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Team LRN generate, create code-128 none on .net projects pdf417 2d barcode XII.3 Representations of visual .net Code 128 Code Set B Orientations = cs + csu u; s 2 u + c2 u csu u = cs + cs; (s 2 + c2 )u = 0; u = u.

. These calculations used the fact that u is a unit vector, and so u u = 1. In addition, c2 + s 2 = 1 because c = cos( /2) and s = sin( /2). We now prove b.

We have v3 = u v2 and v3 u = v2 , where v3 is de ned as in Section II.2.3 (again, see Figure II.

14 on page 37). We will explicitly compute qv2 q 1 . First, qv2 = ( c; su )( 0; v2 ) = 0 su v2 ; cv2 + su v2 = 0; cv2 + sv3 .

Then, qv2 q 1 = ( 0; cv2 + sv3 )( c; su ) = 0 + scv2 u + s 2 v3 u; 0 + c2 v2 + csv3 csv2 u s 2 v3 u = 0; c2 v2 + csv3 + csv3 s 2 v2 = 0; (c2 s 2 )v2 + (2cs)v3 . = 0; (cos )v2 + (sin )v3 = R ,u v2 . The last equality follows from Equation II.

8 on page 42. The next-to-last equality follows from the sine and cosine double angle formulas: cos(2 ) = cos2 sin2 and sin(2 ) = 2 cos sin ,. with = /2. We have th Code 128A for .NET us completed the proof of Theorem XII.

3. Using quaternions to represent rotations makes it easy to calculate the composition of two rotations. Indeed, if the quaternion q1 represents the rotation R 1 ,u1 and if q2 represents R 2 ,u2 , then the product q1 q2 represents the composition R 1 ,u1 R 2 ,u2 .

To prove this, note that. 1 1 1 1 R 1 ,u1 R 2 ,u2 v = q1 (q2 vq2 )q1 = (q1 q2 )v(q2 q1 ) = (q1 q2 )v(q1 q2 ) 1. holds by associativity of multiplication and by Exercise XII.7. Exercise XII.9 Let R1 an d R2 be the two rotations with matrix representations given in the formulas XII.1 on page 295.

a. What are the two unit quaternions that represent R1 What are the two unit quaternions that represent R2 b. Let v = 1, 3, 2 .

Compute R1 v and R2 v using quaternion representations from part a. by the method of Theorem XII.3.

Check your answers by multiplying by the matrices in XII.1..

XII.3.6 Quaternion and Rotation Matrix Conversions Because a quaternion rep resents a rotation, its action on R3 can be represented by a 3 3 matrix. We now show how to convert a quaternion q into a 3 3 matrix. Recall from the.

Team LRN Animation and Kinematics discussion on page 35 th visual .net barcode 128 at a transformation A(v) is represented by the matrix (w1 w2 w3 ), where the columns wi are equal to A(i), A(j), and A(k). Thus, to represent a quaternion q by a matrix, we set w1 = qiq 1 , w2 = qjq 1 , and w3 = qkq 1 ,.

and then the matrix repr esentation will be (w1 w2 w3 ). Let q = d, a, b, c be a unit quaternion. To compute qiq 1 , rst compute ( d, a, b, c )( 0, 1, 0, 0 ) = a, d, c, b .

Then, since q 1 = d, a, b, c , compute qiq 1 by ( a, d, c, b )( d, a, b, c ) = 0, d 2 + a 2 b2 c2 , 2ab + 2cd, 2ac 2bd . Similar computations give qjq 1 = 0, 2ab 2cd, d 2 a 2 + b2 c2 , 2bc + 2ad qkq 1 = 0, 2ac + 2bd, 2bc 2ad, d 2 a 2 b2 + c2 . Thus, the matrix representing the same rotation as the quaternion d, a, b, c is 2 d + a 2 b2 c2 2ab 2cd 2ac + 2bd .

2ab + 2cd d 2 a 2 + b2 c2 2bc 2ad 2 2 2 2 2ac 2bd 2bc + 2ad d a b +c.
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