĄĂk, Petr; Dorst, Leo (February 22, 2021).
4907:
4767:
4716:
4663:
4595:
4557:
4376:
4334:
4277:
3543:
Geometric interpretation of geometric product as "cancelling out" reflections
1528:, since they square to 0. Combining the three basis lines-through-the-origin
64:
operation into a system known as "Projective
Geometric Algebra", see below.
3312:
of some elements of objects in plane-based geometric algebra. For example,
104:". With some rare exceptions described below, the algebra is almost always
4837:
4575:
1239:
sense, like a "line at infinity". The sky itself is a "plane at infinity".
299:
4934:
Geometry, Kinematics, and Rigid Body
Mechanics in Cayley-Klein Geometries
3148:, any element of it can be written as a series of reflections in planes.
1459:
53:
2439:. Thus it can be seen that the inner product is a generalization of the
2248:. It can be used to take angles between most objects: the angle between
4303:"A Galileian formulation of spin. I. Clifford algebras and spin groups"
3663:
1318:
251:(two lower indices) are always notated differently than points such as
4741:
2159:â this formula holds whether the objects are points, lines, or planes.
5037:
5012:
Doran, C.; Hestenes, D.; Sommen, F.; Van Acker, N. (August 1, 1993).
4758:
4537:
4398:
4326:
3747:
171:
representations of rotations in its rotors and bivectors respectively
4463:"Constrained Dynamics in Conformal and Projective Geometric Algebra"
4410:
4973:
4646:
4577:
Foundations of game engine development : Volume 1: mathematics
4187:
3816:
3719:
3662:
3097:
1233:
158:
4170:
Also known as "saddle geometry". Group can perform rotations and
3451:{\displaystyle 0.8{\boldsymbol {e}}_{1}+0.6{\boldsymbol {e}}_{2}}
1310:
can be visualized as the sky. Lying in it are the points called "
4224:
for each group is the grade 2 elements of the
Clifford algebra,
3458:, both of which are planar reflections intersecting at the line
3777:, the 2D version of which is a circle inversion, depicted here.
3039:{\displaystyle 1+{\boldsymbol {e}}_{12}+{\boldsymbol {e}}_{10}}
4863:"Euclidean Geometry and Geometric Algebra | Geometric Algebra"
4511:
Dorst, Leo; Fontijne, Daniel; Manning, Stephen Joseph (2009).
116:, meaning it has three basis grade-1 elements whose square is
4803:, Cambridge University Press, pp. 84â125, May 29, 2003,
3983:; Projective Geometric Algebra is therefore usually based on
3680:
Rotations and translations are transformations that preserve
3083:{\displaystyle {\boldsymbol {e}}_{12}+{\boldsymbol {e}}_{10}}
2525:
is object to be transformed; this is the "sandwich product".
1390:{\displaystyle {\boldsymbol {e}}_{2}+5{\boldsymbol {e}}_{0}}
1224:{\displaystyle {\boldsymbol {e}}_{0}{\boldsymbol {e}}_{0}=0}
4012:
Names for handedness-preserving subgroup (even subalgebra)
3761:
since it allows the treatment to be dimension-independent.
3746:
of 3D euclidean (plane-based) geometric algebra. The word '
879:{\displaystyle {\boldsymbol {e}}_{1}{\boldsymbol {e}}_{23}}
521:{\displaystyle {\boldsymbol {e}}_{2}+{\boldsymbol {e}}_{3}}
3102:
The center of the picture is a point that is performing a
1733:
Derivation of other operations from the geometric product
3560:
A reflection in a plane followed by a reflection in the
2620:
is the logarithm of a transformation being undergone by
4686:"Representations and spinors | Mathematics for Physics"
4220:(after taking the quotient by scalars). The associated
1978:
of objects; for example, the intersection of the plane
532:
An operation that is as fundamental as addition is the
2764:
2666:
2606:
2554:
2511:
2476:
2456:
819:
760:
720:
3891:
3714:
Rotations and translations are both special cases of
3638:
is a 3-reflection, so taking their geometric product
3624:
3570:
3516:
3464:
3414:
3385:
3356:
3318:
3280:
3248:
3219:
3186:
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2783:
2744:
2724:
2704:
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2626:
2534:
2401:
2381:
2358:
2338:
2294:
2274:
2254:
2228:
2192:
2172:
2121:
2101:
2081:
2057:
2024:
2004:
1984:
1960:
1933:
1913:
1880:
1860:
1840:
1808:
1788:
1768:
1702:
1673:
1644:
1621:
1592:
1563:
1534:
1501:
1472:
1435:
1403:
1356:
1327:
1287:
1254:
1190:
1170:
1022:
983:
954:
925:
896:
851:
799:
779:
740:
688:
659:
633:
604:
545:
490:
458:
432:
403:
377:
348:
322:
257:
228:
142:
122:
81:
4247:
4628:Roelfs, Martin; De Keninck, Steven (May 13, 2023).
187:representations of rotations and translations, the
3897:
3730:uclidean (distance-preserving) transformations in
3630:
3585:
3531:
3479:
3450:
3400:
3371:
3342:
3295:
3263:
3234:
3201:
3172:
3136:
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3038:
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2789:
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2730:
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2612:
2588:
2540:
2517:
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2431:
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2240:
2214:
2178:
2151:
2107:
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2010:
1990:
1966:
1939:
1919:
1895:
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1846:
1826:
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1607:
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1516:
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1418:
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1342:
1302:
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1223:
1176:
1148:
998:
969:
940:
911:
878:
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805:
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703:
674:
645:
619:
587:
520:
473:
444:
418:
389:
363:
334:
272:
243:
148:
128:
96:
4936:(Masters thesis). Technische UniversitÀt Berlin.
4105:Also known as "spherical geometry". Analogous to
3750:' is sometimes used to describe this subalgebra.
2432:{\displaystyle \lVert A\rVert =\lVert B\rVert =1}
2075:of objects onto other objects; the projection of
1006:. Transform application is implemented with the
203:model of classical mechanics to be constructed.
2325:{\displaystyle \arccos(\lVert A\cdot B\rVert )}
4884:"Clifford algebra of points, lines and planes"
4353:"Clifford algebra of points, lines and planes"
2816:is the unique line that is orthogonal to both.
