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Copyright 2008 by The McGrawHill Companies Click here for terms of use QR Code JIS X 0510 Creation In None Using Barcode generation for Software Control to generate, create QR Code ISO/IEC18004 image in Software applications. Reading QR In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. APPENDIX D QUATERNIONS
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EAN13 Printer In VS .NET Using Barcode drawer for Reporting Service Control to generate, create EAN / UCC  13 image in Reporting Service applications. Paint GS1 128 In VB.NET Using Barcode encoder for Visual Studio .NET Control to generate, create GS1128 image in VS .NET applications. where b = [b2 , b3 , b4 ] The vector form allows compact representation of quaternion operations For example, the quaternion conjugate is b = b1 b The quaternion product can be written as b c = b1 c1 b c + b1 c + c1 b + b c (D7) Based on the quaternion b we can form the matrices Qb = b1 b b (b1 I + [b ]) and Qb = b1 b b (b1 I [b ]) (D8) From which it is clear that Qb = Qb and Qb = Qb Using the matrices Qb and Qb , the quaternion product can be expressed as b c = Qb c = Qc b (D9) (D10) which are the same matrices written in component form in eqns (D3 D4) It can also be shown, by direct multiplication, that Qb and Qc commute, c = Qc Qb Qb Q Rotations
Let frame a be aligned with frame b by rotating frame a by radians about unit vector E The quaternion b that represents the rotational transformation from frame a to frame b is b= cos( /2) E sin( /2) Note that b has the normality property that b = 1 Therefore, for the representation of rotational transformations, the quaternion b has only three degrees of freedom Let z = Rb v where v is coordinatized in frame a and z is the represena tation of v when coordinatized in frame b Each vector can be expressed as the quaternion form as qv = 0 v and qz = 0 z Using quaternions, the transformation of the vector quantity v from frame a to frame b is qz = b qv b 1 = b qv b (D11)

