Dual quaternion - Knowledge

5235: 4851: 5230:{\displaystyle \cos {\frac {\hat {\gamma }}{2}}+\sin {\frac {\hat {\gamma }}{2}}{\mathsf {C}}=\left(\cos {\frac {\hat {\beta }}{2}}\cos {\frac {\hat {\alpha }}{2}}-\sin {\frac {\hat {\beta }}{2}}\sin {\frac {\hat {\alpha }}{2}}{\mathsf {B}}\cdot {\mathsf {A}}\right)+\left(\sin {\frac {\hat {\beta }}{2}}\cos {\frac {\hat {\alpha }}{2}}{\mathsf {B}}+\sin {\frac {\hat {\alpha }}{2}}\cos {\frac {\hat {\beta }}{2}}{\mathsf {A}}+\sin {\frac {\hat {\beta }}{2}}\sin {\frac {\hat {\alpha }}{2}}{\mathsf {B}}\times {\mathsf {A}}\right).} 5661: 8579: 10566: 4840: 5407: 8021: 1251: 6435: 6080: 5656:{\displaystyle \tan {\frac {\hat {\gamma }}{2}}{\mathsf {C}}={\frac {\tan {\frac {\hat {\beta }}{2}}{\mathsf {B}}+\tan {\frac {\hat {\alpha }}{2}}{\mathsf {A}}+\tan {\frac {\hat {\beta }}{2}}\tan {\frac {\hat {\alpha }}{2}}{\mathsf {B}}\times {\mathsf {A}}}{1-\tan {\frac {\hat {\beta }}{2}}\tan {\frac {\hat {\alpha }}{2}}{\mathsf {B}}\cdot {\mathsf {A}}}}.} 4619: 2005: 5396: 20: 8574:{\displaystyle R={\begin{pmatrix}r_{w}^{2}+r_{x}^{2}-r_{y}^{2}-r_{z}^{2}&2r_{x}r_{y}-2r_{w}r_{z}&2r_{x}r_{z}+2r_{w}r_{y}\\2r_{x}r_{y}+2r_{w}r_{z}&r_{w}^{2}-r_{x}^{2}+r_{y}^{2}-r_{z}^{2}&2r_{y}r_{z}-2r_{w}r_{x}\\2r_{x}r_{z}-2r_{w}r_{y}&2r_{y}r_{z}+2r_{w}r_{x}&r_{w}^{2}-r_{x}^{2}-r_{y}^{2}+r_{z}^{2}\\\end{pmatrix}}.} 973: 6091: 5736: 4596: 693: 7045: 7527: 100:

in three dimensions. Since the space of dual quaternions is 8-dimensional and a rigid transformation has six real degrees of freedom, three for translations and three for rotations, dual quaternions obeying two algebraic constraints are used in this application. Since unit quaternions are subject

3446: 9575:

Rodrigues, O. (1840), Des lois géométriques qui régissent les déplacements d’un système solide dans l’espace, et la variation des coordonnées provenant de ses déplacements considérés indépendamment des causes qui peuvent les produire, Journal de Mathématiques Pures et Appliquées de Liouville 5,

8006: 4835:{\displaystyle {\hat {C}}=\cos {\frac {\hat {\gamma }}{2}}+\sin {\frac {\hat {\gamma }}{2}}{\mathsf {C}}=\left(\cos {\frac {\hat {\beta }}{2}}+\sin {\frac {\hat {\beta }}{2}}{\mathsf {B}}\right)\left(\cos {\frac {\hat {\alpha }}{2}}+\sin {\frac {\hat {\alpha }}{2}}{\mathsf {A}}\right).} 6819: 6643: 1647: 5246: 1636: 1446: 1246:{\displaystyle {\hat {G}}={\hat {A}}{\hat {C}}=({\hat {a}}_{0}+{\mathsf {A}})({\hat {c}}_{0}+{\mathsf {C}})=({\hat {a}}_{0}{\hat {c}}_{0}-{\mathsf {A}}\cdot {\mathsf {C}})+({\hat {c}}_{0}{\mathsf {A}}+{\hat {a}}_{0}{\mathsf {C}}+{\mathsf {A}}\times {\mathsf {C}}).} 6430:{\displaystyle AC=A={\begin{bmatrix}c_{0}&-C_{3}&C_{2}&C_{1}\\C_{3}&c_{0}&-C_{1}&C_{2}\\-C_{2}&C_{1}&c_{0}&C_{3}\\-C_{1}&-C_{2}&-C_{3}&c_{0}\end{bmatrix}}{\begin{Bmatrix}A_{1}\\A_{2}\\A_{3}\\a_{0}\end{Bmatrix}}.} 6075:{\displaystyle AC=C={\begin{bmatrix}a_{0}&A_{3}&-A_{2}&A_{1}\\-A_{3}&a_{0}&A_{1}&A_{2}\\A_{2}&-A_{1}&a_{0}&A_{3}\\-A_{1}&-A_{2}&-A_{3}&a_{0}\end{bmatrix}}{\begin{Bmatrix}C_{1}\\C_{2}\\C_{3}\\c_{0}\end{Bmatrix}}.} 4396: 512: 6945: 4339: 2277: 3064: 7898: 3899: 7671: 2936: 3655: 7351: 3280: 3269: 3169: 7910: 7136: 8669: 5692:

The quaternion product AC is a linear transformation by the operator A of the components of the quaternion C, therefore there is a matrix representation of A operating on the vector formed from the components of C.

2800: 2000:{\displaystyle {\hat {A}}+{\hat {C}}=(A+C,B+D)=(a_{0}+c_{0})+(a_{1}+c_{1})i+(a_{2}+c_{2})j+(a_{3}+c_{3})k+(b_{0}+d_{0})\varepsilon +(b_{1}+d_{1})\varepsilon i+(b_{2}+d_{2})\varepsilon j+(b_{3}+d_{3})\varepsilon k,} 8951:, then the relations defining the dual quaternions are implied by these and vice versa. Second, the dual quaternions are isomorphic to the even part of the Clifford algebra generated by 4 anticommuting elements 6654: 6478: 104:

Similar to the way that rotations in 3D space can be represented by quaternions of unit length, rigid motions in 3D space can be represented by dual quaternions of unit length. This fact is used in theoretical

9121: 5391:{\displaystyle \cos {\frac {\hat {\gamma }}{2}}=\cos {\frac {\hat {\beta }}{2}}\cos {\frac {\hat {\alpha }}{2}}-\sin {\frac {\hat {\beta }}{2}}\sin {\frac {\hat {\alpha }}{2}}{\mathsf {B}}\cdot {\mathsf {A}}} 7770: 3538: 1457: 1267: 5727:. Notice that the components of the vector part of the quaternion are listed first and the scalar is listed last. This is an arbitrary choice, but once this convention is selected we must abide by it. 9154:. Read the article by Joe Rooney linked below for view of a supporter of W.K. Clifford's claim. Since the claims of Clifford and Study are in contention, it is convenient to use the current designation 908: 6918: 2941:

It is useful to introduce the functions Sc(∗) and Vec(∗) that select the scalar and vector parts of a quaternion, or the dual scalar and dual vector parts of a dual quaternion. In particular, if

8730: 2152: 2081: 9756: 7308: 7246: 4591:{\displaystyle {\hat {A}}=\cos({\hat {\alpha }}/2)+\sin({\hat {\alpha }}/2){\mathsf {A}}\quad {\text{and}}\quad {\hat {B}}=\cos({\hat {\beta }}/2)+\sin({\hat {\beta }}/2){\mathsf {B}}.} 315: 9015: 5689:

The matrix representation of the quaternion product is convenient for programming quaternion computations using matrix algebra, which is true for dual quaternion operations as well.

2015:

Multiplication of two dual quaternion follows from the multiplication rules for the quaternion units i, j, k and commutative multiplication by the dual unit ε. In particular, given

6440:

The dual quaternion product ÂĈ = (A, B)(C, D) = (AC, AD+BC) can be formulated as a matrix operation as follows. Assemble the components of Ĉ into the eight dimensional array Ĉ = (C

688:{\displaystyle G=AC=(a_{0}+\mathbf {A} )(c_{0}+\mathbf {C} )=(a_{0}c_{0}-\mathbf {A} \cdot \mathbf {C} )+(c_{0}\mathbf {A} +a_{0}\mathbf {C} +\mathbf {A} \times \mathbf {C} ).} 9421:

Li, Zijia; Schröcker, Hans-Peter; Scharler, Daniel F. (2022-09-07). "A Complete Characterization of Bounded Motion Polynomials Admitting a Factorization with Linear Factors".

7040:{\displaystyle {\hat {x}}=\mathbf {x} +\varepsilon \mathbf {p} \times \mathbf {x} \quad {\text{and}}\quad {\hat {X}}=\mathbf {X} +\varepsilon \mathbf {P} \times \mathbf {X} .} 4016: 8949: 10063: 8858: 4253: 10378: 10301: 10262: 10224: 10196: 10168: 10140: 10028: 9995: 9967: 9939: 7802: 7579: 3941: 8891: 7550: 7050:

Then the dual quaternion of the displacement of this body transforms Plücker coordinates in the moving frame to Plücker coordinates in the fixed frame by the formula

2163: 2964: 8920: 8809: 8789: 8769: 7340: 4056: 4036: 3961: 9466:

W. R. Hamilton, "On quaternions, or on a new system of imaginaries in algebra," Phil. Mag. 18, installments July 1844 – April 1850, ed. by D. E. Wilkins (2000)

3734: 7810: 7587: 7522:{\displaystyle r=r_{w}+r_{x}i+r_{y}j+r_{z}k=\cos \left({\frac {\theta }{2}}\right)+\sin \left({\frac {\theta }{2}}\right)\left({\vec {a}}\cdot (i,j,k)\right)} 2821: 3549: 3441:{\displaystyle {\hat {A}}{\hat {A}}^{*}=({\hat {a}}_{0}+{\mathsf {A}})({\hat {a}}_{0}-{\mathsf {A}})={\hat {a}}_{0}^{2}+{\mathsf {A}}\cdot {\mathsf {A}}.} 9767: 5670:' formula for the screw axis of a composite displacement defined in terms of the screw axes of the two displacements. He derived this formula in 1840. 8001:{\displaystyle T={\begin{pmatrix}1&0&0&0&\\\Delta x&&&\\\Delta y&&R&\\\Delta z&&&\\\end{pmatrix}}} 3180: 3079: 6648:

As we saw for quaternions, the product ÂĈ can be viewed as the operation of Ĉ on the coordinate vector Â, which means ÂĈ can also be formulated as,

9146:

used and wrote about dual quaternions, at times authors refer to dual quaternions as "Study biquaternions" or "Clifford biquaternions". The latter

7056: 9182: 8590: 6814:{\displaystyle {\hat {A}}{\hat {C}}={\hat {A}}={\begin{bmatrix}C^{-}&0\\D^{-}&C^{-}\end{bmatrix}}{\begin{Bmatrix}A\\B\end{Bmatrix}}.} 6638:{\displaystyle {\hat {A}}{\hat {C}}={\hat {C}}={\begin{bmatrix}A^{+}&0\\B^{+}&A^{+}\end{bmatrix}}{\begin{Bmatrix}C\\D\end{Bmatrix}}.} 2704: 5681:

between the common normals that form the sides of this triangle are directly related to the dual angles of the three spatial displacements.

9023: 3660:

In the context of dual quaternions, the term "conjugate" can be used to mean the quaternion conjugate, dual number conjugate, or both.

1631:{\displaystyle {\hat {C}}=(C,D)=c_{0}+c_{1}i+c_{2}j+c_{3}k+d_{0}\varepsilon +d_{1}\varepsilon i+d_{2}\varepsilon j+d_{3}\varepsilon k,} 1441:{\displaystyle {\hat {A}}=(A,B)=a_{0}+a_{1}i+a_{2}j+a_{3}k+b_{0}\varepsilon +b_{1}\varepsilon i+b_{2}\varepsilon j+b_{3}\varepsilon k,} 408:

A convenient way to work with the quaternion product is to write a quaternion as the sum of a scalar and a vector (strictly speaking a

9842: 9826: 7683: 3469: 7904:

These dual quaternions (or actually their transformations on 3D-vectors) can be represented by the homogeneous transformation matrix

9889: 803: 6863: 9739: 9365: 163:

used Ω with Ω = 0 to generate the dual quaternion algebra. However, his terminology of "octonions" did not stick as today's

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Dual quaternions of magnitude 1 are used to represent spatial Euclidean displacements. Notice that the requirement that

472:

is a three dimensional vector. The vector dot and cross operations can now be used to define the quaternion product of

2092: 2021: 9855: 9788:

A beginners guide to dual-quaternions: What they are, how they work, and how to use them for 3d character hierarchies

9637: 9303: 7260: 9177: 8680: 10033: 9748:

E. Pennestri & R. Stefanelli (2007) Linear Algebra and Numerical Algorithms Using Dual Numbers, published in

10451: 7266: 10529: 9644: 7147: 246: 10412: 8954: 9882: 4126: 10038: 9798:

D.P. Chevallier (1996) "On the transference principle in kinematics: its various forms and limitations",

9757:

Dual Quaternions as a Tool for Rigid Body Motion Analysis: A Tutorial with an Application to Biomechanics

9410:, Proc. of the XXIV IEEE Conference on Computer Vision and Pattern Recognition, pp. 2441-2448, June 2011. 81:

and commutes with every element of the algebra. Unlike quaternions, the dual quaternions do not form a

9490:

W. K. Clifford, "Preliminary sketch of bi-quaternions, Proc. London Math. Soc. Vol. 4 (1873) pp. 381–395

9320: 3463:

A second type of conjugate of a dual quaternion is given by taking the dual number conjugate, given by

125:

with coefficients given by (non-zero real norm) dual quaternions have also been used in the context of

4334:{\displaystyle {\hat {S}}=\cos {\frac {\hat {\phi }}{2}}+\sin {\frac {\hat {\phi }}{2}}{\mathsf {S}}.} 3966: 10446: 10402: 8925: 6085:

The product AC can also be viewed as an operation by C on the components of A, in which case we have

10044: 9127: 8814: 10569: 10441: 9143: 190:

called the structure "Study biquaternions", one of three eight-dimensional algebras referred to as

145: 10361: 10284: 10245: 10207: 10179: 10151: 10123: 10011: 9978: 9950: 9922: 9794:. International Conference on Computer Graphics, Visualization and Computer Vision. pp. 1–13. 9405: 3543:

The quaternion and dual number conjugates can be combined into a third form of conjugate given by

9875: 9864:: a standalone open-source library for using dual quaternions within robot modelling and control. 179: 9585: 7778: 7555: 101:

to two algebraic constraints, unit quaternions are standard to represent rigid transformations.

3907: 2272:{\displaystyle {\hat {A}}{\hat {C}}=(A+\varepsilon B)(C+\varepsilon D)=AC+\varepsilon (AD+BC).} 3059:{\displaystyle {\mbox{Sc}}({\hat {A}})={\hat {a}}_{0},{\mbox{Vec}}({\hat {A}})={\mathsf {A}}.} 2698:

The conjugate of a dual quaternion is the extension of the conjugate of a quaternion, that is

10514: 10350: 9670: 8863: 7535: 6837:

that defines the rotation and the vector quaternion constructed from the translation vector

5674: 4145:

A benefit of the dual quaternion formulation of the composition of two spatial displacements

698:

A dual quaternion is usually described as a quaternion with dual numbers as coefficients. A

187: 9392:

Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra — Part I: Fundamentals

7259:

It might be helpful, especially in rigid body motion, to represent unit dual quaternions as

2293:

This gives us the multiplication table (note the multiplication order is row times column):

10600: 10267: 10000: 4117:

of the dual number ring then consists of numbers not in the ideal. The dual numbers form a

913:

It is convenient to write a dual quaternion as the sum of a dual scalar and a dual vector,

198: 126: 97: 36: 8896: 8: 10595: 10478: 10388: 10345: 10327: 10105: 9607: 9516: 9475: 9192: 4103: 3894:{\displaystyle {\hat {A}}{\hat {A}}^{*}=(A,B)(A^{*},B^{*})=(AA^{*},AB^{*}+BA^{*})=(1,0).} 175: 9604:

Application of quaternion algebra and dual numbers to the analysis of spatial mechanisms

9232:"Dual Quaternion Framework for Modeling of Spacecraft-Mounted Multibody Robotic Systems" 9218:

Application of Quaternion Algebra and Dual Numbers to the Analysis of Spatial Mechanisms

7893:{\displaystyle {\hat {v}}'={\hat {q}}\cdot {\hat {v}}\cdot {\overline {{\hat {q}}^{*}}}} 7666:{\displaystyle d=0+{\frac {\Delta x}{2}}i+{\frac {\Delta y}{2}}j+{\frac {\Delta z}{2}}k} 10383: 10095: 9805:

M.A. Gungor (2009) "Dual Lorentzian spherical motions and dual Euler-Savary formulas",

9713: 9686: 9447: 9422: 9371: 9266: 9231: 9151: 8794: 8774: 8754: 7325: 4041: 4021: 3946: 2931:{\displaystyle {\hat {G}}^{*}=({\hat {A}}{\hat {C}})^{*}={\hat {C}}^{*}{\hat {A}}^{*}.} 785:

The result is that a dual quaternion can be written as an ordered pair of quaternions

209:

In order to describe operations with dual quaternions, it is helpful to first consider

4134: 3650:{\displaystyle {\overline {{\hat {A}}^{*}}}=(A^{*},-B^{*})=A^{*}-\varepsilon B^{*}.\!} 10590: 10541: 10504: 10468: 10407: 10393: 10088: 10068: 9735: 9718: 9653: 9633: 9512: 9361: 9346:"Robust kinematic control of manipulator robots using dual quaternion representation" 9299: 9271: 9253: 8012: 160: 110: 19: 10559: 10488: 10463: 10397: 10306: 10272: 10113: 10083: 10005: 9908: 9833: 9813: 9775: 9708: 9698: 9520: 9353: 9261: 9243: 9187: 8748: 8740:

Besides being the tensor product of two Clifford algebras, the quaternions and the

5667: 153: 82: 9375: 8744:, the dual quaternions have two other formulations in terms of Clifford algebras. 6931:

in a moving body and its coordinates in the fixed frame which is in the direction

51:. Thus, they may be constructed in the same way as the quaternions, except using 10436: 9972: 9786: 9779: 9560: 9293: 9172: 4130: 118: 23:

Plaque on Broom bridge (Dublin) commemorating Hamilton's invention of quaternions

9390: 3264:{\displaystyle ({\hat {a}}_{0}+{\mathsf {A}})^{*}={\hat {a}}_{0}-{\mathsf {A}}.} 3164:{\displaystyle {\hat {A}}^{*}={\mbox{Sc}}({\hat {A}})-{\mbox{Vec}}({\hat {A}}).} 10483: 10473: 10458: 10277: 10145: 9916: 9687:"3D kinematics using dual quaternions: theory and applications in neuroscience" 9345: 4114: 201:

developed dual vectors and dual quaternions for use in the study of mechanics.