1466:. But lines that lie in the plane-at-infinity
4746:Bulletin de la Société Mathématique de France
3343:{\displaystyle 0.8+0.6{\boldsymbol {e}}_{12}}
3303:, for example rotations by any angle that is
8:
2420:
2414:
2408:
2402:
2316:
2304:
2235:
2229:
4742:"Essai sur la géométrie à $ n$ dimensions"
1874:again being points, lines or planes; here,
199:of points. Dual Quaternions then allow the
136:and a single basis element whose square is
4630:"Graded Symmetry Groups: Plane and Simple"
4608:: CS1 maint: location missing publisher (
4180:Klein disk model of 2D hyperbolic geometry
3606:in 3D and passing through a certain point
4972:
4757:
4645:
3890:
3773:Planar reflections are a special case of
3623:
3577:
3572:
3569:
3523:
3518:
3515:
3471:
3466:
3463:
3442:
3437:
3424:
3419:
3413:
3392:
3387:
3384:
3363:
3358:
3355:
3334:
3329:
3317:
3279:
3247:
3226:
3221:
3218:
3193:
3188:
3185:
3164:
3159:
3156:
3120:
3074:
3069:
3059:
3054:
3051:
3030:
3025:
3015:
3010:
3001:
2980:
2975:
2965:
2960:
2950:
2945:
2942:
2921:
2916:
2913:
2892:
2887:
2877:
2872:
2862:
2857:
2854:
2833:
2828:
2825:
2801:
2782:
2763:
2743:
2723:
2703:
2684:
2665:
2645:
2625:
2605:
2555:
2553:
2533:
2510:
2484:
2483:
2475:
2455:
2400:
2380:
2357:
2337:
2293:
2273:
2253:
2227:
2199:
2198:
2193:
2191:
2171:
2138:
2137:
2120:
2100:
2080:
2056:
2023:
2003:
1983:
1959:
1932:
1912:
1882:
1881:
1879:
1859:
1839:
1813:
1812:
1807:
1787:
1767:
1709:
1704:
1701:
1680:
1675:
1672:
1651:
1646:
1643:
1638:, with the three basis lines at infinity
1620:
1599:
1594:
1591:
1570:
1565:
1562:
1541:
1536:
1533:
1508:
1503:
1500:
1479:
1474:
1471:
1442:
1437:
1434:
1410:
1405:
1402:
1381:
1376:
1363:
1358:
1355:
1334:
1329:
1326:
1294:
1289:
1286:
1261:
1256:
1253:
1209:
1204:
1197:
1192:
1189:
1169:
1134:
1129:
1122:
1117:
1103:
1098:
1091:
1086:
1072:
1067:
1060:
1055:
1041:
1036:
1029:
1024:
1021:
990:
985:
982:
961:
956:
953:
932:
927:
924:
903:
898:
895:
870:
865:
858:
853:
850:
818:
798:
778:
759:
739:
719:
695:
690:
687:
666:
661:
658:
632:
611:
606:
603:
579:
574:
564:
559:
552:
547:
544:
512:
507:
497:
492:
489:
465:
460:
457:
431:
410:
405:
402:
376:
355:
350:
347:
321:
264:
259:
256:
235:
230:
227:
141:
121:
88:
83:
80:
4009:Apparent "plane at infinity" squares to
3997:
3768:
3546:
3532:{\displaystyle {\boldsymbol {e}}_{0123}}
3094:Interpretation as algebra of reflections
1736:
1462:; in fact they can treated as imaginary
298:
52:, it was developed with applications to
36:to modelling planes, lines, points, and
20:
4237:
3573:
3556:line (which may be a line at infinity).
3519:
3467:
3438:
3420:
3388:
3359:
3330:
3222:
3202:{\displaystyle {\boldsymbol {e}}_{123}}
3189:
3180:are planar reflections, points such as
3160:
3070:
3055:
3026:
3011:
2996:, which results in the dual quaternion
2976:
2961:
2946:
2917:
2888:
2873:
2858:
2829:
2288:, whether they are lines or planes, is
2215:{\displaystyle {\sqrt {A{\tilde {A}}}}}
1705:
1676:
1647:
1595:
1566:
1537:
1504:
1475:
1438:
1406:
1377:
1359:
1330:
1290:
1257:
1205:
1193:
1130:
1118:
1099:
1087:
1068:
1056:
1037:
1025:
986:
957:
928:
899:
866:
854:
704:{\displaystyle {\boldsymbol {e}}_{123}}
691:
662:
607:
575:
560:
548:
508:
493:
461:
406:
351:
273:{\displaystyle {\boldsymbol {e}}_{123}}
260:
231:
84:
4601:
4461:Hadfield, Hugo; Lasenby, Joan (2020),
3942:also maps planes to points, but it is
3539:, which is the highest-grade element.
3480:{\displaystyle {\boldsymbol {e}}_{12}}
3372:{\displaystyle {\boldsymbol {e}}_{12}}
3235:{\displaystyle {\boldsymbol {e}}_{12}}
2152:{\displaystyle (A\cdot B){\tilde {B}}}
1718:{\displaystyle {\boldsymbol {e}}_{30}}
1689:{\displaystyle {\boldsymbol {e}}_{20}}
1660:{\displaystyle {\boldsymbol {e}}_{10}}
1608:{\displaystyle {\boldsymbol {e}}_{12}}
1579:{\displaystyle {\boldsymbol {e}}_{13}}
1550:{\displaystyle {\boldsymbol {e}}_{23}}
1517:{\displaystyle {\boldsymbol {e}}_{30}}
1451:{\displaystyle {\boldsymbol {e}}_{23}}
1419:{\displaystyle {\boldsymbol {e}}_{02}}
970:{\displaystyle {\boldsymbol {e}}_{23}}
912:{\displaystyle {\boldsymbol {e}}_{23}}
675:{\displaystyle {\boldsymbol {e}}_{23}}
627:, which is a planar reflection in the
244:{\displaystyle {\boldsymbol {e}}_{13}}
4961:Advances in Applied Clifford Algebras
4927:
4925:
4634:Advances in Applied Clifford Algebras
4623:
4621:
4619:
4569:
4567:
4266:Porteous, Ian R. (February 5, 1981).
3586:{\displaystyle {\boldsymbol {e}}_{1}}
3401:{\displaystyle {\boldsymbol {e}}_{1}}
3173:{\displaystyle {\boldsymbol {e}}_{1}}
2930:{\displaystyle {\boldsymbol {e}}_{1}}
2842:{\displaystyle {\boldsymbol {e}}_{1}}
2718:. It is also the case that for lines
1488:{\displaystyle {\boldsymbol {e}}_{0}}
1343:{\displaystyle {\boldsymbol {e}}_{2}}
1303:{\displaystyle {\boldsymbol {e}}_{0}}
1270:{\displaystyle {\boldsymbol {e}}_{0}}
999:{\displaystyle {\boldsymbol {e}}_{1}}
941:{\displaystyle {\boldsymbol {e}}_{1}}
620:{\displaystyle {\boldsymbol {e}}_{1}}
474:{\displaystyle {\boldsymbol {e}}_{3}}
419:{\displaystyle {\boldsymbol {e}}_{2}}
364:{\displaystyle {\boldsymbol {e}}_{1}}
193:point normal representation of planes
97:{\displaystyle {\boldsymbol {e}}_{1}}
7:
4456:
4454:
4399:"Geometric Fundamentals of Robotics"
4346:
4344:
3794:(CGA), of which Plane-based GA is a
67:Plane-based geometric algebra takes
4932:Gunn, Charles (December 19, 2011).
3913:Variants of duality and terminology
4797:"Foundations of geometric algebra"
4205:and rotoreflections in the space.
2596:. This is a generalization of the
2589:{\textstyle {\frac {1}{2}}(AB-BA)}
1725:gives the necessary elements for (
1317:Lines at infinity also exist; the
14:
3788:inversions in circles and spheres
2640:, we have that the derivative of
4801:Geometric Algebra for Physicists
4705:Lounesto, Pertti (May 3, 2001).
4228:taking the quotient by scalars.
2450:(dual quaternion) or reflection
316:consists of the plane such that
60:. It is usually combined with a
5018:Journal of Mathematical Physics
4882:Selig, J. M. (September 2000).
4351:Selig, J. M. (September 2000).
