Appendix Composite Point Rotation Sequences

Transcription

1 Appendix Composite Point Rotation Sequences A. Euler Rotations In Chap. 6 we considered composite Euler rotations comprising individual rotations about the x, y and z axes such as R γ,x R β,y R α,z and R γ,z R β,y R α,x. However, there is nothing preventing us from creating other combinations such as R γ,x R β,y R α,x or R γ,z R β,y R α,z that do not include two consecutive rotations about the same axis. In all, there are twelve possible combinations: R γ,x R β,y R α,x R γ,x R β,y R α,z R γ,x R β,z R α,x R γ,x R β,z R α,y R γ,y R β,x R α,y R γ,y R β,x R α,z R γ,y R β,z R α,x R γ,y R β,z R α,y R γ,z R β,x R α,y R γ,z R β,x R α,z R γ,z R β,y R α,x R γ,z R β,y R α,z which we now cover in detail. For each combination there are three Euler rotation matrices, the resulting composite matrix, a matrix where the three angles equal 90, the coordinates of the rotated unit cube, the axis and angle of rotation and a figure illustrating the stages of rotation. To compute the axis of rotation [v v 2 v 3 ] T we use v = (a 22 )(a 33 ) a 23 a 32 v 2 = (a 33 )(a ) a 3 a 3 v 3 = (a )(a 22 ) a 2 a 2 where a a 2 a 3 R = a 2 a 22 a 23, a 3 a 32 a 33 and for the angle of rotation δ we use cos δ = 2( Tr(R) ). J. Vince, Matrix Transforms for Computer Games and Animation, DOI 0.007/ , Springer-Verlag London

2 50 Composite Point Rotation Sequences We begin by defining the three principal Euler rotations: rotate α about the x-axis R α,x = 0 c α s α 0 s α c α c β 0 s β rotate β about the y-axis R β,y = s β 0 c β c γ s γ 0 rotate γ about the z-axis R γ,z = s γ c γ where c α = cos α and s α = sin α,etc. Remember that the right-most transform is applied first and the left-most transform last. In terms of angles, the sequence is always α, β, γ. For each composite transform you can verify that when α = β = γ = 0 the result is the identity transform I. We now examine the twelve combinations in turn. A.2 R γ,x R β,y R α,x c β 0 s β R γ,x R β,y R α,x = 0 c γ s γ 0 c α s α 0 s γ c γ s β 0 c β 0 s α c α c β s β s α s β c α = s γ s β (c γ c α s γ c β s α ) ( c γ s α s γ c β c α ) c γ s β (s γ c α + c γ c β s α ) ( s γ s α + c γ c β c α ) R 90,xR 90,yR 90,x = = This rotation sequence is illustrated in Fig. A., where the axis of rotation is [2 2 0] T and the angle of rotation 80.

3 A.3 R γ,x R β,y R α,z 5 Fig. A. Four views of the unit cube before and during the three rotations R 90,xR 90,yR 90,x A.3 R γ,x R β,y R α,z c β 0 s β c α s α 0 R γ,x R β,y R α,z = 0 c γ s γ s α c α 0 0 s γ c γ s β 0 c β 0 0 c β c α c β s α s β = (c γ s α + s γ s β c α ) (c γ c α s γ s β s α ) s γ c β (s γ s α c γ s β c α ) (s γ c α + c γ s β s α ) c γ c β 0 0 R 90,xR 90,yR 90,z = =

4 52 Composite Point Rotation Sequences Fig. A.2 Four views of the unit cube before and during the three rotations R 90,xR 90,yR 90,z This rotation sequence is illustrated in Fig. A.2, where the axis of rotation is [2 0 2] T and the angle of rotation 80. A.4 R γ,x R β,z R α,x c β s β 0 R γ,x R β,z R α,x = 0 c γ s γ s β c β 0 0 c α s α 0 s γ c γ 0 s α c α c β s β c α s β s α = c γ s β ( s γ s α + c γ c β c α ) ( s γ c α c γ c β s α ) s γ s β (c γ s α + s γ c β c α ) (c γ c α s γ c β s α ) 0 0 R 90,xR 90,zR 90,x =

5 A.5 R γ,x R β,z R α,y 53 Fig. A.3 Four views of the unit cube before and during the three rotations R 90,xR 90,zR 90,x = This rotation sequence is illustrated in Fig. A.3, where the axis of rotation is [2 0 2] T and the angle of rotation 80. A.5 R γ,x R β,z R α,y c β s β 0 c α 0 s α R γ,x R β,z R α,y = 0 c γ s γ s β c β 0 0 s γ c γ 0 0 s α 0 c α c β c α s β c β s α = (s γ s α + c γ s β c α ) c γ c β ( s γ c α + c γ s β s α ) ( c γ s α + s γ s β c α ) s γ c β (c γ c α + s γ s β s α ) 0 0 R 90,xR 90,zR 90,y = 0 0