137: 40: 9357: 10584: 10546: 10519: 10428: 9703: 9257: 9248: 4107: 93: 9321:"Dual-Quaternions: From Classical Mechanics to Computer Graphics and Beyond" 7131:{\displaystyle {\hat {X}}={\hat {S}}{\hat {x}}{\overline {{\hat {S}}^{*}}}.} 10509: 10311: 9722: 9666: 9531: 9442:

Huczala, D.; Siegele, J.; Thimm, D.; Pfurner, M.; Schröcker, H.-P. (2024).

9275: 9167: 9139: 8741: 967:. This notation allows us to write the product of two dual quaternions as 964: 191: 171: 48: 1261:

The addition of dual quaternions is defined componentwise so that given,

10335: 10117: 699: 56: 52: 44: 28: 8664:{\displaystyle v={\begin{pmatrix}1\\v_{x}\\v_{y}\\v_{z}\\\end{pmatrix}}} 6853:

k. Using this notation, the dual quaternion for the displacement D=(,

10316: 10173: 5730:

The quaternion product AC can now be represented as the matrix product

4213: 4204:

In general, the dual quaternion associated with a spatial displacement

4172: 4118: 2805:

As with quaternions, the conjugate of the product of dual quaternions,

2795:{\displaystyle {\hat {A}}^{*}=(A^{*},B^{*})=A^{*}+\varepsilon B^{*}.\!} 210: 141: 122: 106: 9344:

Figueredo, L.F.C.; Adorno, B.V.; Ishihara, J.Y.; Borges, G.A. (2013).

4122: 89: 4129:. Dual quaternions have been used to exhibit transformations in the 797:. Two dual quaternions add componentwise and multiply by the rule, 10424: 10355: 10201: 9659:

Joe Rooney (2007) "William Kingdon Clifford", in Marco Ceccarelli,

9656:, Department of Design and Innovation, the Open University, London. 9618: 9452: 9427: 7141:

Using the matrix form of the dual quaternion product this becomes,

409: 164: 59:

as coefficients. A dual quaternion can be represented in the form

9768:

Linear Dual Algebra Algorithms and their Application to Kinematics

7552:

is the angle of rotation about the direction given by unit vector

7254: 9944: 9867: 9116:{\displaystyle e_{1}^{2}=e_{2}^{2}=e_{3}^{2}=-1,\,\,e_{4}^{2}=0.} 77:

are ordinary quaternions and ε is the dual unit, which satisfies

9861: 7765:{\displaystyle {\hat {v}}:=1+\varepsilon (v_{x}i+v_{y}j+v_{z}k)} 5684: 3533:{\displaystyle {\overline {\hat {A}}}=(A,-B)=A-\varepsilon B.\!} 148:

obtained a broad generalization of these numbers that he called

9898: 9147: 4243:

the slide along this axis, which defines the displacement

4121:

since there is a unique maximal ideal. The group of units is a

717:. Two dual numbers add componentwise and multiply by the rule 9685:

Leclercq, Guillaume; Lefevre, Philippe; Blohm, Gunnar (2013).

9549:

Screw calculus and some applications to geometry and mechanics

9408:

Multiview Registration via Graph Diffusion of Dual Quaternions

6833:) can be constructed from the quaternion S=cos(φ/2) + sin(φ/2) 216:

A quaternion is a linear combination of the basis elements 1,

9350:

2013 IEEE International Conference on Robotics and Automation

9343: 4140: 405:, which links quaternions to the properties of determinants. 114: 7251:

This calculation is easily managed using matrix operations.

4171:) is that the resulting dual quaternion yields directly the 3724:, introduces two algebraic constraints on the components of 7322:

quaternion is known as the real or rotational part and the

4080:

is not zero, then the inverse dual quaternion is given by

3274:

The product of a dual quaternion with its conjugate yields

2286:

term, because the definition of dual numbers requires that

903:{\displaystyle {\hat {A}}{\hat {C}}=(A,B)(C,D)=(AC,AD+BC).} 9441: 6913:{\displaystyle {\hat {S}}=S+\varepsilon {\frac {1}{2}}DS.} 9629: 4382:

is obtained from the product of the dual quaternions of D

9816:(1958). "Anwendung dualer Quaternionen auf Kinematik". 9732:

Dual-Number Methods in Kinematics, Statics and Dynamics

6923:

Let the Plücker coordinates of a line in the direction

2815:, is the product of their conjugates in reverse order, 9661:

Distinguished figures in mechanism and machine science

9444:

Rational Linkages: From Poses to 3D-printed Prototypes

8605: 8036: 7925: 7342:

quaternion is known as the dual or displacement part.

7263:. As given above a dual quaternion can be written as: 6787: 6730: 6611: 6554: 6361: 6131: 6006: 5776: 3134: 3106: 3019: 2969: 10364: 10287: 10248: 10210: 10182: 10154: 10126: 10047: 10014: 9981: 9953: 9925: 9026: 8957: 8928: 8899: 8866: 8817: 8797: 8777: 8757: 8683: 8593: 8024: 7913: 7813: 7781: 7686: 7590: 7558: 7538: 7354: 7328: 7269: 7150: 7059: 6948: 6866: 6657: 6481: 6094: 5739: 5410: 5249: 4854: 4622: 4399: 4256: 4044: 4024: 3969: 3949: 3910: 3737: 3552: 3472: 3283: 3183: 3082: 2967: 2824: 2707: 2166: 2095: 2024: 1650: 1460: 1270: 976: 806: 515: 249: 9684: 9643:

L. Kavan, S. Collins, C. O'Sullivan, J. Zara (2006)

9517:

Octonions: a development of Clifford's Biquaternions

9420: 5240:Divide both sides of this equation by the identity 10372: 10295: 10256: 10218: 10190: 10162: 10134: 10057: 10022: 9989: 9961: 9933: 9646:Dual Quaternions for Rigid Transformation Blending 9586:Dual Quaternions for Rigid Transformation Blending 9115: 9009: 8943: 8914: 8885: 8852: 8803: 8783: 8763: 8735: 8724: 8663: 8573: 8000: 7892: 7796: 7764: 7665: 7573: 7544: 7521: 7334: 7302: 7240: 7130: 7039: 6912: 6813: 6637: 6429: 6074: 5655: 5390: 5229: 4834: 4590: 4333: 4050: 4030: 4010: 3955: 3935: 3893: 3678:| is computed using the conjugate to compute 3649: 3532: 3440: 3263: 3163: 3058: 2930: 2794: 2271: 2146: 2075: 1999: 1630: 1440: 1245: 902: 687: 309: 7677:The dual-quaternion equivalent of a 3D-vector is 4102:do not have inverses. This subspace is called an 3646: 3529: 2791: 2147:{\displaystyle {\hat {C}}=(C,D)=C+\varepsilon D,} 2076:{\displaystyle {\hat {A}}=(A,B)=A+\varepsilon B,} 178:was ideal for describing the group of motions of 10582: 9230:Valverde, Alfredo; Tsiotras, Panagiotis (2018). 9229: 6824: 9183:Conversion between quaternions and Euler angles 7255:Dual quaternions and 4×4 homogeneous transforms 4247:. The associated dual quaternion is given by, 3700:of the dual quaternion. Dual quaternions with 3069:This allows the definition of the conjugate of 2297:Multiplication table for dual quaternion units 16:Eight-dimensional algebra over the real numbers 9840:. Berlin: Deutscher Verlag der Wissenschaften. 9388: 8747:First, dual quaternions are isomorphic to the 6472:), then ÂĈ is given by the 8x8 matrix product 3963:has magnitude 1, while the second constraint, 766:, where ε is the dual unit that commutes with 752:. Dual numbers are often written in the form 152:, which is an example of what is now called a 9883: 8725:{\displaystyle {\vec {v}}'=T\cdot {\vec {v}}} 5685:Matrix form of dual quaternion multiplication 4175:and dual angle of the composite displacement 4613:has the associated dual quaternion given by 4106:in ring theory. It happens to be the unique 9649:, Technical report, Trinity College Dublin. 9503:, (ed. R. Tucker), London: Macmillan, 1882. 7303:{\displaystyle {\hat {q}}=r+d\varepsilon r} 6829:The dual quaternion of a displacement D=(, 10565: 9890: 9876: 9287: 9285: 7581:. The displacement part can be written as 7241:{\displaystyle {\hat {X}}=^{*}{\hat {x}}.} 5696:Assemble the components of the quaternion 4141:Dual quaternions and spatial displacements 109:(see McCarthy), and in applications to 3D 10366: 10289: 10250: 10212: 10184: 10156: 10128: 10016: 9983: 9955: 9927: 9843:Translation in English by D.H. Delphenich 9827:Translation in English by D.H. Delphenich 9784: 9712: 9702: 9626:An Introduction to Theoretical Kinematics 9613:A.T. Yang (1974) "Calculus of Screws" in 9480:, Longmans, Green & Co., London, 1866 9451: 9426: 9406:A. Torsello, E. Rodolà and A. Albarelli, 9295:An Introduction to Theoretical Kinematics 9265: 9247: 9220:, Ph.D thesis, Columbia University, 1963. 9091: 9090: 4344:Let the composition of the displacement D 310:{\displaystyle i^{2}=j^{2}=k^{2}=ijk=-1.} 9812: 9291: 5673:The three screw axes A, B, and C form a 4133:. A typical element can be written as a 3458: 92:, the dual quaternions are applied as a 18: 9729: 9282: 9010:{\displaystyle e_{1},e_{2},e_{3},e_{4}} 4845:Expand this product in order to obtain 10583: 9818:Annales Academiae Scientiarum Fennicae 9807:European Journal of Mechanics A Solids 9669:(1891) "Von Bewegungen und Umlegung", 8751:generated by 3 anticommuting elements 5642: 5632: 5571: 5561: 5505: 5472: 5436: 5383: 5373: 5214: 5204: 5148: 5092: 5026: 5016: 4906: 4819: 4753: 4689: 4580: 4484: 4323: 3430: 3420: 3380: 3345: 3253: 3211: 3048: 1232: 1222: 1212: 1183: 1148: 1138: 1081: 1046: 9871: 9446:. Advances in Robot Kinematics 2024. 9318: 4601:That is, the composite displacement D 4378:. The screw axis and dual angle of D 9832: 9691:Frontiers in Behavioral Neuroscience 8674:the transformation by T is given by 3696:. This is a dual number called the 10079:Set-theoretically definable numbers 9128:Clifford algebras: dual quaternions 3451:This is a dual scalar which is the 182:. He further developed the idea in 13: 10050: 9897: 9774:, October 2008, pp. 207–229, 9766:E. Pennestri and P. P. Valentini, 9755:E. Pennestri and P. P. Valentini, 9678: 7981: 7964: 7951: 7648: 7627: 7606: 7345:The rotation part can be given by 4098:Thus the elements of the subspace 963:is the dual vector that defines a 395:, we see that this product yields 14: 10612: 9849: 9763:, vol. 57, pp. 187–205, 2010 2010: 10564: 9615:Basic Questions of Design Theory 9178:Quaternions and spatial rotation 7030: 7022: 7011: 6984: 6976: 6965: 4011:{\displaystyle AB^{*}+BA^{*}=0,} 3904:The first of these constraints, 675: 667: 659: 641: 617: 609: 572: 548: 9617:, William R. Spillers, editor, 9579: 9569: 9565:, North Holland Publ. Co., 1979 9554: 9541: 9525: 9506: 9493: 9484: 9469: 9460: 9150:has also been used to refer to 8944:{\displaystyle \varepsilon =ek} 8736:Connection to Clifford algebras 6994: 6988: 4495: 4489: 228:. Hamilton's product rule for 197:In 1895, Russian mathematician 10058:{\displaystyle {\mathcal {P}}} 9435: 9414: 9399: 9389:Vilhena Adorno, Bruno (2017). 9382: 9337: 9312: 9223: 9210: 8853:{\displaystyle i^{2}=j^{2}=-1} 8716: 8691: 7873: 7855: 7840: 7821: 7788: 7759: 7711: 7693: 7565: 7511: 7493: 7484: 7276: 7229: 7214: 7201: 7191: 7188: 7176: 7166: 7157: 7108: 7093: 7081: 7066: 7001: 6955: 6873: 6716: 6707: 6695: 6685: 6676: 6664: 6540: 6531: 6519: 6509: 6500: 6488: 6117: 6104: 5762: 5749: 5620: 5597: 5549: 5526: 5493: 5460: 5424: 5361: 5338: 5312: 5289: 5263: 5192: 5169: 5136: 5113: 5080: 5057: 5004: 4981: 4955: 4932: 4894: 4868: 4807: 4781: 4741: 4715: 4677: 4651: 4629: 4575: 4561: 4552: 4540: 4526: 4517: 4502: 4479: 4465: 4456: 4444: 4430: 4421: 4406: 4311: 4285: 4263: 3885: 3873: 3867: 3819: 3813: 3787: 3784: 3772: 3757: 3744: 3611: 3582: 3562: 3508: 3493: 3480: 3398: 3385: 3363: 3353: 3350: 3328: 3318: 3303: 3290: 3236: 3217: 3194: 3184: 3155: 3149: 3140: 3127: 3121: 3112: 3090: 3040: 3034: 3025: 3003: 2990: 2984: 2975: 2913: 2894: 2875: 2868: 2856: 2847: 2832: 2756: 2730: 2715: 2263: 2245: 2227: 2212: 2209: 2194: 2185: 2173: 2123: 2111: 2102: 2052: 2040: 2031: 1985: 1959: 1947: 1921: 1909: 1883: 1874: 1848: 1839: 1813: 1804: 1778: 1769: 1743: 1737: 1711: 1705: 1681: 1672: 1657: 1488: 1476: 1467: 1298: 1286: 1277: 1237: 1198: 1169: 1159: 1153: 1121: 1102: 1092: 1086: 1064: 1054: 1051: 1029: 1019: 1010: 998: 983: 894: 867: 861: 849: 846: 834: 825: 813: 679: 627: 621: 582: 576: 555: 552: 531: 1: 10413:Plane-based geometric algebra 9198: 6825:More on spatial displacements 5677:and the dual angles at these 4125:and can be studied using the 4110:of the ring of dual numbers. 10373:{\displaystyle \mathbb {S} } 10296:{\displaystyle \mathbb {C} } 10257:{\displaystyle \mathbb {R} } 10219:{\displaystyle \mathbb {O} } 10191:{\displaystyle \mathbb {H} } 10163:{\displaystyle \mathbb {C} } 10135:{\displaystyle \mathbb {R} } 10023:{\displaystyle \mathbb {A} } 9990:{\displaystyle \mathbb {Q} } 9962:{\displaystyle \mathbb {Z} } 9934:{\displaystyle \mathbb {N} } 9800:Mechanism and Machine Theory 9780:10.1007/978-1-4020-8829-2_11 9298:. MIT Press. pp. 62–5. 9236:Frontiers in Robotics and AI 7885: 7120: 3574: 3485: 2693: 7: 9161: 3672:of a dual quaternion | 1256: 204: 10: 10617: 9838:Kinematik und Quaternionen 9133: 7797:{\displaystyle {\hat {q}}} 7775:and its transformation by 7574:{\displaystyle {\vec {a}}} 7318:are both quaternions. The 4239:is the rotation about and 4212:) is constructed from its 4076:is a dual quaternion, and 4061: 132: 35:are an 8-dimensional real 10555: 10497: 10423: 10403:Algebra of physical space 10325: 10233: 10104: 9906: 9750:Multibody System Dynamics 9551:, Annal. Imp. Univ. Kazan 9358:10.1109/ICRA.2013.6630836 3936:{\displaystyle AA^{*}=1,} 2650: 2609: 2568: 2530: 2483: 2436: 2389: 2345: 2337: 2331: 2325: 2322: 2317: 2312: 2307: 2304: 2301: 10459:Extended complex numbers 10442:Extended natural numbers 9704:10.3389/fnbeh.2013.00007 9654:William Kingdon Clifford 9561:O. Bottema and B. Roth, 9547:A. P. Kotelnikov (1895) 9292:McCarthy, J. M. (1990). 9249:10.3389/frobt.2018.00128 9144:William Kingdon Clifford 3455:of the dual quaternion. 2282:Notice that there is no 9856:Dual quaternion toolbox 9785:Kenwright, Ben (2012). 9478:Elements of Quaternions 8886:{\displaystyle e^{2}=0} 7545:{\displaystyle \theta } 3663: 180:three-dimensional space 10515:Transcendental numbers 10374: 10351:Hyperbolic quaternions 10297: 10258: 10220: 10192: 10164: 10136: 10059: 10024: 9991: 9963: 9935: 9761:ARCHIWUM BUDOWY MASZYN 9563:Theoretical Kinematics 9352:. pp. 1949–1955. 9117: 9011: 8945: 8916: 8887: 8854: 8805: 8785: 8765: 8726: 8665: 8575: 8002: 7894: 7798: 7766: 7667: 7575: 7546: 7523: 7336: 7304: 7242: 7132: 7041: 6914: 6815: 6639: 6431: 6076: 5657: 5392: 5231: 4836: 4592: 4335: 4227:) and the dual angle ( 4052: 4032: 4012: 3957: 3937: 3895: 3651: 3534: 3442: 3265: 3165: 3060: 2932: 2796: 2273: 2148: 2077: 2001: 1632: 1442: 1247: 904: 689: 311: 24: 10447:Extended real numbers 10375: 10298: 10268:Split-complex numbers 10259: 10221: 10193: 10165: 10137: 10060: 10025: 10001:Constructible numbers 9992: 9964: 9936: 9730:Fischer, Ian (1998). 9671:Mathematische Annalen 9624:J.M. McCarthy (1990) 9536:Geometrie der Dynamen 9118: 9012: 8946: 8917: 8888: 8855: 8806: 8786: 8766: 8727: 8666: 8576: 8003: 7895: 7799: 7767: 7668: 7576: 7547: 7524: 7337: 7305: 7243: 7133: 7042: 6915: 6816: 6640: 6432: 6077: 5658: 5393: 5232: 4837: 4593: 4336: 4208: = (, 4100:{ ε q : q ∈ H } 4053: 4033: 4013: 3958: 3938: 3896: 3712:unit dual quaternions 3652: 3535: 3459:Dual number conjugate 3443: 3266: 3166: 3061: 2933: 2797: 2274: 2149: 2078: 2002: 1633: 1443: 1248: 905: 778:and has the property 690: 436:is a real number and 312: 188:B. L. van der Waerden 184:Geometrie der Dynamen 167:are another algebra. 144:in 1843, and by 1873 98:rigid transformations 22: 10479:Supernatural numbers 10389:Multicomplex numbers 10362: 10346:Dual-complex numbers 10285: 10246: 10208: 10180: 10152: 10124: 10106:Composition algebras 10074:Arithmetical numbers 10045: 10012: 9979: 9951: 9923: 9820:. Gesammelte Werke. 9024: 8955: 8926: 8915:{\displaystyle k=ij} 8897: 8864: 8815: 8795: 8775: 8755: 8681: 8591: 8022: 7911: 7811: 7779: 7684: 7588: 7556: 7536: 7352: 7326: 7267: 7261:homogeneous matrices 7148: 7057: 6946: 6864: 6655: 6479: 6092: 5737: 5408: 5247: 4852: 4620: 4397: 4352:be the displacement 4254: 4135:screw transformation 4042: 4022: 3967: 3947: 3908: 3735: 3550: 3470: 3281: 3181: 3080: 2965: 2822: 2705: 2164: 2093: 2022: 1648: 1458: 1268: 974: 804: 513: 247: 240:is often written as 199:Aleksandr Kotelnikov 10384:Split-biquaternions 10096:Eisenstein integers 10034:Closed-form numbers 9858:, a Matlab toolbox. 9621:, pages 266 to 281. 9608:Columbia University 9501:Mathematical Papers 9193:Dual-complex number 9158:to avoid conflict. 