4307:Journal of Mathematical Physics
4216:All three even subalgebras are
3726:pecial (handedness-preserving)
3350:is a slight rotation about the
2241:{\displaystyle \lVert A\rVert }
1115:
1084:
1053:
948:", it is instead the transform
813:followed by the transformation
189:plĂŒcker representation of lines
16:Application of Clifford algebra
4842:projectivegeometricalgebra.org
4838:"Projective Geometric Algebra"
4785:(in French). Gauthier-Villars.
4711:. Cambridge University Press.
4403:Monographs in Computer Science
4272:. Cambridge University Press.
3673:Rotations and translations as
3290:
3284:
3258:
3252:
3131:
3125:
2583:
2565:
2489:
2319:
2301:
2204:
2143:
2134:
2122:
1887:
1818:
1:
4708:Clifford Algebras and Spinors
4542:. Elsevier, Morgan Kaufmann.
4467:Advances in Computer Graphics
4301:Brooke, J. A. (May 1, 1978).
4119:Gnonomic world map projection
3873:, allows a definition of the
2071:, which is useful for taking
1974:, which is useful for taking
1827:{\displaystyle B{\tilde {A}}}
30:Plane-based geometric algebra
4809:10.1017/cbo9780511807497.006
4475:10.1007/978-3-030-61864-3_39
3813:Projective Geometric Algebra
2937:followed by a reflection in
1896:{\displaystyle {\tilde {A}}}
163:Plane-based GA subsumes the
5014:"Lie groups as spin groups"
4435:"Research â CliffordLayers"
3792:Conformal Geometric Algebra
3707:parallel to the first (the
3695:Rotations and translations
2498:{\textstyle TA{\tilde {T}}}
5082:
4983:10.1007/s00006-021-01118-7
4656:10.1007/s00006-023-01269-9
977:followed by the transform
754:, their geometric product
197:homogeneous representation
4942:10.14279/DEPOSITONCE-3058
4900:10.1017/s0263574799002568
4369:10.1017/S0263574799002568
3839:; this can be written as
2820:For example, recall that
2037:{\displaystyle P\wedge L}
1429:Most lines, for examples
714:For any pair of elements
312:of planes. Its canonical
285:elements of vector spaces
177:axisâangle representation
4781:Michel, Chasles (1875).
4740:Jordan, Camille (1875).
4717:10.1017/cbo9780511526022
4278:10.1017/cbo9780511623943
3146:CartanâDieudonnĂ© theorem
2660:with respect to time is
1762:The transformation from
1754:were extracted from the
1729:) coordinates of lines.
4580:. Lincoln, California.
4117:; provides a model the
3654:and the original point
2790:{\displaystyle \times }
2692:{\displaystyle \times }
2541:{\displaystyle \times }
2528:The commutator product
1967:{\displaystyle \wedge }
1350:to be the ground, then
280:(three lower indices).
201:screw, twist and wrench
4574:Lengyel, Eric (2016).
4201:
3899:
3862:
3778:
3669:
3632:
3618:is a 2-reflection and
3587:
3557:
3533:
3481:
3452:
3402:
3373:
3344:
3297:
3265:
3236:
3203:
3174:
3138:
3108:
3084:
3040:
2990:
2931:
2902:
2843:
2810:
2791:
2772:
2752:
2732:
2712:
2693:
2674:
2654:
2634:
2614:
2590:
2542:
2519:
2499:
2464:
2433:
2389:
2366:
2346:
2326:
2282:
2262:
2242:
2216:
2180:
2153:
2109:
2089:
2065:
2064:{\displaystyle \cdot }
2038:
2012:
1992:
1968:
1941:
1921:
1897:
1868:
1848:
1828:
1796:
1776:
1743:
1719:
1690:
1661:
1632:
1609:
1580:
1551:
1518:
1489:
1458:, can act as axes for
1452:
1420:
1391:
1344:
1304:
1271:
1248:The algebraic element
1240:
1225:
1184:". The statement that
1178:
1157:
1150:
1000:
971:
942:
913:
880:
833:. Note that transform
827:
807:
793:is the transformation
787:
768:
748:
728:
705:
676:
647:
621:
596:
589:
522:
475:
446:
420:
391:
365:
336:
305:
274:
245:
172:
150:
130:
98:
26:
4191:
4178:is equivalent to the
4055:rigid transformations
4006:Transformation group
3900:
3885:(see above) equal to
3820:
3772:
3666:
3633:
3588:
3550:
3534:
3482:
3453:
3403:
3374:
3345:
3298:
3266:
3237:
3204:
3175:
3139:
3101:
3085:
3041:
2991:
2932:
2903:
2844:
2811:
2792:
2773:
2753:
2733:
2713:
2694:
2675:
2655:
2635:
2615:
2591:
2543:
2520:
2500:
2465:
2434:
2390:
2367:
2347:
2327:
2283:
2263:
2243:
2217:
2181:
2154:
2110:
2090:
2066:
2039:
2013:
1993:
1969:
1942:
1922:
1898:
1869:
1849:
1829:
1797:
1777:
1740:
1720:
1691:
1662:
1633:
1610:
1581:
1552:
1519:
1490:
1453:
1421:
1392:
1345:
1305:
1272:
1237:
1226:
1179:
1151:
1015:
1001:
972:
943:
914:
881:
828:
808:
788:
769:
749:
729:
706:
677:
648:
622:
590:
538:
523:
476:
447:
421:
392:
366:
337:
302:
275:
246:
220:contravariant vectors
162:
151:
131:
99:
38:rigid transformations
32:is an application of
24:
4269:Topological Geometry
4218:classical Lie groups
3889:
3622:
3568:
3514:
3462:
3412:
3383:
3354:
3316:
3296:{\displaystyle E(3)}
3278:
3264:{\displaystyle E(3)}
3246:
3217:
3213:, and lines such as
3184:
3155:
3137:{\displaystyle E(3)}
3119:
3050:
3000:
2941:
2912:
2853:
2824:
2800:
2781:
2762:
2742:
2722:
2702:
2683:
2664:
2644:
2624:
2604:
2552:
2532:
2509:
2474:
2454:
2448:rigid transformation
2399:
2379:
2356:
2336:
2332:. This assumes that
2292:
2272:
2252:
2226:
2190:
2170:
2119:
2099:
2079:
2055:
2022:
2002:
1982:
1958:
1931:
1911:
1878:
1858:
1838:
1806:
1786:
1766:
1700:
1671:
1642:
1619:
1590:
1561:
1532:
1499:
1470:
1433:
1401:
1354:
1325:
1285:
1252:
1244:Elements at infinity
1188:
1168:
1020:
981:
952:
923:
894:
849:
817:
797:
777:
758:
738:
718:
686:
657:
631:
602:
543:
488:
456:
430:
401:
397:, which is labelled
375:
346:
342:, which is labelled
320:
255:
226:
140:
120:
79:
5030:1993JMP....34.3642D
4536:Dorst, Leo (2010).