6 54 Composite Point Rotation Sequences Fig. A.4 Four views of the unit cube before and during the three rotations R 90,xR 90,zR 90,y = This rotation sequence is illustrated in Fig. A.4, where the axis of rotation is [0 0 2] T and the angle of rotation 90. A.6 R γ,y R β,x R α,y c γ 0 s γ c α 0 s α R γ,y R β,x R α,y = 0 c β s β s γ 0 c γ 0 s β c β s α 0 c α

7 A.6 R γ,y R β,x R α,y 55 Fig. A.5 Four views of the unit cube before and during the three rotations R 90,yR 90,xR 90,y (c γ c α s γ c β s α ) s γ s β (c γ s α + s γ c β c α ) = s β s α c β s β c α ( s γ c α c γ c β s α ) c γ s β ( s γ s α + c γ c β c α ) R 90,yR 90,xR 90,y = = This rotation sequence is illustrated in Fig. A.5, where the axis of rotation is [2 2 0] T and the angle of rotation 80.

8 56 Composite Point Rotation Sequences Fig. A.6 Four views of the unit cube before and during the three rotations R 90,yR 90,xR 90,z A.7 R γ,y R β,x R α,z c γ 0 s γ c α s α 0 R γ,y R β,x R α,z = 0 c β s β s α c α 0 s γ 0 c γ 0 s β c β 0 0 (c γ c α + s γ s β s α ) ( c γ s α + s γ s β c α ) s γ c β = c β s α c β c α s β ( s γ c α + c γ s β s α ) (s γ s α + c γ s β c α ) c γ c β R 90,yR 90,xR 90,z = = This rotation sequence is illustrated in Fig. A.6, where the axis of rotation is [2 0 0] T and the angle of rotation 90.

9 A.8 R γ,y R β,z R α,x 57 A.8 R γ,y R β,z R α,x c γ 0 s γ c β s β 0 R γ,y R β,z R α,x = s β c β 0 0 c α s α s γ 0 c γ 0 s α c α c γ c β (s γ s α c γ s β c α ) (s γ c α + c γ s β s α ) = s β c β c α c β s α s γ c β (c γ s α + s γ s β c α ) (c γ c α s γ s β s α ) R 90,yR 90,zR 90,x = = This rotation sequence is illustrated in Fig. A.7, where the axis of rotation is [2 2 0] T and the angle of rotation 80. A.9 R γ,y R β,z R α,y c γ 0 s γ c β s β 0 c α 0 s α R γ,y R β,z R α,y = s β c β 0 s γ 0 c γ 0 0 s α 0 c α ( s γ s α + c γ c β c α ) c γ s β (s γ c α + c γ c β s α ) = s β c α c β s β s α ( c γ s α s γ c β c α ) s γ s β (c γ c α s γ c β s α ) 0 0 R 90,yR 90,zR 90,y =

10 58 Composite Point Rotation Sequences Fig. A.7 Four views of the unit cube before and during the three rotations R 90,yR 90,zR 90,x =. 0 0 This rotation sequence is illustrated in Fig. A.8, where the axis of rotation is [0 2 2] T and the angle of rotation 80. A.0 R γ,z R β,x R α,y c γ s γ 0 c α 0 s α R γ,z R β,x R α,y = s γ c γ 0 0 c β s β 0 s β c β s α 0 c α (c γ c α s γ s β s α ) s γ c β (c γ s α + s γ s β c α ) = (s γ c α + c γ s β s α ) c γ c β (s γ s α c γ s β c α ) c β s α s β c β c α 0 0 R 90,zR 90,xR 90,y = 0 0

11 A. R γ,z R β,x R α,z 59 Fig. A.8 Four views of the unit cube before and during the three rotations R 90,yR 90,zR 90,y =. 0 0 This rotation sequence is illustrated in Fig. A.9, where the axis of rotation is [0 2 2] T and the angle of rotation 80. A. R γ,z R β,x R α,z c γ s γ 0 c α s α 0 R γ,z R β,x R α,z = s γ c γ 0 0 c β s β s α c α 0 0 s β c β 0 0

12 60 Composite Point Rotation Sequences Fig. A.9 Four views of the unit cube before and during the three rotations R 90,zR 90,xR 90,y (c γ c α s γ c β s α ) ( c γ s α s γ c β c α ) s γ s β = (s γ c α + c γ c β s α ) ( s γ s α + c γ c β c α ) c γ s β s β s α s β c α c β 0 0 R 90,zR 90,xR 90,z = = This rotation sequence is illustrated in Fig. A.0, where the axis of rotation is [2 0 2] T and the angle of rotation 80.