9152:split-biquaternions 9106: 9077: 9059: 9041: 8559: 8541: 8523: 8505: 8333: 8315: 8297: 8279: 8107: 8089: 8071: 8053: 4127:exponential mapping 3414: 2298: 702:is an ordered pair 176:associative algebra 174:realized that this 127:mechanical linkages 10542:Profinite integers 10505:Irrational numbers 10370: 10293: 10254: 10216: 10188: 10160: 10132: 10089:Gaussian rationals 10069:Computable numbers 10055: 10020: 9987: 9959: 9931: 9772:Multibody Dynamics 9538:, Teubner, Leipzig 9113: 9092: 9063: 9045: 9027: 9007: 8941: 8912: 8883: 8850: 8801: 8781: 8761: 8722: 8661: 8655: 8584:For the 3D-vector 8571: 8562: 8545: 8527: 8509: 8491: 8319: 8301: 8283: 8265: 8093: 8075: 8057: 8039: 7998: 7992: 7890: 7794: 7762: 7663: 7571: 7542: 7519: 7332: 7300: 7238: 7128: 7037: 6935:through the point 6910: 6811: 6802: 6776: 6635: 6626: 6600: 6427: 6418: 6350: 6072: 6063: 5995: 5653: 5388: 5227: 4832: 4588: 4331: 4048: 4028: 4008: 3953: 3933: 3891: 3647: 3530: 3438: 3391: 3261: 3161: 3138: 3110: 3056: 3023: 2973: 2928: 2792: 2296: 2269: 2144: 2073: 1997: 1628: 1438: 1243: 900: 685: 307: 39:isomorphic to the 25: 10578: 10577: 10489:Superreal numbers 10469:Levi-Civita field 10464:Hyperreal numbers 10408:Spacetime algebra 10394:Geometric algebra 10307:Bicomplex numbers 10273:Split-quaternions 10114:Division algebras 10084:Gaussian integers 10006:Algebraic numbers 9909:definable numbers 9834:Blaschke, Wilhelm 9814:Blaschke, Wilhelm 9741:978-0-8493-9115-6 9628:, pp. 62–5, 9602:A.T. Yang (1963) 9513:Alexander McAulay 9367:978-1-4673-5643-5 9126:For details, see 8804:{\displaystyle e} 8784:{\displaystyle j} 8764:{\displaystyle i} 8719: 8694: 8013:orthogonal matrix 7888: 7876: 7858: 7843: 7824: 7791: 7696: 7658: 7637: 7616: 7568: 7487: 7467: 7440: 7335:{\displaystyle d} 7279: 7232: 7204: 7179: 7160: 7123: 7111: 7096: 7084: 7069: 7004: 6992: 6958: 6899: 6876: 6719: 6698: 6679: 6667: 6543: 6522: 6503: 6491: 5648: 5628: 5623: 5605: 5600: 5557: 5552: 5534: 5529: 5501: 5496: 5468: 5463: 5432: 5427: 5369: 5364: 5346: 5341: 5320: 5315: 5297: 5292: 5271: 5266: 5200: 5195: 5177: 5172: 5144: 5139: 5121: 5116: 5088: 5083: 5065: 5060: 5012: 5007: 4989: 4984: 4963: 4958: 4940: 4935: 4902: 4897: 4876: 4871: 4815: 4810: 4789: 4784: 4749: 4744: 4723: 4718: 4685: 4680: 4659: 4654: 4632: 4564: 4529: 4505: 4493: 4468: 4433: 4409: 4319: 4314: 4293: 4288: 4266: 4051:{\displaystyle B} 4031:{\displaystyle A} 3956:{\displaystyle A} 3760: 3747: 3577: 3565: 3488: 3483: 3453:magnitude squared 3401: 3366: 3331: 3306: 3293: 3239: 3197: 3152: 3137: 3124: 3109: 3093: 3037: 3022: 3006: 2987: 2972: 2916: 2897: 2871: 2859: 2835: 2718: 2691: 2690: 2188: 2176: 2105: 2034: 1675: 1660: 1470: 1280: 1201: 1172: 1124: 1105: 1067: 1032: 1013: 1001: 986: 828: 816: 161:Alexander McAulay 111:computer graphics 10608: 10568: 10567: 10535: 10525: 10437:Cardinal numbers 10398:Clifford algebra 10379: 10377: 10376: 10371: 10369: 10341:Dual quaternions 10302: 10300: 10299: 10294: 10292: 10263: 10261: 10260: 10255: 10253: 10225: 10223: 10222: 10217: 10215: 10197: 10195: 10194: 10189: 10187: 10169: 10167: 10166: 10161: 10159: 10141: 10139: 10138: 10133: 10131: 10064: 10062: 10061: 10056: 10054: 10053: 10029: 10027: 10026: 10021: 10019: 9996: 9994: 9993: 9988: 9986: 9973:Rational numbers 9968: 9966: 9965: 9960: 9958: 9940: 9938: 9937: 9932: 9930: 9892: 9885: 9878: 9869: 9868: 9841: 9825: 9795: 9793: 9745: 9726: 9716: 9706: 9589: 9583: 9577: 9573: 9567: 9558: 9552: 9545: 9539: 9529: 9523: 9521:Internet Archive 9510: 9504: 9499:W. K. Clifford, 9497: 9491: 9488: 9482: 9476:W. R. Hamilton, 9473: 9467: 9464: 9458: 9457: 9455: 9439: 9433: 9432: 9430: 9418: 9412: 9403: 9397: 9396: 9386: 9380: 9379: 9341: 9335: 9334: 9332: 9330: 9325: 9319:Kenwright, Ben. 9316: 9310: 9309: 9289: 9280: 9279: 9269: 9251: 9227: 9221: 9214: 9188:Olinde Rodrigues 9122: 9120: 9119: 9114: 9105: 9100: 9076: 9071: 9058: 9053: 9040: 9035: 9016: 9014: 9013: 9008: 9006: 9005: 8993: 8992: 8980: 8979: 8967: 8966: 8950: 8948: 8947: 8942: 8921: 8919: 8918: 8913: 8893:. If we define 8892: 8890: 8889: 8884: 8876: 8875: 8859: 8857: 8856: 8851: 8840: 8839: 8827: 8826: 8810: 8808: 8807: 8802: 8790: 8788: 8787: 8782: 8770: 8768: 8767: 8762: 8749:Clifford algebra 8731: 8729: 8728: 8723: 8721: 8720: 8712: 8700: 8696: 8695: 8687: 8670: 8668: 8667: 8662: 8660: 8659: 8652: 8651: 8638: 8637: 8624: 8623: 8580: 8578: 8577: 8572: 8567: 8566: 8558: 8553: 8540: 8535: 8522: 8517: 8504: 8499: 8488: 8487: 8478: 8477: 8462: 8461: 8452: 8451: 8437: 8436: 8427: 8426: 8411: 8410: 8401: 8400: 8384: 8383: 8374: 8373: 8358: 8357: 8348: 8347: 8332: 8327: 8314: 8309: 8296: 8291: 8278: 8273: 8262: 8261: 8252: 8251: 8236: 8235: 8226: 8225: 8209: 8208: 8199: 8198: 8183: 8182: 8173: 8172: 8158: 8157: 8148: 8147: 8132: 8131: 8122: 8121: 8106: 8101: 8088: 8083: 8070: 8065: 8052: 8047: 8007: 8005: 8004: 7999: 7997: 7996: 7990: 7989: 7988: 7977: 7971: 7960: 7959: 7958: 7947: 7899: 7897: 7896: 7891: 7889: 7884: 7883: 7878: 7877: 7869: 7865: 7860: 7859: 7851: 7845: 7844: 7836: 7830: 7826: 7825: 7817: 7803: 7801: 7800: 7795: 7793: 7792: 7784: 7771: 7769: 7768: 7763: 7755: 7754: 7739: 7738: 7723: 7722: 7698: 7697: 7689: 7672: 7670: 7669: 7664: 7659: 7654: 7646: 7638: 7633: 7625: 7617: 7612: 7604: 7580: 7578: 7577: 7572: 7570: 7569: 7561: 7551: 7549: 7548: 7543: 7528: 7526: 7525: 7520: 7518: 7514: 7489: 7488: 7480: 7472: 7468: 7460: 7445: 7441: 7433: 7415: 7414: 7399: 7398: 7383: 7382: 7370: 7369: 7341: 7339: 7338: 7333: 7309: 7307: 7306: 7301: 7281: 7280: 7272: 7247: 7245: 7244: 7239: 7234: 7233: 7225: 7222: 7221: 7212: 7211: 7206: 7205: 7197: 7187: 7186: 7181: 7180: 7172: 7162: 7161: 7153: 7137: 7135: 7134: 7129: 7124: 7119: 7118: 7113: 7112: 7104: 7100: 7098: 7097: 7089: 7086: 7085: 7077: 7071: 7070: 7062: 7046: 7044: 7043: 7038: 7033: 7025: 7014: 7006: 7005: 6997: 6993: 6990: 6987: 6979: 6968: 6960: 6959: 6951: 6927:through a point 6919: 6917: 6916: 6911: 6900: 6892: 6878: 6877: 6869: 6841:, given by D = d 6820: 6818: 6817: 6812: 6807: 6806: 6781: 6780: 6773: 6772: 6761: 6760: 6742: 6741: 6721: 6720: 6712: 6706: 6705: 6700: 6699: 6691: 6681: 6680: 6672: 6669: 6668: 6660: 6644: 6642: 6641: 6636: 6631: 6630: 6605: 6604: 6597: 6596: 6585: 6584: 6566: 6565: 6545: 6544: 6536: 6530: 6529: 6524: 6523: 6515: 6505: 6504: 6496: 6493: 6492: 6484: 6436: 6434: 6433: 6428: 6423: 6422: 6415: 6414: 6401: 6400: 6387: 6386: 6373: 6372: 6355: 6354: 6347: 6346: 6335: 6334: 6320: 6319: 6305: 6304: 6288: 6287: 6276: 6275: 6264: 6263: 6252: 6251: 6235: 6234: 6223: 6222: 6208: 6207: 6196: 6195: 6182: 6181: 6170: 6169: 6158: 6157: 6143: 6142: 6116: 6115: 6081: 6079: 6078: 6073: 6068: 6067: 6060: 6059: 6046: 6045: 6032: 6031: 6018: 6017: 6000: 5999: 5992: 5991: 5980: 5979: 5965: 5964: 5950: 5949: 5933: 5932: 5921: 5920: 5909: 5908: 5894: 5893: 5880: 5879: 5868: 5867: 5856: 5855: 5844: 5843: 5827: 5826: 5815: 5814: 5800: 5799: 5788: 5787: 5761: 5760: 5726: 5706: 5675:spatial triangle 5662: 5660: 5659: 5654: 5649: 5647: 5646: 5645: 5636: 5635: 5629: 5624: 5616: 5614: 5606: 5601: 5593: 5591: 5576: 5575: 5574: 5565: 5564: 5558: 5553: 5545: 5543: 5535: 5530: 5522: 5520: 5509: 5508: 5502: 5497: 5489: 5487: 5476: 5475: 5469: 5464: 5456: 5454: 5445: 5440: 5439: 5433: 5428: 5420: 5418: 5397: 5395: 5394: 5389: 5387: 5386: 5377: 5376: 5370: 5365: 5357: 5355: 5347: 5342: 5334: 5332: 5321: 5316: 5308: 5306: 5298: 5293: 5285: 5283: 5272: 5267: 5259: 5257: 5236: 5234: 5233: 5228: 5223: 5219: 5218: 5217: 5208: 5207: 5201: 5196: 5188: 5186: 5178: 5173: 5165: 5163: 5152: 5151: 5145: 5140: 5132: 5130: 5122: 5117: 5109: 5107: 5096: 5095: 5089: 5084: 5076: 5074: 5066: 5061: 5053: 5051: 5035: 5031: 5030: 5029: 5020: 5019: 5013: 5008: 5000: 4998: 4990: 4985: 4977: 4975: 4964: 4959: 4951: 4949: 4941: 4936: 4928: 4926: 4910: 4909: 4903: 4898: 4890: 4888: 4877: 4872: 4864: 4862: 4841: 4839: 4838: 4833: 4828: 4824: 4823: 4822: 4816: 4811: 4803: 4801: 4790: 4785: 4777: 4775: 4762: 4758: 4757: 4756: 4750: 4745: 4737: 4735: 4724: 4719: 4711: 4709: 4693: 4692: 4686: 4681: 4673: 4671: 4660: 4655: 4647: 4645: 4634: 4633: 4625: 4597: 4595: 4594: 4589: 4584: 4583: 4571: 4566: 4565: 4557: 4536: 4531: 4530: 4522: 4507: 4506: 4498: 4494: 4491: 4488: 4487: 4475: 4470: 4469: 4461: 4440: 4435: 4434: 4426: 4411: 4410: 4402: 4340: 4338: 4337: 4332: 4327: 4326: 4320: 4315: 4307: 4305: 4294: 4289: 4281: 4279: 4268: 4267: 4259: 4154: = (, 4101: 4075: 4058:are orthogonal. 4057: 4055: 4054: 4049: 4037: 4035: 4034: 4029: 4017: 4015: 4014: 4009: 3998: 3997: 3982: 3981: 3962: 3960: 3959: 3954: 3942: 3940: 3939: 3934: 3923: 3922: 3900: 3898: 3897: 3892: 3866: 3865: 3850: 3849: 3834: 3833: 3812: 3811: 3799: 3798: 3768: 3767: 3762: 3761: 3753: 3749: 3748: 3740: 3723: 3709: 3707: 3695: 3694: 3693: 3685: 3677: 3656: 3654: 3653: 3648: 3642: 3641: 3626: 3625: 3610: 3609: 3594: 3593: 3578: 3573: 3572: 3567: 3566: 3558: 3554: 3539: 3537: 3536: 3531: 3489: 3484: 3476: 3474: 3447: 3445: 3444: 3439: 3434: 3433: 3424: 3423: 3413: 3408: 3403: 3402: 3394: 3384: 3383: 3374: 3373: 3368: 3367: 3359: 3349: 3348: 3339: 3338: 3333: 3332: 3324: 3314: 3313: 3308: 3307: 3299: 3295: 3294: 3286: 3270: 3268: 3267: 3262: 3257: 3256: 3247: 3246: 3241: 3240: 3232: 3225: 3224: 3215: 3214: 3205: 3204: 3199: 3198: 3190: 3170: 3168: 3167: 3162: 3154: 3153: 3145: 3139: 3135: 3126: 3125: 3117: 3111: 3107: 3101: 3100: 3095: 3094: 3086: 3065: 3063: 3062: 3057: 3052: 3051: 3039: 3038: 3030: 3024: 3020: 3014: 3013: 3008: 3007: 2999: 2989: 2988: 2980: 2974: 2970: 2957: 2937: 2935: 2934: 2929: 2924: 2923: 2918: 2917: 2909: 2905: 2904: 2899: 2898: 2890: 2883: 2882: 2873: 2872: 2864: 2861: 2860: 2852: 2843: 2842: 2837: 2836: 2828: 2814: 2801: 2799: 2798: 2793: 2787: 2786: 2771: 2770: 2755: 2754: 2742: 2741: 2726: 2725: 2720: 2719: 2711: 2299: 2295: 2289: 2278: 2276: 2275: 2270: 2190: 2189: 2181: 2178: 2177: 2169: 2153: 2151: 2150: 2145: 2107: 2106: 2098: 2082: 2080: 2079: 2074: 2036: 2035: 2027: 2006: 2004: 2003: 1998: 1984: 1983: 1971: 1970: 1946: 1945: 1933: 1932: 1908: 1907: 1895: 1894: 1873: 1872: 1860: 1859: 1838: 1837: 1825: 1824: 1803: 1802: 1790: 1789: 1768: 1767: 1755: 1754: 1736: 1735: 1723: 1722: 1677: 1676: 1668: 1662: 1661: 1653: 1637: 1635: 1634: 1629: 1618: 1617: 1599: 1598: 1580: 1579: 1564: 1563: 1548: 1547: 1532: 1531: 1516: 1515: 1503: 1502: 1472: 1471: 1463: 1447: 1445: 1444: 1439: 1428: 1427: 1409: 1408: 1390: 1389: 1374: 1373: 1358: 1357: 1342: 1341: 1326: 1325: 1313: 1312: 1282: 1281: 1273: 1252: 1250: 1249: 1244: 1236: 1235: 1226: 1225: 1216: 1215: 1209: 1208: 1203: 1202: 1194: 1187: 1186: 1180: 1179: 1174: 1173: 1165: 1152: 1151: 1142: 1141: 1132: 1131: 1126: 1125: 1117: 1113: 1112: 1107: 1106: 1098: 1085: 1084: 1075: 1074: 1069: 1068: 1060: 1050: 1049: 1040: 1039: 1034: 1033: 1025: 1015: 1014: 1006: 1003: 1002: 994: 988: 987: 979: 962: 947: 929: 909: 907: 906: 901: 830: 829: 821: 818: 817: 809: 796: 781: 765: 751: 716: 694: 692: 691: 686: 678: 670: 662: 657: 656: 644: 639: 638: 620: 612: 604: 603: 594: 593: 575: 567: 566: 551: 543: 542: 505: 488: 471: 428: 404: 394: 376: 366: 347: 337: 316: 314: 313: 308: 285: 284: 272: 271: 259: 258: 154:Clifford algebra 83:division algebra 80: 68: 33:dual quaternions 10616: 10615: 10611: 10610: 10609: 10607: 10606: 10605: 10581: 10580: 10579: 10574: 10551: 10530: 10520: 10493: 10484:Surreal numbers 10474:Ordinal numbers 10419: 10365: 10363: 10360: 10359: 10321: 10288: 10286: 10283: 10282: 10280: 10278:Split-octonions 10249: 10247: 10244: 10243: 10235: 10229: 10211: 10209: 10206: 10205: 10183: 10181: 10178: 10177: 10155: 10153: 10150: 10149: 10146:Complex numbers 10127: 10125: 10122: 10121: 10100: 10049: 10048: 10046: 10043: 10042: 10015: 10013: 10010: 10009: 9982: 9980: 9977: 9976: 9954: 9952: 9949: 9948: 9926: 9924: 9921: 9920: 9917:Natural numbers 9902: 9896: 9852: 9791: 9742: 9681: 9679:Further reading 9676: 9606:, Ph.D thesis, 9593: 9592: 9584: 9580: 9574: 9570: 9559: 9555: 9546: 9542: 9530: 9526: 9511: 9507: 9498: 9494: 9489: 9485: 9474: 9470: 9465: 9461: 9440: 9436: 9419: 9415: 9404: 9400: 9387: 9383: 9368: 9342: 9338: 9328: 9326: 9323: 9317: 9313: 9306: 9290: 9283: 9228: 9224: 9215: 9211: 9201: 9173:Rational motion 9164: 9156:dual quaternion 9136: 9101: 9096: 9072: 9067: 9054: 9049: 9036: 9031: 9025: 9022: 9021: 9001: 8997: 8988: 8984: 8975: 8971: 8962: 8958: 8956: 8953: 8952: 8927: 8924: 8923: 8898: 8895: 8894: 8871: 8867: 8865: 8862: 8861: 8835: 8831: 8822: 8818: 8816: 8813: 8812: 8796: 8793: 8792: 8776: 8773: 8772: 8756: 8753: 8752: 8738: 8711: 8710: 8686: 8685: 8684: 8682: 8679: 8678: 8654: 