4439:microsoft.github.io
4319:1978JMP....19..952B
4090:Split-biquaternions
3999:
2446:Application of any
1495:, such as the line
646:{\displaystyle x=0}
445:{\displaystyle z=0}
390:{\displaystyle y=0}
335:{\displaystyle x=0}
287:; however they are
4202:
4155:Complex quaternion
3998:
3895:
3875:regressive product
3871:Grassmann algebras
3863:
3859:regressive product
3782:Inversive Geometry
3779:
3670:
3628:
3583:
3558:
3529:
3477:
3448:
3398:
3369:
3340:
3310:linear combination
3293:
3261:
3232:
3199:
3170:
3134:
3109:
3080:
3036:
2986:
2927:
2898:
2849:is a plane, as is
2839:
2806:
2787:
2768:
2748:
2728:
2708:
2689:
2670:
2650:
2630:
2610:
2586:
2538:
2515:
2495:
2460:
2429:
2385:
2362:
2342:
2322:
2278:
2258:
2238:
2212:
2176:
2149:
2105:
2085:
2061:
2034:
2008:
1988:
1964:
1937:
1917:
1893:
1864:
1844:
1824:
1792:
1772:
1756:quaternion product
1744:
1715:
1686:
1657:
1631:{\displaystyle -1}
1628:
1615:, which square to
1605:
1576:
1547:
1514:
1485:
1448:
1416:
1387:
1340:
1300:
1267:
1241:
1221:
1174:
1146:
996:
967:
938:
909:
876:
823:
803:
783:
764:
744:
724:
701:
672:
643:
617:
585:
518:
471:
442:
416:
387:
361:
332:
306:
270:
241:
179:of rotations, the
173:
146:
126:
94:
69:planar reflections
27:
5066:Geometric algebra
4726:978-0-521-00551-7
4587:978-0-9858117-4-7
4549:978-0-12-374942-0
4522:978-0-12-374942-0
4484:978-3-030-61863-6
4420:978-0-387-20874-9
4287:978-0-521-23160-2
4186:
4185:
4174:, a.k.a. boosts.
3898:{\displaystyle 1}
3775:sphere inversions
3631:{\displaystyle P}
3504:point reflections
3500:glide reflections
3274:Some elements of
3211:point reflections
2809:{\displaystyle B}
2751:{\displaystyle B}
2731:{\displaystyle A}
2711:{\displaystyle B}
2653:{\displaystyle B}
2633:{\displaystyle B}
2563:
2492:
2388:{\displaystyle 1}
2365:{\displaystyle B}
2345:{\displaystyle A}
2281:{\displaystyle B}
2261:{\displaystyle A}
2210:
2207:
2179:{\displaystyle A}
2146:
2108:{\displaystyle B}
2088:{\displaystyle A}
2011:{\displaystyle L}
1991:{\displaystyle P}
1940:{\displaystyle B}
1920:{\displaystyle A}
1890:
1867:{\displaystyle B}
1847:{\displaystyle A}
1821:
1795:{\displaystyle B}
1775:{\displaystyle A}
1758:. These include:
1279:plane at infinity
1177:{\displaystyle 1}
1162:identity function
806:{\displaystyle B}
786:{\displaystyle B}
747:{\displaystyle B}
534:geometric product
149:{\displaystyle 0}
129:{\displaystyle 1}
58:computer graphics
5073:
5050:
5049:
5038:10.1063/1.530050
5024:(8): 3642â3669.
5009:
5003:
5002:
4976:
4952:
4946:
4945:
4929:
4920:
4919:
4879:
4873:
4872:
4870:
4869:
4861:Doran |, Chris.
4858:
4852:
4851:
4849:
4848:
4834:
4828:
4827:
4826:
4825:
4793:
4787:
4786:
4778:
4772:
4771:
4761:
4759:10.24033/bsmf.90
4737:
4731:
4730:
4702:
4696:
4695:
4693:
4692:
4682:
4676:
4675:
4649:
4625:
4614:
4613:
4607:
4599:
4571:
4562:
4561:
4533:
4527:
4526:
4508:
4502:
4501:
4500:
4499:
4458:
4449:
4448:
4446:
4445:
4431:
4425:
4424:
4395:
4389:
4388:
4348:
4339:
4338:
4327:10.1063/1.523798
4298:
4292:
4291:
4263:
4257:
4256:
4255:
4254:
4242:
4211:hyperbolic angle
4199:
4195:
4177:
4172:spacetime boosts
4163:double cover of
4160:
4147:
4135:
4123:Poincaré duality
4116:
4098:double cover of
4095:
4082:
4070:
4053:double cover of
4050:
4045:Dual quaternions
4037:
4025:
4003:Geometric space
4000:
3994:
3904:
3902:
3901:
3896:
3856:
3852:
3838:
3833:define the line
3832:
3826:
3736:Dual Quaternions
3659:
3653:
3643:
3637:
3635:
3634:
3629:
3617:
3611:
3605:
3596:
3592:
3590:
3589:
3584:
3582:
3581:
3576:
3538:
3536:
3535:
3530:
3528:
3527:
3522:
3486:
3484:
3483:
3478:
3476:
3475:
3470:
3457:
3455:
3454:
3449:
3447:
3446:
3441:
3429:
3428:
3423:
3407:
3405:
3404:
3399:
3397:
3396:
3391:
3378:
3376:
3375:
3370:
3368:
3367:
3362:
3349:
3347:
3346:
3341:
3339:
3338:
3333:
3302:
3300:
3299:
3294:
3270:
3268:
3267:
3262:
3241:
3239:
3238:
3233:
3231:
3230:
3225:
3208:
3206:
3205:
3200:
3198:
3197:
3192:
3179:
3177:
3176:
3171:
3169:
3168:
3163:
3143:
3141:
3140:
3135:
3104:point reflection
3089:
3087:
3086:
3081:
3079:
3078:
3073:
3064:
3063:
3058:
3045:
3043:
3042:
3037:
3035:
3034:
3029:
3020:
3019:
3014:
2995:
2993:
2992:
2987:
2985:
2984:
2979:
2970:
2969:
2964:
2955:
2954:
2949:
2936:
2934:
2933:
2928:
2926:
2925:
2920:
2907:
2905:
2904:
2899:
2897:
2896:
2891:
2882:
2881:
2876:
2867:
2866:
2861:
2848:
2846:
2845:
2840:
2838:
2837:
2832:
2815:
2813:
2812:
2807:
2796:
2794:
2793:
2788:
2777:
2775:
2774:
2769:
2757:
2755:
2754:
2749:
2737:
2735:
2734:
2729:
2717:
2715:
2714:
2709:
2698:
2696:
2695:
2690:
2679:
2677:
2676:
2671:
2659:
2657:
2656:
2651:
2639:
2637:
2636:
2631:
2619:
2617:
2616:
2611:
2595:
2593:
2592:
2587:
2564:
2556:
2547:
2545:
2544:
2539:
2524:
2522:
2521:
2516:
2504:
2502:
2501:
2496:
2494:
2493:
2485:
2469:
2467:
2466:
2461:
2438:
2436:
2435:
2430:
2394:
2392:
2391:
2386:
2371:
2369:
2368:
2363:
2351:
2349:
2348:
2343:
2331:
2329:
2328:
2323:
2287:
2285:
2284:
2279:
2267:
2265:
2264:
2259:
2247:
2245:
2244:
2239:
2221:
2219:
2218:
2213:
2211:
2209:
2208:
2200:
2194:
2185:
2183:
2182:
2177:
2158:
2156:
2155:
2150:
2148:
2147:
2139:
2114:
2112:
2111:
2106:
2094:
2092:
2091:
2086:
2070:
2068:
2067:
2062:
2043:
2041:
2040:
2035:
2017:
2015:
2014:
2009:
1997:
1995:
1994:
1989:
1973:
1971:
1970:
1965:
1946:
1944:
1943:
1938:
1926:
1924:
1923:
1918:
1902:
1900:
1899:
1894:
1892:
1891:
1883:
1873:
1871:
1870:
1865:
1853:
1851:
1850:
1845:
1833:
1831:
1830:
1825:
1823:
1822:
1814:
1801:
1799:
1798:
1793:
1781:
1779:
1778:
1773:
1724:
1722:
1721:
1716:
1714:
1713:
1708:
1695:
1693:
1692:
1687:
1685:
1684:
1679:
1666:
1664:
1663:
1658:
1656:
1655:
1650:
1637:
1635:
1634:
1629:
1614:
1612:
1611:
1606:
1604:
1603:
1598:
1585:
1583:
1582:
1577:
1575:
1574:
1569:
1556:
1554:
1553:
1548:
1546:
1545:
1540:
1523:
1521:
1520:
1515:
1513:
1512:
1507:
1494:
1492:
1491:
1486:
1484:
1483:
1478:
1457:
1455:
1454:
1449:
1447:
1446:
1441:
1425:
1423:
1422:
1417:
1415:
1414:
1409:
1396:
1394:
1393:
1388:
1386:
1385:
1380:
1368:
1367:
1362:
1349:
1347:
1346:
1341:
1339:
1338:
1333:
1312:vanishing points
1309:
1307:
1306:
1301:
1299:
1298:
1293:
1276:
1274:
1273:
1268:
1266:
1265:
1260:
1230:
1228:
1227:
1222:
1214:
1213:
1208:
1202:
1201:
1196:
1183:
1181:
1180:
1175:
1155:
1153:
1152:
1147:
1139:
1138:
1133:
1127:
1126:
1121:
1108:
1107:
1102:
1096:
1095:
1090:
1077:
1076:
1071:
1065:
1064:
1059:
1046:
1045:
1040:
1034:
1033:
1028:
1008:sandwich product
1005:
1003:
1002:
997:
995:
994:
989:
976:
974:
973:
968:
966:
965:
960:
947:
945:
944:
939:
937:
936:
931:
918:
916:
915:
910:
908:
907:
902:
885:
883:
882:
877:
875:
874:
869:
863:
862:
857:
832:
830:
829:
824:
812:
810:
809:
804:
792:
790:
789:
784:
773:
771:
770:
765:
753:
751:
750:
745:
733:
731:
730:
725:
710:
708:
707:
702:
700:
699:
694:
681:
679:
678:
673:
671:
670:
665:
652:
650:
649:
644:
626:
624:
623:
618:
616:
615:
610:
594:
592:
591:
586:
584:
583:
578:
569:
568:
563:
557:
556:
551:
527:
525:
524:
519:
517:
516:
511:
502:
501:
496:
480:
478:
477:
472:
470:
469:
464:
451:
449:
448:
443:
425:
423:
422:
417:
415:
414:
409:
396:
394:
393:
388:
370:
368:
367:
362:
360:
359:
354:
341:
339:
338:
333:
279:
277:
276:
271:
269:
268:
263:
250:
248:
247:
242:
240:
239:
234:
155:
153:
152:
147:
135:
133:
132:
127:
115:
103:
101:
100:
95:
93:
92:
87:
34:Clifford algebra
5081:
5080:
5076:
5075:
5074:
5072:
5071:
5070:
5056:
5055:
5054:
5053:
5011:
5010:
5006:
4954:
4953:
4949:
4931:
4930:
4923:
4881:
4880:
4876:
4867:
4865:
4860:
4859:
4855:
4846:
4844:
4836:
4835:
4831:
4823:
4821:
4819:
4795:
4794:
4790:
4780:
4779:
4775:
4739:
4738:
4734:
4727:
4704:
4703:
4699:
4690:
4688:
4684:
4683:
4679:
4627:
4626:
4617:
4600:
4588:
4573:
4572:
4565:
4550:
4535:
4534:
4530:
4523:
4510:
4509:
4505:
4497:
4495:
4485:
4460:
4459:
4452:
4443:
4441:
4433:
4432:
4428:
4421:
4411:10.1007/b138859
4397:
4396:
4392:
4350:
4349:
4342:
4300:
4299:
4295:
4288:
4265:
4264:
4260:
4252:
4250:
4244:
4243:
4239:
4234:
4197:
4193:
4175:
4158:
4141:
4137:
4133:
4110:
4106:
4093:
4076:
4072:
4068:
4048:
4031:
4027:
4023:
3988:
3984:
3977:
3915:
3887:
3886:
3854:
3840:
3834:
3828:
3822:
3815:
3784:
3767:
3765:Generalizations
3744:even subalgebra
3722:, the group of
3678:
3675:even subalgebra
3655:
3649:
3639:
3620:
3619:
3613:
3607:
3601:
3594:
3571:
3566:
3565:
3545:
3517:
3512:
3511:
3496:Rotoreflections
3465:
3460:
3459:
3436:
3418:
3410:
3409:
3386:
3381:
3380:
3357:
3352:
3351:
3328:
3314:
3313:
3276:
3275:
3244:
3243:
3220:
3215:
3214:
3187:
3182:
3181:
3158:
3153:
3152:
3117:
3116:
3113:Euclidean Group
3096:
3068:
3053:
3048:
3047:
3024:
3009:
2998:
2997:
2974:
2959:
2944:
2939:
2938:
2915:
2910:
2909:
2886:
2871:
2856:
2851:
2850:
2827:
2822:
2821:
2798:
2797:
2779:
2778:
2760:
2759:
2740:
2739:
2720:
2719:
2700:
2699:
2681:
2680:
2662:
2661:
2642:
2641:
2622:
2621:
2602:
2601:
2550:
2549:
2530:
2529:
2507:
2506:
2472:
2471:
2452:
2451:
2397:
2396:
2377:
2376:
2354:
2353:
2334:
2333:
2290:
2289:
2270:
2269:
2250:
2249:
2224:
2223:
2222:and is denoted
2188:
2187:
2168:
2167:
2117:
2116:
2097:
2096:
2077:
2076:
2053:
2052:
2020:
2019:
2000:
1999:
1980:
1979:
1956:
1955:
1929:
1928:
1909:
1908:
1876:
1875:
1856:
1855:
1836:
1835:
1804:
1803:
1784:
1783:
1764:
1763:
1735:
1703:
1698:
1697:
1674:
1669:
1668:
1645:
1640:
1639:
1617:
1616:
1593:
1588:
1587:
1564:
1559:
1558:
1535:
1530:
1529:
1502:
1497:
1496:
1473:
1468:
1467:
1436:
1431:
1430:
1404:
1399:
1398:
1375:
1357:
1352:
1351:
1328:
1323:
1322:
1288:
1283:
1282:
1277:represents the
1255:
1250:
1249:
1246:
1231:is more subtle.