13 A.2 R γ,z R β,y R α,x 6 Fig. A.0 Four views of the unit cube before and during the three rotations R 90,zR 90,xR 90,z A.2 R γ,z R β,y R α,x c γ s γ 0 c β 0 s β R γ,z R β,y R α,x = s γ c γ 0 0 c α s α 0 0 s β 0 c β 0 s α c α c γ c β ( s γ c α + c γ s β s α ) (s γ s α + c γ s β c α ) = s γ c β (c γ c α + s γ s β s α ) ( c γ s α + s γ s β c α ) s β c β s α c β c α ) 0 0 R 90,zR 90,yR 90,x = =

14 62 Composite Point Rotation Sequences Fig. A. Four views of the unit cube before and during the three rotations R 90,zR 90,yR 90,x This rotation sequence is illustrated in Fig. A., where the axis of rotation is [0 2 0] T and the angle of rotation 90. A.3 R γ,z R β,y R α,z c γ s γ 0 c β 0 s β c α s α 0 R γ,z R β,y R α,z = s γ c γ 0 s α c α s β 0 c β 0 0 ( s γ s α + c γ c β c α ) ( s γ c α c γ c β s α ) c γ s β = (c γ s α + s γ c β c α ) (c γ c α s γ c β s α ) s γ s β s β c α s β s α c β 0 0 R 90,zR 90,yR 90,z = 0 0

15 A.3 R γ,z R β,y R α,z 63 Fig. A.2 Four views of the unit cube before and during the three rotations R 90,zR 90,yR 90,z =. 0 0 This rotation sequence is illustrated in Fig. A.2, where the axis of rotation is [0 2 2] T and the angle of rotation 80.

16 Index Symbols 2D matrix transforms, 55 2D reflection transform, 63 2D scaling transform, 60 2D shearing transform, 67 2D translation transform, 58 3D reflection transform, 85 3D rotation transform, 87 3D scaling transform, 83 3D shearing transform, 85 3D translation transform, 84 A Adding quaternions, 8 Additive identity, Affine transform, 7 Antisymmetric matrix, 45 Areal coordinates, 57 B Barycentric coordinates, 57 C Cayley, Arthur, 3 Change of axes, 73 Characteristic equation, 75, 95 Cofactor matrix, 48 Cofactors, 26 Column index, 3 Column matrix, 6, 35 Columnvector,6,36 Composite rotations,, 89 Conformable matrices, 35 D Determinant, 6, 9 Diagonal matrix, 38 E Eigenvalue, 75, 34 Eigenvector, 75, 94, 34 Euler angles, 39 Euler rotations,, 87, 89 F Frames of reference, 38 G Gauss, Carl, 3 Gibbs, 7 Gimbal lock, 02 H Hamilton, 7 Hamilton s rules, 7 Homogeneous coordinates, 57 I Identity matrix,, 39 Inverse matrix,, 47 Inverse quaternion, 9 L Laplace expansion, 24 Linear equations, 3 M Matrices, 3 Matrix addition, 32 Matrix multiplication, 8, 34 Matrix notation, 5 Matrix scaling, 33 Matrix subtraction, 32 Minor determinant, 26 Möbius, 57 Multiplicative identity, J. Vince, Matrix Transforms for Computer Games and Animation, DOI 0.007/ , Springer-Verlag London

17 66 Index Multiplicative inverse, Multiplying quaternions, 8 N Negative matrix, 38 O Order of a matrix, 32 Orthogonal matrix, 50 P Pitch, 03 Pitch quaternion, 23 Pure quaternion, 9 Q Quaternion matrix, 25 Quaternions, 7 R Rectangular matrix, 3 Rodrigues, Olinde, Roll, 03 Roll quaternion, 23 Rotating a point about an axis, 20 Rotating about an axis, 88, 06 Row index, 3 Row vector, 36 S Sarrus s rule, 23 Shearing, 85 Skew symmetric matrix, 45 Square matrix, 6, 3, 38, 43 Subtracting quaternions, 8 Symmetric matrix, 44 T Trace of a matrix, 43 Transforms, 55 Transposed matrix, 40 Transposed vector, 36 U Unit matrix, Unit-norm quaternion, 20 V Vector scalar product, 35 Y Yaw, 03 Yaw quaternion, 23 Z Zero matrix, 37