8653: 8647: 8643: 8640: 8639: 8633: 8629: 8626: 8625: 8619: 8615: 8612: 8611: 8601: 8600: 8592: 8589: 8588: 8561: 8560: 8554: 8549: 8536: 8531: 8518: 8513: 8500: 8495: 8489: 8483: 8479: 8473: 8469: 8457: 8453: 8447: 8443: 8438: 8432: 8428: 8422: 8418: 8406: 8402: 8396: 8392: 8386: 8385: 8379: 8375: 8369: 8365: 8353: 8349: 8343: 8339: 8334: 8328: 8323: 8310: 8305: 8292: 8287: 8274: 8269: 8263: 8257: 8253: 8247: 8243: 8231: 8227: 8221: 8217: 8211: 8210: 8204: 8200: 8194: 8190: 8178: 8174: 8168: 8164: 8159: 8153: 8149: 8143: 8139: 8127: 8123: 8117: 8113: 8108: 8102: 8097: 8084: 8079: 8066: 8061: 8048: 8043: 8032: 8031: 8023: 8020: 8019: 7991: 7987: 7978: 7976: 7970: 7961: 7957: 7948: 7946: 7941: 7936: 7931: 7921: 7920: 7912: 7909: 7908: 7879: 7868: 7867: 7866: 7864: 7850: 7849: 7835: 7834: 7816: 7815: 7814: 7812: 7809: 7808: 7783: 7782: 7780: 7777: 7776: 7750: 7746: 7734: 7730: 7718: 7714: 7688: 7687: 7685: 7682: 7681: 7647: 7645: 7626: 7624: 7605: 7603: 7589: 7586: 7585: 7560: 7559: 7557: 7554: 7553: 7537: 7534: 7533: 7479: 7478: 7477: 7473: 7459: 7455: 7432: 7428: 7410: 7406: 7394: 7390: 7378: 7374: 7365: 7361: 7353: 7350: 7349: 7327: 7324: 7323: 7271: 7270: 7268: 7265: 7264: 7257: 7224: 7223: 7217: 7213: 7207: 7196: 7195: 7194: 7182: 7171: 7170: 7169: 7152: 7151: 7149: 7146: 7145: 7114: 7103: 7102: 7101: 7099: 7088: 7087: 7076: 7075: 7061: 7060: 7058: 7055: 7054: 7029: 7021: 7010: 6996: 6995: 6989: 6983: 6975: 6964: 6950: 6949: 6947: 6944: 6943: 6891: 6868: 6867: 6865: 6862: 6861: 6852: 6848: 6844: 6827: 6801: 6800: 6794: 6793: 6783: 6782: 6775: 6774: 6768: 6764: 6762: 6756: 6752: 6749: 6748: 6743: 6737: 6733: 6726: 6725: 6711: 6710: 6701: 6690: 6689: 6688: 6671: 6670: 6659: 6658: 6656: 6653: 6652: 6625: 6624: 6618: 6617: 6607: 6606: 6599: 6598: 6592: 6588: 6586: 6580: 6576: 6573: 6572: 6567: 6561: 6557: 6550: 6549: 6535: 6534: 6525: 6514: 6513: 6512: 6495: 6494: 6483: 6482: 6480: 6477: 6476: 6471: 6467: 6463: 6459: 6455: 6451: 6447: 6443: 6417: 6416: 6410: 6406: 6403: 6402: 6396: 6392: 6389: 6388: 6382: 6378: 6375: 6374: 6368: 6364: 6357: 6356: 6349: 6348: 6342: 6338: 6336: 6330: 6326: 6321: 6315: 6311: 6306: 6300: 6296: 6290: 6289: 6283: 6279: 6277: 6271: 6267: 6265: 6259: 6255: 6253: 6247: 6243: 6237: 6236: 6230: 6226: 6224: 6218: 6214: 6209: 6203: 6199: 6197: 6191: 6187: 6184: 6183: 6177: 6173: 6171: 6165: 6161: 6159: 6153: 6149: 6144: 6138: 6134: 6127: 6126: 6111: 6107: 6093: 6090: 6089: 6062: 6061: 6055: 6051: 6048: 6047: 6041: 6037: 6034: 6033: 6027: 6023: 6020: 6019: 6013: 6009: 6002: 6001: 5994: 5993: 5987: 5983: 5981: 5975: 5971: 5966: 5960: 5956: 5951: 5945: 5941: 5935: 5934: 5928: 5924: 5922: 5916: 5912: 5910: 5904: 5900: 5895: 5889: 5885: 5882: 5881: 5875: 5871: 5869: 5863: 5859: 5857: 5851: 5847: 5845: 5839: 5835: 5829: 5828: 5822: 5818: 5816: 5810: 5806: 5801: 5795: 5791: 5789: 5783: 5779: 5772: 5771: 5756: 5752: 5738: 5735: 5734: 5724: 5720: 5716: 5712: 5708: 5707:into the array 5701: 5697: 5687: 5641: 5640: 5631: 5630: 5615: 5613: 5592: 5590: 5577: 5570: 5569: 5560: 5559: 5544: 5542: 5521: 5519: 5504: 5503: 5488: 5486: 5471: 5470: 5455: 5453: 5446: 5444: 5435: 5434: 5419: 5417: 5409: 5406: 5405: 5382: 5381: 5372: 5371: 5356: 5354: 5333: 5331: 5307: 5305: 5284: 5282: 5258: 5256: 5248: 5245: 5244: 5213: 5212: 5203: 5202: 5187: 5185: 5164: 5162: 5147: 5146: 5131: 5129: 5108: 5106: 5091: 5090: 5075: 5073: 5052: 5050: 5043: 5039: 5025: 5024: 5015: 5014: 4999: 4997: 4976: 4974: 4950: 4948: 4927: 4925: 4918: 4914: 4905: 4904: 4889: 4887: 4863: 4861: 4853: 4850: 4849: 4818: 4817: 4802: 4800: 4776: 4774: 4767: 4763: 4752: 4751: 4736: 4734: 4710: 4708: 4701: 4697: 4688: 4687: 4672: 4670: 4646: 4644: 4624: 4623: 4621: 4618: 4617: 4612: 4608: 4604: 4579: 4578: 4567: 4556: 4555: 4532: 4521: 4520: 4497: 4496: 4490: 4483: 4482: 4471: 4460: 4459: 4436: 4425: 4424: 4401: 4400: 4398: 4395: 4394: 4389: 4385: 4381: 4377: 4369: 4360: 4351: 4347: 4322: 4321: 4306: 4304: 4280: 4278: 4258: 4257: 4255: 4252: 4251: 4200: 4192: 4183: 4167: = (, 4166: 4153: 4143: 4131:Euclidean group 4099: 4067: 4064: 4043: 4040: 4039: 4023: 4020: 4019: 3993: 3989: 3977: 3973: 3968: 3965: 3964: 3948: 3945: 3944: 3918: 3914: 3909: 3906: 3905: 3861: 3857: 3845: 3841: 3829: 3825: 3807: 3803: 3794: 3790: 3763: 3752: 3751: 3750: 3739: 3738: 3736: 3733: 3732: 3718: 3703: 3701: 3689: 3687: 3681: 3679: 3673: 3666: 3637: 3633: 3621: 3617: 3605: 3601: 3589: 3585: 3568: 3557: 3556: 3555: 3553: 3551: 3548: 3547: 3475: 3473: 3471: 3468: 3467: 3461: 3429: 3428: 3419: 3418: 3409: 3404: 3393: 3392: 3379: 3378: 3369: 3358: 3357: 3356: 3344: 3343: 3334: 3323: 3322: 3321: 3309: 3298: 3297: 3296: 3285: 3284: 3282: 3279: 3278: 3252: 3251: 3242: 3231: 3230: 3229: 3220: 3216: 3210: 3209: 3200: 3189: 3188: 3187: 3182: 3179: 3178: 3144: 3143: 3133: 3116: 3115: 3105: 3096: 3085: 3084: 3083: 3081: 3078: 3077: 3047: 3046: 3029: 3028: 3018: 3009: 2998: 2997: 2996: 2979: 2978: 2968: 2966: 2963: 2962: 2952: 2942: 2919: 2908: 2907: 2906: 2900: 2889: 2888: 2887: 2878: 2874: 2863: 2862: 2851: 2850: 2838: 2827: 2826: 2825: 2823: 2820: 2819: 2806: 2782: 2778: 2766: 2762: 2750: 2746: 2737: 2733: 2721: 2710: 2709: 2708: 2706: 2703: 2702: 2696: 2302:(Row x Column) 2287: 2180: 2179: 2168: 2167: 2165: 2162: 2161: 2097: 2096: 2094: 2091: 2090: 2026: 2025: 2023: 2020: 2019: 2013: 1979: 1975: 1966: 1962: 1941: 1937: 1928: 1924: 1903: 1899: 1890: 1886: 1868: 1864: 1855: 1851: 1833: 1829: 1820: 1816: 1798: 1794: 1785: 1781: 1763: 1759: 1750: 1746: 1731: 1727: 1718: 1714: 1667: 1666: 1652: 1651: 1649: 1646: 1645: 1613: 1609: 1594: 1590: 1575: 1571: 1559: 1555: 1543: 1539: 1527: 1523: 1511: 1507: 1498: 1494: 1462: 1461: 1459: 1456: 1455: 1423: 1419: 1404: 1400: 1385: 1381: 1369: 1365: 1353: 1349: 1337: 1333: 1321: 1317: 1308: 1304: 1272: 1271: 1269: 1266: 1265: 1259: 1231: 1230: 1221: 1220: 1211: 1210: 1204: 1193: 1192: 1191: 1182: 1181: 1175: 1164: 1163: 1162: 1147: 1146: 1137: 1136: 1127: 1116: 1115: 1114: 1108: 1097: 1096: 1095: 1080: 1079: 1070: 1059: 1058: 1057: 1045: 1044: 1035: 1024: 1023: 1022: 1005: 1004: 993: 992: 978: 977: 975: 972: 971: 949: 937: 931: 924: 914: 820: 819: 808: 807: 805: 802: 801: 786: 779: 753: 718: 703: 674: 666: 658: 652: 648: 640: 634: 630: 616: 608: 599: 595: 589: 585: 571: 562: 558: 547: 538: 534: 514: 511: 510: 500: 490: 483: 473: 467: 457: 447: 437: 435: 423: 413: 396: 378: 368: 349: 339: 321: 280: 276: 267: 263: 254: 250: 248: 245: 244: 207: 135: 119:computer vision 78: 60: 17: 12: 11: 5: 10614: 10604: 10603: 10598: 10593: 10576: 10575: 10573: 10572: 10562: 10560:Classification 10556: 10553: 10552: 10550: 10549: 10547:Normal numbers 10544: 10539: 10517: 10512: 10507: 10501: 10499: 10495: 10494: 10492: 10491: 10486: 10481: 10476: 10471: 10466: 10461: 10456: 10455: 10454: 10444: 10439: 10433: 10431: 10429:infinitesimals 10421: 10420: 10418: 10417: 10416: 10415: 10410: 10405: 10391: 10386: 10381: 10368: 10353: 10348: 10343: 10338: 10332: 10330: 10323: 10322: 10320: 10319: 10314: 10309: 10304: 10291: 10275: 10270: 10265: 10252: 10239: 10237: 10231: 10230: 10228: 10227: 10214: 10199: 10186: 10171: 10158: 10143: 10130: 10110: 10108: 10102: 10101: 10099: 10098: 10093: 10092: 10091: 10081: 10076: 10071: 10066: 10052: 10036: 10031: 10018: 10003: 9998: 9985: 9970: 9957: 9942: 9929: 9913: 9911: 9904: 9903: 9895: 9894: 9887: 9880: 9872: 9866: 9865: 9859: 9851: 9850:External links 9848: 9847: 9846: 9830: 9810: 9803: 9796: 9782: 9764: 9753: 9752:18(3):323–349. 9746: 9740: 9727: 9680: 9677: 9675: 9674: 9664: 9657: 9650: 9641: 9622: 9611: 9599: 9591: 9590: 9578: 9568: 9553: 9540: 9524: 9505: 9492: 9483: 9468: 9459: 9434: 9413: 9398: 9381: 9366: 9336: 9311: 9304: 9281: 9222: 9208: 9207: 9200: 9197: 9196: 9195: 9190: 9185: 9180: 9175: 9170: 9163: 9160: 9135: 9132: 9124: 9123: 9112: 9109: 9104: 9099: 9095: 9089: 9086: 9083: 9080: 9075: 9070: 9066: 9062: 9057: 9052: 9048: 9044: 9039: 9034: 9030: 9004: 9000: 8996: 8991: 8987: 8983: 8978: 8974: 8970: 8965: 8961: 8940: 8937: 8934: 8931: 8911: 8908: 8905: 8902: 8882: 8879: 8874: 8870: 8849: 8846: 8843: 8838: 8834: 8830: 8825: 8821: 8800: 8780: 8760: 8737: 8734: 8733: 8732: 8718: 8715: 8709: 8706: 8703: 8699: 8693: 8690: 8672: 8671: 8658: 8650: 8646: 8642: 8641: 8636: 8632: 8628: 8627: 8622: 8618: 8614: 8613: 8610: 8607: 8606: 8604: 8599: 8596: 8582: 8581: 8570: 8565: 8557: 8552: 8548: 8544: 8539: 8534: 8530: 8526: 8521: 8516: 8512: 8508: 8503: 8498: 8494: 8490: 8486: 8482: 8476: 8472: 8468: 8465: 8460: 8456: 8450: 8446: 8442: 8439: 8435: 8431: 8425: 8421: 8417: 8414: 8409: 8405: 8399: 8395: 8391: 8388: 8387: 8382: 8378: 8372: 8368: 8364: 8361: 8356: 8352: 8346: 8342: 8338: 8335: 8331: 8326: 8322: 8318: 8313: 8308: 8304: 8300: 8295: 8290: 8286: 8282: 8277: 8272: 8268: 8264: 8260: 8256: 8250: 8246: 8242: 8239: 8234: 8230: 8224: 8220: 8216: 8213: 8212: 8207: 8203: 8197: 8193: 8189: 8186: 8181: 8177: 8171: 8167: 8163: 8160: 8156: 8152: 8146: 8142: 8138: 8135: 8130: 8126: 8120: 8116: 8112: 8109: 8105: 8100: 8096: 8092: 8087: 8082: 8078: 8074: 8069: 8064: 8060: 8056: 8051: 8046: 8042: 8038: 8037: 8035: 8030: 8027: 8011:where the 3×3 8009: 8008: 7995: 7986: 7983: 7980: 7979: 7975: 7972: 7969: 7966: 7963: 7962: 7956: 7953: 7950: 7949: 7945: 7942: 7940: 7937: 7935: 7932: 7930: 7927: 7926: 7924: 7919: 7916: 7902: 7901: 7887: 7882: 7875: 7872: 7863: 7857: 7854: 7848: 7842: 7839: 7833: 7829: 7823: 7820: 7790: 7787: 7773: 7772: 7761: 7758: 7753: 7749: 7745: 7742: 7737: 7733: 7729: 7726: 7721: 7717: 7713: 7710: 7707: 7704: 7701: 7695: 7692: 7675: 7674: 7662: 7657: 7653: 7650: 7644: 7641: 7636: 7632: 7629: 7623: 7620: 7615: 7611: 7608: 7602: 7599: 7596: 7593: 7567: 7564: 7541: 7530: 7529: 7517: 7513: 7510: 7507: 7504: 7501: 7498: 7495: 7492: 7486: 7483: 7476: 7471: 7466: 7463: 7458: 7454: 7451: 7448: 7444: 7439: 7436: 7431: 7427: 7424: 7421: 7418: 7413: 7409: 7405: 7402: 7397: 7393: 7389: 7386: 7381: 7377: 7373: 7368: 7364: 7360: 7357: 7331: 7299: 7296: 7293: 7290: 7287: 7284: 7278: 7275: 7256: 7253: 7249: 7248: 7237: 7231: 7228: 7220: 7216: 7210: 7203: 7200: 7193: 7190: 7185: 7178: 7175: 7168: 7165: 7159: 7156: 7139: 7138: 7127: 7122: 7117: 7110: 7107: 7095: 7092: 7083: 7080: 7074: 7068: 7065: 7048: 7047: 7036: 7032: 7028: 7024: 7020: 7017: 7013: 7009: 7003: 7000: 6986: 6982: 6978: 6974: 6971: 6967: 6963: 6957: 6954: 6921: 6920: 6909: 6906: 6903: 6898: 6895: 6890: 6887: 6884: 6881: 6875: 6872: 6857:) is given by 6850: 6846: 6842: 6826: 6823: 6822: 6821: 6810: 6805: 6799: 6796: 6795: 6792: 6789: 6788: 6786: 6779: 6771: 6767: 6763: 6759: 6755: 6751: 6750: 6747: 6744: 6740: 6736: 6732: 6731: 6729: 6724: 6718: 6715: 6709: 6704: 6697: 6694: 6687: 6684: 6678: 6675: 6666: 6663: 6646: 6645: 6634: 6629: 6623: 6620: 6619: 6616: 6613: 6612: 6610: 6603: 6595: 6591: 6587: 6583: 6579: 6575: 6574: 6571: 6568: 6564: 6560: 6556: 6555: 6553: 6548: 6542: 6539: 6533: 6528: 6521: 6518: 6511: 6508: 6502: 6499: 6490: 6487: 6469: 6465: 6461: 6457: 6453: 6449: 6445: 6441: 6438: 6437: 6426: 6421: 6413: 6409: 6405: 6404: 6399: 6395: 6391: 6390: 6385: 6381: 6377: 6376: 6371: 6367: 6363: 6362: 6360: 6353: 6345: 6341: 6337: 6333: 6329: 6325: 6322: 6318: 6314: 6310: 6307: 6303: 6299: 6295: 6292: 6291: 6286: 6282: 6278: 6274: 6270: 6266: 6262: 6258: 6254: 6250: 6246: 6242: 6239: 6238: 6233: 6229: 6225: 6221: 6217: 6213: 6210: 6206: 6202: 6198: 6194: 6190: 6186: 6185: 6180: 6176: 6172: 6168: 6164: 6160: 6156: 6152: 6148: 6145: 6141: 6137: 6133: 6132: 6130: 6125: 6122: 6119: 6114: 6110: 6106: 6103: 6100: 6097: 6083: 6082: 6071: 6066: 6058: 6054: 6050: 6049: 6044: 6040: 6036: 6035: 6030: 6026: 6022: 6021: 6016: 6012: 6008: 6007: 6005: 5998: 5990: 5986: 5982: 5978: 5974: 5970: 5967: 5963: 5959: 5955: 5952: 5948: 5944: 5940: 5937: 5936: 5931: 5927: 5923: 5919: 5915: 5911: 5907: 5903: 5899: 