1203:
1191:
1186:
1185:
1166:
1165:
1128:
1116:
1097:
1085:
1066:
1054:
1035:
1023:
1018:
1017:
984:
979:
978:
955:
950:
949:
926:
921:
920:
919:transformed by
897:
892:
891:
864:
852:
847:
846:
815:
814:
795:
794:
775:
774:
756:
755:
736:
735:
716:
715:
689:
684:
683:
660:
655:
654:
629:
628:
605:
600:
599:
573:
558:
546:
541:
540:
536:. For example:
506:
491:
486:
485:
459:
454:
453:
428:
427:
404:
399:
398:
373:
372:
349:
344:
343:
318:
317:
297:
258:
253:
252:
229:
224:
223:
185:dual quaternion
138:
137:
118:
117:
109:
105:
82:
77:
76:
17:
12:
11:
5:
5079:
5077:
5069:
5068:
5058:
5057:
5052:
5051:
5004:
4947:
4921:
4894:(5): 545â556.
4874:
4853:
4829:
4817:
4788:
4773:
4732:
4725:
4697:
4677:
4615:
4586:
4563:
4548:
4528:
4521:
4503:
4483:
4450:
4426:
4419:
4390:
4363:(5): 545â556.
4340:
4313:(5): 952â959.
4293:
4286:
4258:
4236:
4235:
4233:
4230:
4184:
4183:
4168:
4152:
4149:
4139:
4131:
4127:
4126:
4108:
4103:
4087:
4084:
4074:
4066:
4062:
4061:
4058:
4042:
4039:
4029:
4021:
4017:
4016:
4013:
4010:
4007:
4004:
3986:
3976:
3973:
3972:
3971:
3959:
3933:
3914:
3911:
3894:
3814:
3811:
3783:
3780:
3766:
3763:
3759:Michel Chasles
3755:Camille Jordan
3677:
3671:
3646:rotoreflection
3627:
3580:
3575:
3544:
3541:
3526:
3521:
3474:
3469:
3445:
3440:
3435:
3432:
3427:
3422:
3417:
3395:
3390:
3366:
3361:
3337:
3332:
3327:
3324:
3321:
3292:
3289:
3286:
3283:
3260:
3257:
3254:
3251:
3229:
3224:
3196:
3191:
3167:
3162:
3133:
3130:
3127:
3124:
3095:
3092:
3077:
3072:
3067:
3062:
3057:
3033:
3028:
3023:
3018:
3013:
3008:
3005:
2983:
2978:
2973:
2968:
2963:
2958:
2953:
2948:
2924:
2919:
2895:
2890:
2885:
2880:
2875:
2870:
2865:
2860:
2836:
2831:
2818:
2817:
2805:
2786:
2771:{\textstyle A}
2767:
2747:
2727:
2707:
2688:
2673:{\textstyle A}
2669:
2649:
2629:
2613:{\textstyle A}
2609:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2562:
2559:
2537:
2526:
2518:{\textstyle A}
2514:
2491:
2488:
2482:
2479:
2463:{\textstyle T}
2459:
2444:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2384:
2361:
2341:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2277:
2257:
2237:
2234:
2231:
2206:
2203:
2197:
2175:
2160:
2145:
2142:
2136:
2133:
2130:
2127:
2124:
2104:
2084:
2060:
2045:
2033:
2030:
2027:
2007:
1998:with the line
1987:
1963:
1950:The meet (or "
1948:
1936:
1916:
1889:
1886:
1863:
1843:
1820:
1817:
1811:
1791:
1771:
1734:
1731:
1712:
1707:
1683:
1678:
1654:
1649:
1627:
1624:
1602:
1597:
1573:
1568:
1544:
1539:
1511:
1506:
1482:
1477:
1445:
1440:
1413:
1408:
1384:
1379:
1374:
1371:
1366:
1361:
1337:
1332:
1297:
1292:
1264:
1259:
1245:
1242:
1220:
1217:
1212:
1207:
1200:
1195:
1173:
1145:
1142:
1137:
1132:
1125:
1120:
1114:
1111:
1106:
1101:
1094:
1089:
1083:
1080:
1075:
1070:
1063:
1058:
1052:
1049:
1044:
1039:
1032:
1027:
993:
988:
964:
959:
935:
930:
906:
901:
873:
868:
861:
856:
845:; for example
826:{\textstyle A}
822:
802:
782:
767:{\textstyle A}
763:
743:
727:{\textstyle A}
723:
698:
693:
669:
664:
642:
639:
636:
614:
609:
582:
577:
572:
567:
562:
555:
550:
515:
510:
505:
500:
495:
468:
463:
441:
438:
435:
413:
408:
386:
383:
380:
358:
353:
331:
328:
325:
296:
293:
267:
262:
238:
233:
145:
125:
107:
91:
86:
15:
13:
10:
9:
6:
4:
3:
2:
5078:
5067:
5064:
5063:
5061:
5047:
5043:
5039:
5035:
5031:
5027:
5023:
5019:
5015:
5008:
5005:
5000:
4996:
4992:
4988:
4984:
4980:
4975:
4970:
4966:
4962:
4958:
4951:
4948:
4943:
4939:
4935:
4928:
4926:
4922:
4917:
4913:
4909:
4905:
4901:
4897:
4893:
4889:
4885:
4878:
4875:
4864:
4857:
4854:
4843:
4839:
4833:
4830:
4820:
4818:9780521480222
4814:
4810:
4806:
4802:
4798:
4792:
4789:
4784:
4777:
4774:
4769:
4765:
4760:
4755:
4751:
4747:
4743:
4736:
4733:
4728:
4722:
4718:
4714:
4710:
4709:
4701:
4698:
4687:
4681:
4678:
4673:
4669:
4665:
4661:
4657:
4653:
4648:
4643:
4639:
4635:
4631:
4624:
4622:
4620:
4616:
4611:
4605:
4597:
4593:
4589:
4583:
4579:
4578:
4570:
4568:
4564:
4559:
4555:
4551:
4545:
4541:
4540:
4532:
4529:
4524:
4518:
4514:
4507:
4504:
4494:
4490:
4486:
4480:
4476:
4472:
4468:
4464:
4457:
4455:
4451:
4440:
4436:
4430:
4427:
4422:
4416:
4412:
4408:
4404:
4400:
4394:
4391:
4386:
4382:
4378:
4374:
4370:
4366:
4362:
4358:
4354:
4347:
4345:
4341:
4336:
4332:
4328:
4324:
4320:
4316:
4312:
4308:
4304:
4297:
4294:
4289:
4283:
4279:
4275:
4271:
4270:
4262:
4259:
4249:
4248:
4241:
4238:
4231:
4229:
4227:
4223:
4219:
4214:
4212:
4206:
4190:
4181:
4173:
4169:
4167:
4166:
4165:Lorentz group
4159:Spin(3, 1, 0)
4156:
4153:
4150:
4148:
4145:
4132:
4129:
4128:
4124:
4120:
4114:
4104:
4102:
4101:
4094:Spin(4, 0, 0)
4091:
4088:
4085:
4083:
4080:
4067:
4064:
4063:
4059:
4057:
4056:
4049:Spin(3, 0, 1)
4046:
4043:
4040:
4038:
4035:
4022:
4019:
4018:
4014:
4011:
4008:
4005:
4002:
4001:
3996:
3992:
3982:
3974:
3969:
3965:
3960:
3957:
3956:collineations
3953:
3949:
3948:non-euclidean
3945:
3941:
3939:
3934:
3930:
3927:
3926:
3921:
3920:
3919:
3912:
3910:
3908:
3892:
3884:
3880:
3876:
3872:
3868:
3860:
3851:
3847:
3843:
3837:
3831:
3825:
3819:
3812:
3810:
3808:
3807:Poincare disk
3804:
3799:
3797:
3793:
3789:
3781:
3776:
3771:
3764:
3762:
3760:
3756:
3751:
3749:
3745:
3741:
3737:
3733:
3729:
3725:
3721:
3717:
3716:screw motions
3712:
3710:
3706:
3702:
3698:
3693:
3691:
3687:
3683:
3676:
3672:
3665:
3661:
3658:
3652:
3647:
3642:
3625:
3616:
3610:
3604:
3598:
3578:
3563:
3554:
3549:
3542:
3540:
3524:
3509:
3505:
3501:
3497:
3493:
3488:
3472:
3443:
3433:
3430:
3425:
3415:
3393:
3364:
3335:
3325:
3322:
3319:
3311:
3306:
3287:
3281:
3272:
3255:
3249:
3227:
3212:
3194:
3165:
3149:
3147:
3128:
3122:
3114:
3105:
3100:
3093:
3091:
3075:
3065:
3060:
3031:
3021:
3016:
3006:
3003:
2981:
2971:
2966:
2956:
2951:
2922:
2893:
2883:
2878:
2868:
2863:
2834:
2803:
2784:
2765:
2758:we have that
2745:
2725:
2705:
2686:
2667:
2647:
2627:
2607:
2599:
2580:
2577:
2574:
2571:
2568:
2560:
2557:
2548:, defined as
2535:
2527:
2512:
2486:
2480:
2477:
2457:
2449:
2445:
2442:
2426:
2423:
2417:
2411:
2405:
2382:
2375:
2359:
2339:
2313:
2310:
2307:
2298:
2295:
2275:
2255:
2232:
2201:
2195:
2173:
2165:
2161:
2140:
2131:
2128:
2125:
2102:
2082:
2074:
2058:
2050:
2046:
2031:
2028:
2025:
2018:is the point
2005:
1985:
1977:
1976:intersections
1961:
1953:
1949:
1934:
1914:
1906:
1884:
1861:
1841:
1815:
1809:
1789:
1769:
1761:
1760:
1759:
1757:
1753:
1752:cross product
1749:
1739:
1732:
1730:
1728:
1710:
1681:
1652:
1625:
1622:
1600:
1571:
1542:
1527:
1509:
1480:
1465:
1461:
1443:
1427:
1411:
1382:
1372:
1369:
1364:
1335:
1320:
1315:
1313:
1295:
1280:
1262:
1243:
1236:
1232:
1218:
1215:
1210:
1198:
1171:
1163:
1156:
1143:
1140:
1135:
1123:
1112:
1109:
1104:
1092:
1081:
1078:
1073:
1061:
1050:
1047:
1042:
1030:
1014:
1011:
1010:, see below.
1009:
991:
962:
933:
904:
889:
871:
859:
844:
840:
836:
820:
800:
780:
761:
741:
721:
712:
696:
667:
640:
637:
634:
612:
598:Here we take
595:
580:
570:
565:
553:
537:
535:
530:
513:
503:
498:
482:
466:
439:
436:
433:
411:
384:
381:
378:
356:
329:
326:
323:
315:
311:
310:intersections
301:
294:
292:
290:
286:
281:
265:
236:
221:
217:
216:pseudovectors
213:
212:Gibbs vectors
209:
208:cross product
204:
202:
198:
194:
190:
186:
182:
178:
170:
166:
161:
157:
143:
123:
113:
89:
74:
70:
65:
63:
59:
55:
51:
47:
43:
42:intersections
39:
35:
31:
23:
19:
5021:
5017:
5007:
4964:
4960:
4950:
4933:
4891:
4887:
4877:
4866:. Retrieved
4856:
4845:. Retrieved
4841:
4832:
4822:, retrieved
4800:
4791:
4782:
4776:
4749:
4745:
4735:
4707:
4700:
4689:. Retrieved
4680:
4637:
4633:
4576:
4538:
4531:
4512:
4506:
4496:, retrieved
4466:
4442:. Retrieved
4438:
4429:
4402:
4393:
4360:
4356:
4310:
4306:
4296:
4268:
4261:
4251:, retrieved
4246:
4240:
4225:
4215:
4207:
4203:
4162:
4143:
4136:
4134:Pin(3, 1, 0)
4112:
4100:4D rotations
4097:
4078:
4071:
4069:Pin(4, 0, 0)
4052:
4033:
4026:
4024:Pin(3, 0, 1)
3990:
3980:
3978:
3967:
3963:
3943:
3937:
3923:
3916:
3882:
3878:
3866:
3864:
3858:
3849:
3845:
3841:
3835:
3829:
3823:
3800:
3785:
3752:
3743:
3740:double cover
3739:
3735:
3731:
3727:
3723:
3715:
3713:
3704:
3700:
3696:
3694:
3685:
3681:
3679:
3674:
3668:"cancelled".
3656:
3650:
3640:
3614:
3608:
3602:
3599:
3561:
3559:
3552:
3508:screw motion
3492:2-reflection
3491:
3489:
3304:
3273:
3150:
3110:
2819:
2163:
2048:
1951:
1745:
1526:dual numbers
1428:
1319:horizon line
1316:
1247:
1158:
1016:
1012:
1007:
887:
842:
838:
834:
713:
597:
539:
533:
531:
483:
309:
307:
295:Construction
288:
284:
282:
205:
174:
111:
72:
68:
66:
61:
29:
28:
18:
4752:: 103â174.
4222:Lie algebra
4130:Hyperbolic
4121:. Includes
3968:point pairs
3950:space, see
3907:affine hull
3821:The points
3709:quaternions
3701:even number
2598:Lie Bracket
2441:dot product
2073:projections
1748:dot product
1464:quaternions
843:application
835:composition
653:plane, and
50:spin groups
46:projections
4974:2002.05993
4868:2023-09-08
4847:2023-09-08
4824:2023-09-23
4691:2023-09-08
4647:2107.03771
4498:2023-09-09
4444:2023-08-10
4253:2023-09-09
4232:References
4020:Euclidean
3952:dual space
3938:Projective
3857:being the
3796:subalgebra
3686:handedness
3593:square to
2372:both have
1954:product")
841:transform
426:, and the
195:, and the
181:quaternion
169:axis-angle
165:quaternion
5046:0022-2488
4999:211126515
4991:0188-7009
4908:0263-5747
4768:0037-9484
4672:235765240
4664:0188-7009
4604:cite book
4596:972909098
4558:846456514
4493:224820480
4377:0263-5747
4335:0022-2488
4065:Elliptic
3803:conformal
3690:chirality
3682:distances
3144:. By the
3107:element".
2785:×
2687:×
2575:−
2536:×
2490:~
2421:‖
2415:‖
2409:‖
2403:‖
2317:‖
2311:⋅
2305:‖
2299:
2236:‖
2230:‖
2205:~
2144:~
2129:⋅
2059:⋅
2029:∧
1962:∧
1888:~
1819:~
1623:−
1460:rotations
5060:Category
4916:28929170
4888:Robotica
4405:. 2005.