5896: 5892: 5888: 5884: 5883: 5878: 5874: 5870: 5866: 5862: 5858: 5854: 5850: 5846: 5842: 5838: 5834: 5831: 5830: 5825: 5821: 5817: 5813: 5809: 5805: 5802: 5798: 5794: 5790: 5786: 5782: 5778: 5777: 5775: 5770: 5767: 5764: 5759: 5755: 5751: 5748: 5745: 5742: 5722: 5718: 5714: 5710: 5699: 5686: 5683: 5664: 5663: 5652: 5644: 5639: 5634: 5627: 5622: 5619: 5612: 5609: 5604: 5599: 5596: 5589: 5586: 5583: 5580: 5573: 5568: 5563: 5556: 5551: 5548: 5541: 5538: 5533: 5528: 5525: 5518: 5515: 5512: 5507: 5500: 5495: 5492: 5485: 5482: 5479: 5474: 5467: 5462: 5459: 5452: 5449: 5443: 5438: 5431: 5426: 5423: 5416: 5413: 5399: 5398: 5385: 5380: 5375: 5368: 5363: 5360: 5353: 5350: 5345: 5340: 5337: 5330: 5327: 5324: 5319: 5314: 5311: 5304: 5301: 5296: 5291: 5288: 5281: 5278: 5275: 5270: 5265: 5262: 5255: 5252: 5238: 5237: 5226: 5222: 5216: 5211: 5206: 5199: 5194: 5191: 5184: 5181: 5176: 5171: 5168: 5161: 5158: 5155: 5150: 5143: 5138: 5135: 5128: 5125: 5120: 5115: 5112: 5105: 5102: 5099: 5094: 5087: 5082: 5079: 5072: 5069: 5064: 5059: 5056: 5049: 5046: 5042: 5038: 5034: 5028: 5023: 5018: 5011: 5006: 5003: 4996: 4993: 4988: 4983: 4980: 4973: 4970: 4967: 4962: 4957: 4954: 4947: 4944: 4939: 4934: 4931: 4924: 4921: 4917: 4913: 4908: 4901: 4896: 4893: 4886: 4883: 4880: 4875: 4870: 4867: 4860: 4857: 4843: 4842: 4831: 4827: 4821: 4814: 4809: 4806: 4799: 4796: 4793: 4788: 4783: 4780: 4773: 4770: 4766: 4761: 4755: 4748: 4743: 4740: 4733: 4730: 4727: 4722: 4717: 4714: 4707: 4704: 4700: 4696: 4691: 4684: 4679: 4676: 4669: 4666: 4663: 4658: 4653: 4650: 4643: 4640: 4637: 4631: 4628: 4610: 4606: 4602: 4599: 4598: 4587: 4582: 4577: 4574: 4570: 4563: 4560: 4554: 4551: 4548: 4545: 4542: 4539: 4535: 4528: 4525: 4519: 4516: 4513: 4510: 4504: 4501: 4486: 4481: 4478: 4474: 4467: 4464: 4458: 4455: 4452: 4449: 4446: 4443: 4439: 4432: 4429: 4423: 4420: 4417: 4414: 4408: 4405: 4387: 4383: 4379: 4373: 4365: 4356: 4349: 4345: 4342: 4341: 4330: 4325: 4318: 4313: 4310: 4303: 4300: 4297: 4292: 4287: 4284: 4277: 4274: 4271: 4265: 4262: 4219: = ( 4196: 4188: 4179: 4162: 4149: 4142: 4139: 4115:group of units 4096: 4095: 4063: 4060: 4047: 4027: 4007: 4004: 4001: 3996: 3992: 3988: 3985: 3980: 3976: 3972: 3952: 3932: 3929: 3926: 3921: 3917: 3913: 3902: 3901: 3890: 3887: 3884: 3881: 3878: 3875: 3872: 3869: 3864: 3860: 3856: 3853: 3848: 3844: 3840: 3837: 3832: 3828: 3824: 3821: 3818: 3815: 3810: 3806: 3802: 3797: 3793: 3789: 3786: 3783: 3780: 3777: 3774: 3771: 3766: 3759: 3756: 3746: 3743: 3665: 3662: 3658: 3657: 3645: 3640: 3636: 3632: 3629: 3624: 3620: 3616: 3613: 3608: 3604: 3600: 3597: 3592: 3588: 3584: 3581: 3576: 3571: 3564: 3561: 3541: 3540: 3528: 3525: 3522: 3519: 3516: 3513: 3510: 3507: 3504: 3501: 3498: 3495: 3492: 3487: 3482: 3479: 3460: 3457: 3449: 3448: 3437: 3432: 3427: 3422: 3417: 3412: 3407: 3400: 3397: 3390: 3387: 3382: 3377: 3372: 3365: 3362: 3355: 3352: 3347: 3342: 3337: 3330: 3327: 3320: 3317: 3312: 3305: 3302: 3292: 3289: 3272: 3271: 3260: 3255: 3250: 3245: 3238: 3235: 3228: 3223: 3219: 3213: 3208: 3203: 3196: 3193: 3186: 3172: 3171: 3160: 3157: 3151: 3148: 3142: 3132: 3129: 3123: 3120: 3114: 3104: 3099: 3092: 3089: 3067: 3066: 3055: 3050: 3045: 3042: 3036: 3033: 3027: 3017: 3012: 3005: 3002: 2995: 2992: 2986: 2983: 2977: 2950: 2939: 2938: 2927: 2922: 2915: 2912: 2903: 2896: 2893: 2886: 2881: 2877: 2870: 2867: 2858: 2855: 2849: 2846: 2841: 2834: 2831: 2803: 2802: 2790: 2785: 2781: 2777: 2774: 2769: 2765: 2761: 2758: 2753: 2749: 2745: 2740: 2736: 2732: 2729: 2724: 2717: 2714: 2695: 2692: 2689: 2688: 2685: 2682: 2679: 2676: 2673: 2667: 2661: 2655: 2648: 2647: 2644: 2641: 2638: 2635: 2629: 2626: 2620: 2614: 2607: 2606: 2603: 2600: 2597: 2594: 2588: 2582: 2579: 2573: 2566: 2565: 2562: 2559: 2556: 2553: 2547: 2541: 2535: 2532: 2528: 2527: 2524: 2518: 2512: 2506: 2503: 2497: 2492: 2487: 2481: 2480: 2474: 2471: 2465: 2459: 2454: 2451: 2445: 2440: 2434: 2433: 2427: 2421: 2418: 2412: 2406: 2401: 2398: 2393: 2387: 2386: 2380: 2374: 2368: 2365: 2360: 2355: 2350: 2347: 2343: 2342: 2336: 2330: 2324: 2321: 2316: 2311: 2306: 2303: 2280: 2279: 2268: 2265: 2262: 2259: 2256: 2253: 2250: 2247: 2244: 2241: 2238: 2235: 2232: 2229: 2226: 2223: 2220: 2217: 2214: 2211: 2208: 2205: 2202: 2199: 2196: 2193: 2187: 2184: 2175: 2172: 2155: 2154: 2143: 2140: 2137: 2134: 2131: 2128: 2125: 2122: 2119: 2116: 2113: 2110: 2104: 2101: 2084: 2083: 2072: 2069: 2066: 2063: 2060: 2057: 2054: 2051: 2048: 2045: 2042: 2039: 2033: 2030: 2012: 2011:Multiplication 2009: 2008: 2007: 1996: 1993: 1990: 1987: 1982: 1978: 1974: 1969: 1965: 1961: 1958: 1955: 1952: 1949: 1944: 1940: 1936: 1931: 1927: 1923: 1920: 1917: 1914: 1911: 1906: 1902: 1898: 1893: 1889: 1885: 1882: 1879: 1876: 1871: 1867: 1863: 1858: 1854: 1850: 1847: 1844: 1841: 1836: 1832: 1828: 1823: 1819: 1815: 1812: 1809: 1806: 1801: 1797: 1793: 1788: 1784: 1780: 1777: 1774: 1771: 1766: 1762: 1758: 1753: 1749: 1745: 1742: 1739: 1734: 1730: 1726: 1721: 1717: 1713: 1710: 1707: 1704: 1701: 1698: 1695: 1692: 1689: 1686: 1683: 1680: 1674: 1671: 1665: 1659: 1656: 1639: 1638: 1627: 1624: 1621: 1616: 1612: 1608: 1605: 1602: 1597: 1593: 1589: 1586: 1583: 1578: 1574: 1570: 1567: 1562: 1558: 1554: 1551: 1546: 1542: 1538: 1535: 1530: 1526: 1522: 1519: 1514: 1510: 1506: 1501: 1497: 1493: 1490: 1487: 1484: 1481: 1478: 1475: 1469: 1466: 1449: 1448: 1437: 1434: 1431: 1426: 1422: 1418: 1415: 1412: 1407: 1403: 1399: 1396: 1393: 1388: 1384: 1380: 1377: 1372: 1368: 1364: 1361: 1356: 1352: 1348: 1345: 1340: 1336: 1332: 1329: 1324: 1320: 1316: 1311: 1307: 1303: 1300: 1297: 1294: 1291: 1288: 1285: 1279: 1276: 1258: 1255: 1254: 1253: 1242: 1239: 1234: 1229: 1224: 1219: 1214: 1207: 1200: 1197: 1190: 1185: 1178: 1171: 1168: 1161: 1158: 1155: 1150: 1145: 1140: 1135: 1130: 1123: 1120: 1111: 1104: 1101: 1094: 1091: 1088: 1083: 1078: 1073: 1066: 1063: 1056: 1053: 1048: 1043: 1038: 1031: 1028: 1021: 1018: 1012: 1009: 1000: 997: 991: 985: 982: 935: 922: 911: 910: 899: 896: 893: 890: 887: 884: 881: 878: 875: 872: 869: 866: 863: 860: 857: 854: 851: 848: 845: 842: 839: 836: 833: 827: 824: 815: 812: 696: 695: 684: 681: 677: 673: 669: 665: 661: 655: 651: 647: 643: 637: 633: 629: 626: 623: 619: 615: 611: 607: 602: 598: 592: 588: 584: 581: 578: 574: 570: 565: 561: 557: 554: 550: 546: 541: 537: 533: 530: 527: 524: 521: 518: 498: 481: 465: 455: 445: 433: 421: 377:. Now because 318: 317: 306: 303: 300: 297: 294: 291: 288: 283: 279: 275: 270: 266: 262: 257: 253: 206: 203: 146:W. K. Clifford 138:W. R. Hamilton 134: 131: 41:tensor product 15: 9: 6: 4: 3: 2: 10613: 10602: 10599: 10597: 10594: 10592: 10589: 10588: 10586: 10571: 10563: 10561: 10558: 10557: 10554: 10548: 10545: 10543: 10540: 10537: 10533: 10527: 10523: 10518: 10516: 10513: 10511: 10510:Fuzzy numbers 10508: 10506: 10503: 10502: 10500: 10496: 10490: 10487: 10485: 10482: 10480: 10477: 10475: 10472: 10470: 10467: 10465: 10462: 10460: 10457: 10453: 10450: 10449: 10448: 10445: 10443: 10440: 10438: 10435: 10434: 10432: 10430: 10426: 10422: 10414: 10411: 10409: 10406: 10404: 10401: 10400: 10399: 10395: 10392: 10390: 10387: 10385: 10382: 10357: 10354: 10352: 10349: 10347: 10344: 10342: 10339: 10337: 10334: 10333: 10331: 10329: 10324: 10318: 10315: 10313: 10312:Biquaternions 10310: 10308: 10305: 10279: 10276: 10274: 10271: 10269: 10266: 10241: 10240: 10238: 10232: 10203: 10200: 10175: 10172: 10147: 10144: 10119: 10115: 10112: 10111: 10109: 10107: 10103: 10097: 10094: 10090: 10087: 10086: 10085: 10082: 10080: 10077: 10075: 10072: 10070: 10067: 10040: 10037: 10035: 10032: 10007: 10004: 10002: 9999: 9974: 9971: 9946: 9943: 9918: 9915: 9914: 9912: 9910: 9905: 9900: 9893: 9888: 9886: 9881: 9879: 9874: 9873: 9870: 9863: 9860: 9857: 9854: 9853: 9844: 9839: 9835: 9831: 9828: 9823: 9819: 9815: 9811: 9808: 9804: 9801: 9797: 9790: 9789: 9783: 9781: 9777: 9773: 9769: 9765: 9762: 9758: 9754: 9751: 9747: 9743: 9737: 9734:. CRC Press. 9733: 9728: 9724: 9720: 9715: 9710: 9705: 9700: 9696: 9692: 9688: 9683: 9682: 9672: 9668: 9665: 9662: 9658: 9655: 9651: 9648: 9647: 9642: 9639: 9638:0-262-13252-4 9635: 9631: 9627: 9623: 9620: 9616: 9612: 9609: 9605: 9601: 9600: 9598: 9597: 9587: 9582: 9572: 9566: 9564: 9557: 9550: 9544: 9537: 9533: 9528: 9522: 9518: 9514: 9509: 9502: 9496: 9487: 9481: 9479: 9472: 9463: 9454: 9449: 9445: 9438: 9429: 9424: 9417: 9411: 9409: 9402: 9394: 9393: 9385: 9377: 9373: 9369: 9363: 9359: 9355: 9351: 9347: 9340: 9322: 9315: 9307: 9305:9780262132527 9301: 9297: 9296: 9288: 9286: 9277: 9273: 9268: 9263: 9259: 9255: 9250: 9245: 9241: 9237: 9233: 9226: 9219: 9213: 9209: 9206: 9205: 9194: 9191: 9189: 9186: 9184: 9181: 9179: 9176: 9174: 9171: 9169: 9166: 9165: 9159: 9157: 9153: 9149: 9145: 9141: 9131: 9129: 9110: 9107: 9102: 9097: 9093: 9087: 9084: 9081: 9078: 9073: 9068: 9064: 9060: 9055: 9050: 9046: 9042: 9037: 9032: 9028: 9020: 9019: 9018: 9002: 8998: 8994: 8989: 8985: 8981: 8976: 8972: 8968: 8963: 8959: 8938: 8935: 8932: 8929: 8909: 8906: 8903: 8900: 8880: 8877: 8872: 8868: 8847: 8844: 8841: 8836: 8832: 8828: 8823: 8819: 8798: 8778: 8758: 8750: 8745: 8743: 8713: 8707: 8704: 8701: 8697: 8688: 8677: 8676: 8675: 8656: 8648: 8644: 8634: 8630: 8620: 8616: 8608: 8602: 8597: 8594: 8587: 8586: 8585: 8568: 8563: 8555: 8550: 8546: 8542: 8537: 8532: 8528: 8524: 8519: 8514: 8510: 8506: 8501: 8496: 8492: 8484: 8480: 8474: 8470: 8466: 8463: 8458: 8454: 8448: 8444: 8440: 8433: 8429: 8423: 8419: 8415: 8412: 8407: 8403: 8397: 8393: 8389: 8380: 8376: 8370: 8366: 8362: 8359: 8354: 8350: 8344: 8340: 8336: 8329: 8324: 8320: 8316: 8311: 8306: 8302: 8298: 8293: 8288: 8284: 8280: 8275: 8270: 8266: 8258: 8254: 8248: 8244: 8240: 8237: 8232: 8228: 8222: 8218: 8214: 8205: 8201: 8195: 8191: 8187: 8184: 8179: 8175: 8169: 8165: 8161: 8154: 8150: 8144: 8140: 8136: 8133: 8128: 8124: 8118: 8114: 8110: 8103: 8098: 8094: 8090: 8085: 8080: 8076: 8072: 8067: 8062: 8058: 8054: 8049: 8044: 8040: 8033: 8028: 8025: 8018: 8017: 8016: 8014: 7993: 7984: 7973: 7967: 7954: 7943: 7938: 7933: 7928: 7922: 7917: 7914: 7907: 7906: 7905: 7880: 7870: 7861: 7852: 7846: 7837: 7831: 7827: 7818: 7807: 7806: 7805: 7785: 7756: 7751: 7747: 7743: 7740: 7735: 7731: 7727: 7724: 7719: 7715: 7708: 7705: 7702: 7699: 7690: 7680: 7679: 7678: 7660: 7655: 7651: 7642: 7639: 7634: 7630: 7621: 7618: 7613: 7609: 7600: 7597: 7594: 7591: 7584: 7583: 7582: 7562: 7539: 7515: 7508: 7505: 7502: 7499: 7496: 7490: 7481: 7474: 7469: 7464: 7461: 7456: 7452: 7449: 7446: 7442: 7437: 7434: 7429: 7425: 7422: 7419: 7416: 7411: 7407: 7403: 7400: 7395: 7391: 7387: 7384: 7379: 7375: 7371: 7366: 7362: 7358: 7355: 7348: 7347: 7346: 7343: 7329: 7321: 7317: 7313: 7297: 7294: 7291: 7288: 7285: 7282: 7273: 7262: 7252: 7235: 7226: 7218: 7208: 7198: 7183: 7173: 7163: 7154: 7144: 7143: 7142: 7125: 7115: 7105: 7090: 7078: 7072: 7063: 7053: 7052: 7051: 7034: 7026: 7018: 7015: 7007: 6998: 6980: 6972: 6969: 6961: 6952: 6942: 6941: 6940: 6939:be given by, 6938: 6934: 6930: 6926: 6907: 6904: 6901: 6896: 6893: 6888: 6885: 6882: 6879: 6870: 6860: 6859: 6858: 6856: 6840: 6836: 6832: 6808: 6803: 6797: 6790: 6784: 6777: 6769: 6765: 6757: 6753: 6745: 6738: 6734: 6727: 6722: 6713: 6702: 6692: 6682: 6673: 6661: 6651: 6650: 6649: 6632: 6627: 6621: 6614: 6608: 6601: 6593: 6589: 6581: 6577: 6569: 6562: 6558: 6551: 6546: 6537: 6526: 6516: 6506: 6497: 6485: 6475: 6474: 6473: 6424: 6419: 6411: 6407: 6397: 6393: 6383: 6379: 6369: 6365: 6358: 6351: 6343: 6339: 6331: 6327: 6323: 6316: 6312: 6308: 6301: 6297: 6293: 6284: 6280: 6272: 6268: 6260: 6256: 6248: 6244: 6240: 6231: 6227: 6219: 6215: 6211: 6204: 6200: 6192: 6188: 6178: 6174: 6166: 6162: 6154: 6150: 6146: 6139: 6135: 6128: 6123: 6120: 6112: 6108: 6101: 6098: 6095: 6088: 6087: 6086: 6069: 6064: 6056: 6052: 6042: 6038: 6028: 6024: 6014: 6010: 6003: 5996: 5988: 5984: 5976: 5972: 5968: 5961: 5957: 5953: 5946: 5942: 