4385:28929170
4357:Robotica
2505:, where
2051:product
54:robotics
5026:Bibcode
4315:Bibcode
4176:(2,1,0)
3853:, with
1905:reverse
1903:is the
1834:, with
1782:toward
1727:PlĂŒcker
452:plane,
73:grade-1
62:duality
5044:
4997:
4989:
4914:
4906:
4815:
4766:
4723:
4670:
4662:
4594:
4584:
4556:
4546:
4519:
4491:
4481:
4417:
4383:
4375:
4333:
4284:
4015:Notes
3964:circle
3757:. and
3748:spinor
3502:, and
2296:arccos
371:, the
191:, the
4995:S2CID
4969:arXiv
4967:(2).
4912:S2CID
4668:S2CID
4642:arXiv
4640:(3).
4489:S2CID
4381:S2CID
4140:3,1,0
4109:3,0,0
4075:4,0,0
4030:3,0,1
3987:3,0,1
3981:rigid
3932:dual.
3925:Hodge
3720:SE(3)
3553:gauge
2600:: if
2395:, eg
2095:onto
2049:inner
1952:wedge
314:basis
108:3,0,1
5042:ISSN
4987:ISSN
4904:ISSN
4813:ISBN
4764:ISSN
4721:ISBN
4660:ISSN
4610:link
4592:OCLC
4582:ISBN
4554:OCLC
4544:ISBN
4517:ISBN
4479:ISBN
4415:ISBN
4373:ISSN
4331:ISSN
4282:ISBN
3940:dual
3935:The
3929:dual
3922:The
3883:norm
3879:join
3867:dual
3827:and
3684:and
3562:same
3525:0123
3408:and
3209:are
2738:and
2374:norm
2352:and
2268:and
2164:norm
2162:The
2047:The
1927:and
1854:and
1750:and
1742:etc.
734:and
218:and
183:and
167:and
5034:doi
4979:doi
4938:doi
4896:doi
4805:doi
4754:doi
4713:doi
4652:doi
4471:doi
4407:doi
4365:doi
4323:doi
4274:doi
4226:not
4196:or
4151:â1
3944:not
3705:not
3434:0.6
3416:0.8
3326:0.6
3320:0.8
3305:not
3195:123
2186:is
2166:of
2115:is
1802:is
1164:, "
888:not
886:is
839:not
837:is
697:123
581:123
289:not
266:123
5062::
5040:.
5032:.
5022:34
5020:.
5016:.
4993:.
4985:.
4977:.
4965:31
4963:.
4959:.
4924:^
4910:.
4902:.
4892:18
4890:.
4886:.
4840:.
4811:,
4799:,
4762:.
4748:.
4744:.
4719:.
4666:.
4658:.
4650:.
4638:33
4636:.
4632:.
4618:^
4606:}}
4602:{{
4590:.
4566:^
4552:.
4487:,
4477:,
4465:,
4453:^
4437:.
4413:.
4401:.
4379:.
4371:.
4361:18
4359:.
4355:.
4343:^
4329:.
4321:.
4311:19
4309:.
4305:.
4280:.
4213:.
4198:â1
4182:.
4161:;
4157:;
4138:Cl
4125:.
4107:Cl
4096:;
4092:;
4086:1
4073:Cl
4051:;
4047:;
4041:0
4028:Cl
3985:Cl
3962:a
3909:.
3848:=
3844:âš
3798:.
3697:do
3641:PL
3612:.
3498:,
3494:.
3487:.
3473:12
3365:12
3336:12
3271:.
3228:12
3115:,
3076:10
3061:12
3032:10
3017:12
1711:30
1696:,
1682:20
1667:,
1653:10
1601:12
1586:,
1572:13
1557:,
1543:23
1510:30
1444:23
1426:.
1412:02
963:23
905:23
872:23
668:23
566:23
481:.
237:13
214:,
156:.
106:Cl
44:,
5048:.
5036::
5028::
5001:.
4981::
4971::
4944:.
4940::
4918:.
4898::
4871:.
4850:.
4807::
4770:.
4756::
4750:2
4729:.
4715::
4694:.
4674:.
4654::
4644::
4612:)
4598:.
4560:.
4525:.
4473::
4447:.
4423:.
4409::
4387:.
4367::
4337:.
4325::
4317::
4290:.
4276::
4194:1
4146:)
4144:R
4142:(
4115:)
4113:R
4111:(
4081:)
4079:R
4077:(
4036:)
4034:R
4032:(
3993:)
3991:R
3989:(
3893:1
3855:âš
3850:g
3846:Q
3842:P
3836:g
3830:Q
3824:P
3732:3
3728:E
3724:S
3688:(
3657:P
3651:L
3626:P
3615:L
3609:P
3603:L
3595:1
3579:1
3574:e
3520:e
3468:e
3444:2
3439:e
3431:+
3426:1
3421:e
3394:1
3389:e
3360:e
3331:e
3323:+
3291:)
3288:3
3285:(
3282:E
3259:)
3256:3
3253:(
3250:E
3223:e
3190:e
3166:1
3161:e
3132:)
3129:3
3126:(
3123:E
3071:e
3066:+
3056:e
3027:e
3022:+
3012:e
3007:+
3004:1
2982:0
2977:e
2972:+
2967:2
2962:e
2957:+
2952:1
2947:e
2923:1
2918:e
2894:0
2889:e
2884:+
2879:2
2874:e
2869:+
2864:1
2859:e
2835:1
2830:e
2804:B
2766:A
2746:B
2726:A
2706:B
2668:A
2648:B
2628:B
2608:A
2584:)
2581:A
2578:B
2572:B
2569:A
2566:(
2561:2
2558:1
2513:A
2487:T
2481:A
2478:T
2458:T
2443:.
2427:1
2424:=
2418:B
2412:=
2406:A
2383:1
2360:B
2340:A
2320:)
2314:B
2308:A
2302:(
2276:B
2256:A
2233:A
2202:A
2196:A
2174:A
2141:B
2135:)
2132:B
2126:A
2123:(
2103:B
2083:A
2044:.
2032:L
2026:P
2006:L
1986:P
1935:B
1915:A
1885:A
1862:B
1842:A
1816:A
1810:B
1790:B
1770:A
1706:e
1677:e
1648:e
1626:1
1596:e
1567:e
1538:e
1505:e
1481:0
1476:e
1439:e
1407:e
1383:0
1378:e
1373:5
1370:+
1365:2
1360:e
1336:2
1331:e
1296:0
1291:e
1263:0
1258:e
1219:0
1216:=
1211:0
1206:e
1199:0
1194:e
1172:1
1144:0
1141:=
1136:0
1131:e
1124:0
1119:e
1113:1
1110:=
1105:3
1100:e
1093:3
1088:e
1082:1
1079:=
1074:2
1069:e
1062:2
1057:e
1051:1
1048:=
1043:1
1038:e
1031:1
1026:e
992:1
987:e
958:e
934:1
929:e
900:e
890:"
867:e
860:1
855:e
821:A
801:B
781:B
762:A
742:B
722:A
692:e
663:e
641:0
638:=
635:x
613:1
608:e
576:e
571:=
561:e
554:1
549:e
514:3
509:e
504:+
499:2
494:e
467:3
462:e
440:0
437:=
434:z
412:2
407:e
385:0
382:=
379:y
357:1
352:e
330:0
327:=
324:x
261:e
232:e
144:0
124:1
114:)
112:R
110:(
90:1
85:e
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