5938: 5929: 5925: 5917: 5913: 5905: 5901: 5897: 5890: 5886: 5876: 5872: 5864: 5860: 5852: 5848: 5840: 5836: 5832: 5823: 5819: 5811: 5807: 5803: 5796: 5792: 5784: 5780: 5773: 5768: 5765: 5757: 5753: 5746: 5743: 5740: 5733: 5732: 5731: 5728: 5705: 5694: 5690: 5682: 5680: 5676: 5671: 5669: 5650: 5637: 5625: 5617: 5610: 5607: 5602: 5594: 5587: 5584: 5581: 5578: 5566: 5554: 5546: 5539: 5536: 5531: 5523: 5516: 5513: 5510: 5498: 5490: 5483: 5480: 5477: 5465: 5457: 5450: 5447: 5441: 5429: 5421: 5414: 5411: 5404: 5403: 5402: 5378: 5366: 5358: 5351: 5348: 5343: 5335: 5328: 5325: 5322: 5317: 5309: 5302: 5299: 5294: 5286: 5279: 5276: 5273: 5268: 5260: 5253: 5250: 5243: 5242: 5241: 5224: 5220: 5209: 5197: 5189: 5182: 5179: 5174: 5166: 5159: 5156: 5153: 5141: 5133: 5126: 5123: 5118: 5110: 5103: 5100: 5097: 5085: 5077: 5070: 5067: 5062: 5054: 5047: 5044: 5040: 5036: 5032: 5021: 5009: 5001: 4994: 4991: 4986: 4978: 4971: 4968: 4965: 4960: 4952: 4945: 4942: 4937: 4929: 4922: 4919: 4915: 4911: 4899: 4891: 4884: 4881: 4878: 4873: 4865: 4858: 4855: 4848: 4847: 4846: 4829: 4825: 4812: 4804: 4797: 4794: 4791: 4786: 4778: 4771: 4768: 4764: 4759: 4746: 4738: 4731: 4728: 4725: 4720: 4712: 4705: 4702: 4698: 4694: 4682: 4674: 4667: 4664: 4661: 4656: 4648: 4641: 4638: 4635: 4626: 4616: 4615: 4614: 4585: 4572: 4568: 4558: 4549: 4546: 4543: 4537: 4533: 4523: 4514: 4511: 4508: 4499: 4476: 4472: 4462: 4453: 4450: 4447: 4441: 4437: 4427: 4418: 4415: 4412: 4403: 4393: 4392: 4391: 4376: 4372: 4368: 4364: 4361: = 4359: 4355: 4328: 4316: 4308: 4301: 4298: 4295: 4290: 4282: 4275: 4272: 4269: 4260: 4250: 4249: 4248: 4246: 4242: 4238: 4234: 4230: 4226: 4222: 4218: 4215: 4211: 4207: 4202: 4199: 4195: 4191: 4187: 4184: = 4182: 4178: 4174: 4170: 4165: 4161: 4157: 4152: 4148: 4138: 4136: 4132: 4128: 4124: 4120: 4116: 4111: 4109: 4108:maximal ideal 4105: 4093: 4090: 4086: 4083: 4082: 4081: 4079: 4074: 4070: 4059: 4045: 4025: 4018:implies that 4005: 4002: 3999: 3994: 3990: 3986: 3983: 3978: 3974: 3970: 3950: 3943:implies that 3930: 3927: 3924: 3919: 3915: 3911: 3888: 3882: 3879: 3876: 3870: 3862: 3858: 3854: 3851: 3846: 3842: 3838: 3835: 3830: 3826: 3822: 3816: 3808: 3804: 3800: 3795: 3791: 3781: 3778: 3775: 3769: 3764: 3754: 3741: 3731: 3730: 3729: 3727: 3721: 3715: 3713: 3706: 3699: 3692: 3684: 3676: 3671: 3661: 3643: 3638: 3634: 3630: 3627: 3622: 3618: 3614: 3606: 3602: 3598: 3595: 3590: 3586: 3579: 3569: 3559: 3546: 3545: 3544: 3526: 3523: 3520: 3517: 3514: 3511: 3505: 3502: 3499: 3496: 3490: 3477: 3466: 3465: 3464: 3456: 3454: 3435: 3425: 3415: 3410: 3405: 3395: 3388: 3375: 3370: 3360: 3340: 3335: 3325: 3315: 3310: 3300: 3287: 3277: 3276: 3275: 3258: 3248: 3243: 3233: 3226: 3221: 3206: 3201: 3191: 3177: 3176: 3175: 3158: 3146: 3130: 3118: 3102: 3097: 3087: 3076: 3075: 3074: 3072: 3053: 3043: 3031: 3015: 3010: 3000: 2993: 2981: 2961: 2960: 2959: 2956: 2949: 2945: 2925: 2920: 2910: 2901: 2891: 2884: 2879: 2865: 2853: 2844: 2839: 2829: 2818: 2817: 2816: 2813: 2809: 2788: 2783: 2779: 2775: 2772: 2767: 2763: 2759: 2751: 2747: 2743: 2738: 2734: 2727: 2722: 2712: 2701: 2700: 2699: 2686: 2683: 2680: 2677: 2674: 2672: 2668: 2666: 2662: 2660: 2656: 2654: 2649: 2645: 2642: 2639: 2636: 2634: 2630: 2627: 2625: 2621: 2619: 2615: 2613: 2608: 2604: 2601: 2598: 2595: 2593: 2589: 2587: 2583: 2580: 2578: 2574: 2572: 2567: 2563: 2560: 2557: 2554: 2552: 2548: 2546: 2542: 2540: 2536: 2533: 2529: 2525: 2523: 2519: 2517: 2513: 2511: 2507: 2504: 2502: 2498: 2496: 2493: 2491: 2488: 2486: 2482: 2479: 2475: 2472: 2470: 2466: 2464: 2460: 2458: 2455: 2452: 2450: 2446: 2444: 2441: 2439: 2435: 2432: 2428: 2426: 2422: 2419: 2417: 2413: 2411: 2407: 2405: 2402: 2399: 2397: 2394: 2392: 2388: 2385: 2381: 2379: 2375: 2373: 2369: 2366: 2364: 2361: 2359: 2356: 2354: 2351: 2348: 2344: 2341: 2335: 2329: 2320: 2315: 2310: 2300: 2294: 2291: 2285: 2266: 2260: 2257: 2254: 2251: 2248: 2242: 2239: 2236: 2233: 2230: 2224: 2221: 2218: 2215: 2206: 2203: 2200: 2197: 2191: 2182: 2170: 2160: 2159: 2158: 2141: 2138: 2135: 2132: 2129: 2126: 2120: 2117: 2114: 2108: 2099: 2089: 2088: 2087: 2070: 2067: 2064: 2061: 2058: 2055: 2049: 2046: 2043: 2037: 2028: 2018: 2017: 2016: 1994: 1991: 1988: 1980: 1976: 1972: 1967: 1963: 1956: 1953: 1950: 1942: 1938: 1934: 1929: 1925: 1918: 1915: 1912: 1904: 1900: 1896: 1891: 1887: 1880: 1877: 1869: 1865: 1861: 1856: 1852: 1845: 1842: 1834: 1830: 1826: 1821: 1817: 1810: 1807: 1799: 1795: 1791: 1786: 1782: 1775: 1772: 1764: 1760: 1756: 1751: 1747: 1740: 1732: 1728: 1724: 1719: 1715: 1708: 1702: 1699: 1696: 1693: 1690: 1687: 1684: 1678: 1669: 1663: 1654: 1644: 1643: 1642: 1625: 1622: 1619: 1614: 1610: 1606: 1603: 1600: 1595: 1591: 1587: 1584: 1581: 1576: 1572: 1568: 1565: 1560: 1556: 1552: 1549: 1544: 1540: 1536: 1533: 1528: 1524: 1520: 1517: 1512: 1508: 1504: 1499: 1495: 1491: 1485: 1482: 1479: 1473: 1464: 1454: 1453: 1452: 1435: 1432: 1429: 1424: 1420: 1416: 1413: 1410: 1405: 1401: 1397: 1394: 1391: 1386: 1382: 1378: 1375: 1370: 1366: 1362: 1359: 1354: 1350: 1346: 1343: 1338: 1334: 1330: 1327: 1322: 1318: 1314: 1309: 1305: 1301: 1295: 1292: 1289: 1283: 1274: 1264: 1263: 1262: 1240: 1227: 1217: 1205: 1195: 1188: 1176: 1166: 1156: 1143: 1133: 1128: 1118: 1109: 1099: 1089: 1076: 1071: 1061: 1041: 1036: 1026: 1016: 1007: 995: 989: 980: 970: 969: 968: 966: 960: 956: 952: 945: 941: 934: 928: 921: 917: 897: 891: 888: 885: 882: 879: 876: 873: 870: 864: 858: 855: 852: 843: 840: 837: 831: 822: 810: 800: 799: 798: 794: 790: 783: 777: 773: 769: 764: 760: 756: 749: 745: 741: 737: 733: 729: 725: 721: 714: 710: 706: 701: 682: 671: 663: 653: 649: 645: 635: 631: 624: 613: 605: 600: 596: 590: 586: 579: 568: 563: 559: 544: 539: 535: 528: 525: 522: 519: 516: 509: 508: 507: 504: 497: 493: 487: 480: 476: 470: 464: 460: 454: 450: 444: 440: 432: 427: 420: 416: 411: 406: 403: 399: 393: 389: 385: 381: 375: 371: 365: 361: 357: 353: 346: 342: 336: 332: 328: 324: 304: 301: 298: 295: 292: 289: 286: 281: 277: 273: 268: 264: 260: 255: 251: 243: 242: 241: 239: 235: 231: 227: 223: 219: 214: 212: 202: 200: 195: 193: 192:biquaternions 189: 185: 181: 177: 173: 168: 166: 162: 157: 155: 151: 150:biquaternions 147: 143: 139: 130: 128: 124: 120: 116: 112: 108: 102: 99: 96:to represent 95: 94:number system 91: 86: 84: 76: 72: 67: 63: 58: 54: 50: 46: 42: 38: 34: 30: 21: 10531: 10521: 10340: 10336:Dual numbers 10328:hypercomplex 10118:Real numbers 9837: 9821: 9817: 9809:28(4):820–6. 9806: 9802:31(1):57–76. 9799: 9787: 9771: 9760: 9749: 9731: 9694: 9690: 9667:Eduard Study 9660: 9645: 9625: 9614: 9603: 9595: 9594: 9581: 9571: 9562: 9556: 9548: 9543: 9535: 9532:Eduard Study 9527: 9519:, link from 9508: 9500: 9495: 9486: 9477: 9471: 9462: 9443: 9437: 9416: 9407: 9401: 9391: 9384: 9349: 9339: 9329:December 24, 9327:. Retrieved 9314: 9294: 9239: 9235: 9225: 9217: 9212: 9203: 9202: 9168:Screw theory 9155: 9140:Eduard Study 9138:Since both 9137: 9125: 8746: 8742:dual numbers 8739: 8673: 8583: 8015:is given by 8010: 7903: 7804:is given by 7774: 7676: 7531: 7344: 7319: 7315: 7311: 7258: 7250: 7140: 7049: 6936: 6932: 6928: 6924: 6922: 6854: 6838: 6834: 6830: 6828: 6647: 6439: 6084: 5729: 5703: 5695: 5691: 5688: 5678: 5672: 5665: 5400: 5239: 4844: 4600: 4374: 4370: 4366: 4362: 4357: 4353: 4343: 4244: 4240: 4236: 4232: 4228: 4224: 4220: 4216: 4209: 4205: 4203: 4197: 4193: 4189: 4185: 4180: 4176: 4168: 4163: 4159: 4155: 4150: 4146: 4144: 4112: 4097: 4091: 4088: 4084: 4077: 4072: 4068: 4065: 3903: 3725: 3719: 3716: 3711: 3704: 3697: 3690: 3682: 3674: 3669: 3667: 3659: 3542: 3462: 3452: 3450: 3273: 3173: 3070: 3068: 2954: 2947: 2943: 2940: 2811: 2807: 2804: 2697: 2670: 2664: 2658: 2652: 2632: 2623: 2617: 2611: 2591: 2585: 2576: 2570: 2550: 2544: 2538: 2521: 2515: 2509: 2500: 2494: 2489: 2484: 2477: 2468: 2462: 2456: 2448: 2442: 2437: 2430: 2424: 2415: 2409: 2403: 2395: 2390: 2383: 2377: 2371: 2362: 2357: 2352: 2339: 2333: 2327: 2318: 2313: 2308: 2292: 2283: 2281: 2156: 2085: 2014: 1640: 1450: 1260: 958: 954: 950: 943: 939: 932: 926: 919: 915: 912: 792: 788: 784: 775: 771: 767: 762: 758: 754: 747: 743: 739: 735: 731: 727: 723: 719: 712: 708: 704: 697: 502: 495: 491: 485: 478: 474: 468: 462: 458: 452: 448: 442: 438: 430: 425: 418: 414: 407: 401: 397: 391: 387: 383: 379: 373: 369: 363: 359: 355: 351: 344: 340: 338:, to obtain 334: 330: 326: 322: 319: 237: 233: 229: 225: 221: 217: 215: 208: 196: 183: 172:Eduard Study 169: 158: 149: 136: 103: 87: 74: 70: 65: 61: 57:real numbers 53:dual numbers 49:dual numbers 32: 26: 10601:Quaternions 10498:Other types 10317:Bioctonions 10174:Quaternions 9663:, Springer. 9652:Joe Rooney 9216:A.T. Yang, 4390:, given by 700:dual number 412:), that is 211:quaternions 142:quaternions 140:introduced 123:Polynomials 55:instead of 45:quaternions 29:mathematics 10596:Kinematics 10585:Categories 10452:Projective 10425:Infinities 9862:DQrobotics 9453:2403.00558 9428:2209.02306 9199:References 5401:to obtain 4214:screw axis 4173:screw axis 4119:local ring 3728:, that is 3708:| = 1 186:in 1901. 107:kinematics 10536:solenoids 10356:Sedenions 10202:Octonions 9258:2296-9144 9082:− 8930:ε 8845:− 8717:→ 8708:⋅ 8692:→ 8525:− 8507:− 8413:− 8360:− 8317:− 8281:− 8134:− 8091:− 8073:− 7982:Δ 7965:Δ 7952:Δ 7886:¯ 7881:∗ 7874:^ 7862:⋅ 7856:^ 7847:⋅ 7841:^ 7822:^ 7789:^ 7709:ε 7694:^ 7649:Δ 7628:Δ 7607:Δ 7566:→ 7540:θ 7491:⋅ 7485:→ 7462:θ 7453:⁡ 7435:θ 7426:⁡ 7295:ε 7277:^ 7230:^ 7219:∗ 7209:− 7202:^ 7177:^ 7158:^ 7121:¯ 7116:∗ 7109:^ 7094:^ 7082:^ 7067:^ 7027:× 7019:ε 7002:^ 6981:× 6973:ε 6956:^ 6889:ε 6874:^ 6770:− 6758:− 6739:− 6717:^ 6703:− 6696:^ 6677:^ 6665:^ 6541:^ 6520:^ 6501:^ 6489:^ 6324:− 6309:− 6294:− 6241:− 6212:− 6147:− 6113:− 5969:− 5954:− 5939:− 5898:− 5833:− 5804:− 5668:Rodrigues 5638:⋅ 5621:^ 5618:α 5611:⁡ 5598:^ 5595:β 5588:⁡ 5582:− 5567:× 5550:^ 5547:α 5540:⁡ 5527:^ 5524:β 5517:⁡ 5494:^ 5491:α 5484:⁡ 5461:^ 5458:β 5451:⁡ 5425:^ 5422:γ 5415:⁡ 5379:⋅ 5362:^ 5359:α 5352:⁡ 5339:^ 5336:β 5329:⁡ 5323:− 5313:^ 5310:α 5303:⁡ 5290:^ 5287:β 5280:⁡ 5264:^ 5261:γ 5254:⁡ 5210:× 5193:^ 5190:α 5183:⁡ 5170:^ 5167:β 5160:⁡ 5137:^ 5134:β 5127:⁡ 5114:^ 5111:α 5104:⁡ 5081:^ 5078:α 5071:⁡ 5058:^ 5055:β 5048:⁡ 5022:⋅ 5005:^ 5002:α 4995:⁡ 4982:^ 4979:β 4972:⁡ 4966:− 4956:^ 4953:α 4946:⁡ 4933:^ 4930:β 4923:⁡ 4895:^ 4892:γ 4885:⁡ 4869:^ 4866:γ 4859:⁡ 4808:^ 4805:α 4798:⁡ 4782:^ 4779:α 4772:⁡ 4742:^ 4739:β 4732:⁡ 4716:^ 4713:β 4706:⁡ 4678:^ 4675:γ 4668:⁡ 4652:^ 4649:γ 4642:⁡ 4630:^ 4562:^ 4559:β 4550:⁡ 4527:^ 4524:β 4515:⁡ 4503:^ 4466:^ 4463:α 4454:⁡ 4431:^ 4428:α 4419:⁡ 4407:^ 4312:^ 4309:ϕ 4302:⁡ 4286:^ 4283:ϕ 4276:⁡ 4264:^ 4123:Lie group 3995:∗ 3979:∗ 3920:∗ 3863:∗ 3847:∗ 3831:∗ 3809:∗ 3796:∗ 3765:∗ 3758:^ 3745:^ 3698:magnitude 3686:| = 3639:∗ 3631:ε 3628:− 3623:∗ 3607:∗ 3599:− 3591:∗ 3575:¯ 3570:∗ 3563:^ 3521:ε 3518:− 3503:− 3486:¯ 3481:^ 3426:⋅ 3399:^ 3376:− 3364:^ 3329:^ 3311:∗ 3304:^ 3291:^ 3249:− 3237:^ 3222:∗ 3195:^ 3150:^ 3131:− 3122:^ 3098:∗ 3091:^ 3035:^ 3004:^ 2985:^ 2921:∗ 2914:^ 2902:∗ 2895:^ 2880:∗ 2869:^ 2857:^ 2840:∗ 2833:^ 2784:∗ 2776:ε 2768:∗ 2752:∗ 2739:∗ 2723:∗ 2716:^ 2694:Conjugate 2243:ε 2222:ε 2204:ε 2186:^ 2174:^ 2136:ε 2103:^ 2065:ε 2032:^ 1989:ε 1951:ε 1913:ε 1878:ε 1673:^ 1658:^ 1620:ε 1601:ε 1582:ε 1566:ε 1468:^ 1430:ε 1411:ε 1392:ε 1376:ε 1278:^ 1228:× 1199:^ 1170:^ 1144:⋅ 1134:− 1122:^ 1103:^ 1065:^ 1030:^ 1011:^ 999:^ 984:^ 826:^ 814:^ 672:× 614:⋅ 606:− 302:− 165:octonions 90:mechanics 10591:Machines 9945:Integers 9907:Sets of 9723:23443667 9619:Elsevier 9576:380–440. 9276:33501006 9162:See also 8698:′ 7828:′ 5679:vertices 5666:This is 4235:) where 1257:Addition 930:, where 429:, where 410:bivector 320:Compute 205:Formulas 170:In 1891 159:In 1898 129:design. 115:robotics 69:, where 47:and the 10526:numbers 10358: ( 10204: ( 10176: ( 10148: ( 10120: ( 10041: ( 10039:Periods 10008: ( 9975: ( 9947: ( 9919: ( 9901:systems 9824:: 1–13. 9714:3576712 9673:39:520. 9596:Sources 9588:, p. 4. 9534:(1901) 9515:(1898) 9267:7805728 9242:: 128. 9134:Eponyms 4231:, 4223:, 4087:(1 − ε 4062:Inverse 3688:√ 2958:, then 133:History 43:of the 37:algebra 10326:Other 9899:Number 9738: 9721: 9711: 9636: 9632:Press 9376:531000 9374: 9364: 9302: 9274: 9264: 9256: 9148:eponym 7532:where 7310:where 5709:C = (C 4348:with D 4158:) and 3702:| 3680:| 348:, and 236:, and 224:, and 31:, the 10534:-adic 10524:-adic 10281:Over 10242:Over 10236:types 10234:Split 9792:(PDF) 9697:: 7. 9448:arXiv 9423:arXiv 9372:S2CID 9324:(PDF) 9204:Notes 9017:with 8811:with 6849:j + d 6845:i + d 5698:C = c 4386:and D 4104:ideal 3174:or, 2288:ε = 0 2157:then 1641:then 965:screw 780:ε = 0 738:) = ( 352:i j k 329:) = − 327:i j k 79:ε = 0 10570:List 10427:and 9736:ISBN 9719:PMID 9634:ISBN 9362:ISBN 9331:2022 9300:ISBN 9272:PMID 9254:ISSN 9142:and 8922:and 8860:and 7314:and 4113:The 4071:+ ε 4038:and 3710:are 3670:norm 3668:The 3664:Norm 2086:and 1451:and 953:= ( 948:and 938:= ( 730:) ( 722:= ( 707:= ( 489:and 386:) = 117:and 73:and 9776:doi 9709:PMC 9699:doi 9630:MIT 9354:doi 9262:PMC 9244:doi 7450:sin 7423:cos 6991:and 6468:, d 6464:, D 6460:, D 6456:, D 6452:, c 6448:, C 6444:, C 5721:, c 5717:, C 5713:, C 5608:tan 5585:tan 5537:tan 5514:tan 5481:tan 5448:tan 5412:tan 5349:sin 5326:sin 5300:cos 5277:cos 5251:cos 5180:sin 5157:sin 5124:cos 5101:sin 5068:cos 5045:sin 4992:sin 4969:sin 4943:cos 4920:cos 4882:sin 4856:cos 4795:sin 4769:cos 4729:sin 4703:cos 4665:sin 4639:cos 4547:sin 4512:cos 4492:and 4451:sin 4416:cos 4299:sin 4273:cos 4066:If 3722:= 1 3720:Â Â 3691:Â Â 3136:Vec 3073:as 3021:Vec 2526:−ε 761:+ ε 748:b c 744:a d 740:a c 720:â ĉ 506:as 402:j i 400:= − 398:i j 390:= − 388:j i 384:j k 370:i j 367:or 362:= − 360:i j 358:= − 341:j k 333:= − 331:j k 88:In 64:+ ε 27:In 10587:: 10116:: 9836:. 9770:, 9759:, 9717:. 9707:. 9693:. 9689:. 9370:. 9360:. 9348:. 9284:^ 9270:. 9260:. 9252:. 9238:. 9234:. 9130:. 9111:0. 8791:, 8771:, 7700::= 5702:+ 4605:=D 4201:. 4137:. 4094:). 3714:. 3108:Sc 2971:Sc 2953:+ 2946:= 2812:ÂĈ 2810:= 2687:0 2675:−ε 2669:−ε 2646:0 2628:−ε 2622:−ε 2605:0 2590:−ε 2581:−ε 2564:0 2531:ε 2520:−ε 2505:−1 2473:−ε 2467:−ε 2453:−1 2429:−ε 2420:−ε 2400:−1 2346:1 2323:ε 2305:1 2290:. 2284:BD 957:, 942:, 925:+ 918:= 791:, 787:( 782:. 774:, 770:, 757:= 746:+ 742:, 734:, 726:, 711:, 501:+ 494:= 484:+ 477:= 461:+ 451:+ 441:= 424:+ 417:= 382:( 372:= 354:) 350:( 343:= 325:( 305:1. 232:, 220:, 213:. 194:. 156:. 121:. 113:, 85:. 10538:) 10532:p 10528:( 10522:p 10396:/ 10380:) 10367:S 10303:: 10290:C 10264:: 10251:R 10226:) 10213:O 10198:) 10185:H 10170:) 10157:C 10142:) 10129:R 10065:) 10051:P 10030:) 10017:A 9997:) 9984:Q 9969:) 9956:Z 9941:) 9928:N 9891:e 9884:t 9877:v 9845:. 9829:. 9822:2 9778:: 9744:. 9725:. 9701:: 9695:7 9640:. 9610:. 9456:. 9450:: 9431:. 9425:: 9395:. 9378:. 9356:: 9333:. 9308:. 9278:. 9246:: 9240:5 9108:= 9103:2 9098:4 9094:e 9088:, 9085:1 9079:= 9074:2 9069:3 9065:e 9061:= 9056:2 9051:2 9047:e 9043:= 9038:2 9033:1 9029:e 9003:4 8999:e 8995:, 8990:3 8986:e 8982:, 8977:2 8973:e 8969:, 8964:1 8960:e 8939:k 8936:e 8933:= 8910:j 8907:i 8904:= 8901:k 8881:0 8878:= 8873:2 8869:e 8848:1 8842:= 8837:2 8833:j 8829:= 8824:2 8820:i 8799:e 8779:j 8759:i 8714:v 8705:T 8702:= 8689:v 8657:) 8649:z 8645:v 8635:y 8631:v 8621:x 8617:v 8609:1 8603:( 8598:= 8595:v 8569:. 8564:) 8556:2 8551:z 8547:r 8543:+ 8538:2 8533:y 8529:r 8520:2 8515:x 8511:r 8502:2 8497:w 8493:r 8485:x 8481:r 8475:w 8471:r 8467:2 8464:+ 8459:z 8455:r 8449:y 8445:r 8441:2 8434:y 8430:r 8424:w 8420:r 8416:2 8408:z 8404:r 8398:x 8394:r 8390:2 8381:x 8377:r 8371:w 8367:r 8363:2 8355:z 8351:r 8345:y 8341:r 8337:2 8330:2 8325:z 8321:r 8312:2 8307:y 8303:r 8299:+ 8294:2 8289:x 8285:r 8276:2 8271:w 8267:r 8259:z 8255:r 8249:w 8245:r 8241:2 8238:+ 8233:y 8229:r 8223:x 8219:r 8215:2 8206:y 8202:r 8196:w 8192:r 8188:2 8185:+ 8180:z 8176:r 8170:x 8166:r 8162:2 8155:z 8151:r 8145:w 8141:r 8137:2 8129:y 8125:r 8119:x 8115:r 8111:2 8104:2 8099:z 8095:r 8086:2 8081:y 8077:r 8068:2 8063:x 8059:r 8055:+ 8050:2 8045:w 8041:r 8034:( 8029:= 8026:R 7994:) 7985:z 7974:R 7968:y 7955:x 7944:0 7939:0 7934:0 7929:1 7923:( 7918:= 7915:T 7900:. 7871:q 7853:v 7838:q 7832:= 7819:v 7786:q 7760:) 7757:k 7752:z 7748:v 7744:+ 7741:j 7736:y 7732:v 7728:+ 7725:i 7720:x 7716:v 7712:( 7706:+ 7703:1 7691:v 7673:. 7661:k 7656:2 7652:z 7643:+ 7640:j 7635:2 7631:y 7622:+ 7619:i 7614:2 7610:x 7601:+ 7598:0 7595:= 7592:d 7563:a 7516:) 7512:) 7509:k 7506:, 7503:j 7500:, 7497:i 7494:( 7482:a 7475:( 7470:) 7465:2 7457:( 7447:+ 7443:) 7438:2 7430:( 7420:= 7417:k 7412:z 7408:r 7404:+ 7401:j 7396:y 7392:r 7388:+ 7385:i 7380:x 7376:r 7372:+ 7367:w 7363:r 7359:= 7356:r 7330:d 7320:r 7316:d 7312:r 7298:r 7292:d 7289:+ 7286:r 7283:= 7274:q 7236:. 7227:x 7215:] 7199:S 7192:[ 7189:] 7184:+ 7174:S 7167:[ 7164:= 7155:X 7126:. 7106:S 7091:x 7079:S 7073:= 7064:X 7035:. 7031:X 7023:P 7016:+ 7012:X 7008:= 6999:X 6985:x 6977:p 6970:+ 6966:x 6962:= 6953:x 6937:P 6933:X 6929:p 6925:x 6908:. 6905:S 6902:D 6897:2 6894:1 6886:+ 6883:S 6880:= 6871:S 6855:d 6851:3 6847:2 6843:1 6839:d 6835:S 6831:d 6809:. 6804:} 6798:B 6791:A 6785:{ 6778:] 6766:C 6754:D 6746:0 6735:C 6728:[ 6723:= 6714:A 6708:] 6693:C 6686:[ 6683:= 6674:C 6662:A 6633:. 6628:} 6622:D 6615:C 6609:{ 6602:] 6594:+ 6590:A 6582:+ 6578:B 6570:0 6563:+ 6559:A 6552:[ 6547:= 6538:C 6532:] 6527:+ 6517:A 6510:[ 6507:= 6498:C 6486:A 6470:0 6466:3 6462:2 6458:1 6454:0 6450:3 6446:2 6442:1 6425:. 6420:} 6412:0 6408:a 6398:3 6394:A 6384:2 6380:A 6370:1 6366:A 6359:{ 6352:] 6344:0 6340:c 6332:3 6328:C 6317:2 6313:C 6302:1 6298:C 6285:3 6281:C 6273:0 6269:c 6261:1 6257:C 6249:2 6245:C 6232:2 6228:C 6220:1 6216:C 6205:0 6201:c 6193:3 6189:C 6179:1 6175:C 6167:2 6163:C 6155:3 6151:C 6140:0 6136:c 6129:[ 6124:= 6121:A 6118:] 6109:C 6105:[ 6102:= 6099:C 6096:A 6070:. 6065:} 6057:0 6053:c 6043:3 6039:C 6029:2 6025:C 6015:1 6011:C 6004:{ 5997:] 5989:0 5985:a 5977:3 5973:A 5962:2 5958:A 5947:1 5943:A 5930:3 5926:A 5918:0 5914:a 5906:1 5902:A 5891:2 5887:A 5877:2 5873:A 5865:1 5861:A 5853:0 5849:a 5841:3 5837:A 5824:1 5820:A 5812:2 5808:A 5797:3 5793:A 5785:0 5781:a 5774:[ 5769:= 5766:C 5763:] 5758:+ 5754:A 5750:[ 5747:= 5744:C 5741:A 5725:) 5723:0 5719:3 5715:2 5711:1 5704:C 5700:0 5651:. 5643:A 5633:B 5626:2 5603:2 5579:1 5572:A 5562:B 5555:2 5532:2 5511:+ 5506:A 5499:2 5478:+ 5473:B 5466:2 5442:= 5437:C 5430:2 5384:A 5374:B 5367:2 5344:2 5318:2 5295:2 5274:= 5269:2 5225:. 5221:) 5215:A 5205:B 5198:2 5175:2 5154:+ 5149:A 5142:2 5119:2 5098:+ 5093:B 5086:2 5063:2 5041:( 5037:+ 5033:) 5027:A 5017:B 5010:2 4987:2 4961:2 4938:2 4916:( 4912:= 4907:C 4900:2 4879:+ 4874:2 4830:. 4826:) 4820:A 4813:2 4792:+ 4787:2 4765:( 4760:) 4754:B 4747:2 4726:+ 4721:2 4699:( 4695:= 4690:C 4683:2 4662:+ 4657:2 4636:= 4627:C 4611:A 4609:D 4607:B 4603:C 4586:. 4581:B 4576:) 4573:2 4569:/ 4553:( 4544:+ 4541:) 4538:2 4534:/ 4518:( 4509:= 4500:B 4485:A 4480:) 4477:2 4473:/ 4457:( 4448:+ 4445:) 4442:2 4438:/ 4422:( 4413:= 4404:A 4388:B 4384:A 4380:C 4375:A 4371:D 4367:B 4363:D 4358:C 4354:D 4350:A 4346:B 4329:. 4324:S 4317:2 4296:+ 4291:2 4270:= 4261:S 4245:D 4241:d 4237:φ 4233:d 4229:φ 4225:V 4221:S 4217:S 4210:d 4206:D 4198:A 4194:D 4190:B 4186:D 4181:C 4177:D 4169:a 4164:A 4160:D 4156:b 4151:B 4147:D 4092:p 4089:q 4085:p 4078:p 4073:q 4069:p 4046:B 4026:A 4006:, 4003:0 4000:= 3991:A 3987:B 3984:+ 3975:B 3971:A 3951:A 3931:, 3928:1 3925:= 3916:A 3912:A 3889:. 3886:) 3883:0 3880:, 3877:1 3874:( 3871:= 3868:) 3859:A 3855:B 3852:+ 3843:B 3839:A 3836:, 3827:A 3823:A 3820:( 3817:= 3814:) 3805:B 3801:, 3792:A 3788:( 3785:) 3782:B 3779:, 3776:A 3773:( 3770:= 3755:A 3742:A 3726:Â 3705:Â 3683:Â 3675:Â 3644:. 3635:B 3619:A 3615:= 3612:) 3603:B 3596:, 3587:A 3583:( 3580:= 3560:A 3527:. 3524:B 3515:A 3512:= 3509:) 3506:B 3500:, 3497:A 3494:( 3491:= 3478:A 3436:. 3431:A 3421:A 3416:+ 3411:2 3406:0 3396:a 3389:= 3386:) 3381:A 3371:0 3361:a 3354:( 3351:) 3346:A 3341:+ 3336:0 3326:a 3319:( 3316:= 3301:A 3288:A 3259:. 3254:A 3244:0 3234:a 3227:= 3218:) 3212:A 3207:+ 3202:0 3192:a 3185:( 3159:. 3156:) 3147:A 3141:( 3128:) 3119:A 3113:( 3103:= 3088:A 3071:Â 3054:. 3049:A 3044:= 3041:) 3032:A 3026:( 3016:, 3011:0 3001:a 2994:= 2991:) 2982:A 2976:( 2955:A 2951:0 2948:â 2944:Â 2926:. 2911:A 2892:C 2885:= 2876:) 2866:C 2854:A 2848:( 2845:= 2830:G 2808:Ĝ 2789:. 2780:B 2773:+ 2764:A 2760:= 2757:) 2748:B 2744:, 2735:A 2731:( 2728:= 2713:A 2684:0 2681:0 2678:0 2671:i 2665:j 2663:ε 2659:k 2657:ε 2653:k 2651:ε 2643:0 2640:0 2637:0 2633:i 2631:ε 2624:k 2618:j 2616:ε 2612:j 2610:ε 2602:0 2599:0 2596:0 2592:j 2586:k 2584:ε 2577:i 2575:ε 2571:i 2569:ε 2561:0 2558:0 2555:0 2551:k 2549:ε 2545:j 2543:ε 2539:i 2537:ε 2534:ε 2522:i 2516:j 2514:ε 2510:k 2508:ε 2501:i 2499:− 2495:j 2490:k 2485:k 2478:i 2476:ε 2469:k 2463:j 2461:ε 2457:i 2449:k 2447:− 2443:j 2438:j 2431:j 2425:k 2423:ε 2416:i 2414:ε 2410:j 2408:− 2404:k 2396:i 2391:i 2384:k 2382:ε 2378:j 2376:ε 2372:i 2370:ε 2367:ε 2363:k 2358:j 2353:i 2349:1 2340:k 2338:ε 2334:j 2332:ε 2328:i 2326:ε 2319:k 2314:j 2309:i 2267:. 2264:) 2261:C 2258:B 2255:+ 2252:D 2249:A 2246:( 2240:+ 2237:C 2234:A 2231:= 2228:) 2225:D 2219:+ 2216:C 2213:( 2210:) 2207:B 2201:+ 2198:A 2195:( 2192:= 2183:C 2171:A 2142:, 2139:D 2133:+ 2130:C 2127:= 2124:) 2121:D 2118:, 2115:C 2112:( 2109:= 2100:C 2071:, 2068:B 2062:+ 2059:A 2056:= 2053:) 2050:B 2047:, 2044:A 2041:( 2038:= 2029:A 1995:, 1992:k 1986:) 1981:3 1977:d 1973:+ 1968:3 1964:b 1960:( 1957:+ 1954:j 1948:) 1943:2 1939:d 1935:+ 1930:2 1926:b 1922:( 1919:+ 1916:i 1910:) 1905:1 1901:d 1897:+ 1892:1 1888:b 1884:( 1881:+ 1875:) 1870:0 1866:d 1862:+ 1857:0 1853:b 1849:( 1846:+ 1843:k 1840:) 1835:3 1831:c 1827:+ 1822:3 1818:a 1814:( 1811:+ 1808:j 1805:) 1800:2 1796:c 1792:+ 1787:2 1783:a 1779:( 1776:+ 1773:i 1770:) 1765:1 1761:c 1757:+ 1752:1 1748:a 1744:( 1741:+ 1738:) 1733:0 1729:c 1725:+ 1720:0 1716:a 1712:( 1709:= 1706:) 1703:D 1700:+ 1697:B 1694:, 1691:C 1688:+ 1685:A 1682:( 1679:= 1670:C 1664:+ 1655:A 1626:, 1623:k 1615:3 1611:d 1607:+ 1604:j 1596:2 1592:d 1588:+ 1585:i 1577:1 1573:d 1569:+ 1561:0 1557:d 1553:+ 1550:k 1545:3 1541:c 1537:+ 1534:j 1529:2 1525:c 1521:+ 1518:i 1513:1 1509:c 1505:+ 1500:0 1496:c 1492:= 1489:) 1486:D 1483:, 1480:C 1477:( 1474:= 1465:C 1436:, 1433:k 1425:3 1421:b 1417:+ 1414:j 1406:2 1402:b 1398:+ 1395:i 1387:1 1383:b 1379:+ 1371:0 1367:b 1363:+ 1360:k 1355:3 1351:a 1347:+ 1344:j 1339:2 1335:a 1331:+ 1328:i 1323:1 1319:a 1315:+ 1310:0 1306:a 1302:= 1299:) 1296:B 1293:, 1290:A 1287:( 1284:= 1275:A 1241:. 1238:) 1233:C 1223:A 1218:+ 1213:C 1206:0 1196:a 1189:+ 1184:A 1177:0 1167:c 1160:( 1157:+ 1154:) 1149:C 1139:A 1129:0 1119:c 1110:0 1100:a 1093:( 1090:= 1087:) 1082:C 1077:+ 1072:0 1062:c 1055:( 1052:) 1047:A 1042:+ 1037:0 1027:a 1020:( 1017:= 1008:C 996:A 990:= 981:G 961:) 959:B 955:A 951:A 946:) 944:b 940:a 936:0 933:â 927:A 923:0 920:â 916:Â 898:. 895:) 892:C 889:B 886:+ 883:D 880:A 877:, 874:C 871:A 868:( 865:= 862:) 859:D 856:, 853:C 850:( 847:) 844:B 841:, 838:A 835:( 832:= 823:C 811:A 795:) 793:B 789:A 776:k 772:j 768:i 763:b 759:a 755:â 750:) 736:d 732:c 728:b 724:a 715:) 713:b 709:a 705:â 683:. 680:) 676:C 668:A 664:+ 660:C 654:0 650:a 646:+ 642:A 636:0 632:c 628:( 625:+ 622:) 618:C 610:A 601:0 597:c 591:0 587:a 583:( 580:= 577:) 573:C 569:+ 564:0 560:c 556:( 553:) 549:A 545:+ 540:0 536:a 532:( 529:= 526:C 523:A 520:= 517:G 503:C 499:0 496:c 492:C 486:A 482:0 479:a 475:A 469:k 466:3 463:A 459:j 456:2 453:A 449:i 446:1 443:A 439:A 434:0 431:a 426:A 422:0 419:a 415:A 392:k 380:j 374:k 364:k 356:k 345:i 335:i 323:i 299:= 296:k 293:j 290:i 287:= 282:2 278:k 274:= 269:2 265:j 261:= 256:2 252:i 238:k 234:j 230:i 226:k 222:j 218:i 75:B 71:A 66:B 62:A

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Index