zB B yB zA B0 VA0B0 A xB yA A0 xA Figure 2.1: Two Coordinate Systems.

B A zB 1 yB VB01 zA VA01 B0 VA0B0 xB yA A0 xA Figure 2.2: Depiction of Point Transformation Problem.

zB 1 B b3 yB b1 b2 xB Figure 2.3: Point 1 Projected onto the B Coordinate System

C B A zC 1 VC01 yC C0 xC VA01 zB VB0C0 zA yB B0 VA0B0 xB A0 yA xA Figure 2.4: Compound Transformation

zA zB yB  yA xA Figure 2.5: Rotation About the X Axis

Coordinate system B is initially aligned with coordinate system A. It is then translated to the point [5, 4, 1]T. It is then rotated 30 about its X axis. It is then rotated 60 about an axis that passes through the point [2, 0, 2]T, measured in the current coordinate system, which is parallel to the current Y axis. Find

zB zA yB B A xA yA xB Figure 2.6: Initial and Final Coordinate Systems

A C C initially aligned with A ; translated to [5,4,1]T zA zC xA yA xC yC

C D D initially aligned with C ; rotated 30° about the X axis zC zD yD xC yC

B D B initially aligned with D ; rotated 60° about an axis that passes through point M, [2, 0, 2]T and is parallel to the Y axis zB yB B zD 60° D M yD xB xD

E initially aligned with D ; translate to point M, i.e. [2, 0, 2]T zE zB yB E B 60° zF zD yE F xE D M yD xB xF xD

E B F D F initially aligned with E ; rotate 60° about the Y axis zE zB yB E B 60° zF zD yE F xE D M yD xB xF xD

E B F D B initially aligned with F ; translate to [-2, 0, -2]T zE zB yB E B 60° zF zD yE F xE D M yD xB xF xD

m  zA zB yB yA xA xB Figure 2.8: Rotation of Angle  about axis m

C and A will rotate together by an angle  about m A moves to B ; C moves to D m zA the relationship between C and A is the same as the relationship between D and B zC yC yA xA xC

Coordinate system B is initially aligned with coordinate system A. It is then rotated an angle  about the axis m (unit vector). Find . m zA  yA xA

Coordinate system C is created such that its Z axis is along the direction of m. zC yA yC xA xC

Coordinate systems A and C rotate together about m by the angle . A rotates to become B and C rotates to become D. The relationship between B and D must be the same as the relationship between A and C. m zA zC  yA yC xA xC

Lastly, the rotation matrix that relates the B and A coordinate systems may be written as zA zC  yA yC xA xC

1st row is a unit vector ax2 + bx2 + mx2 = 1 ax2 + bx2 = 1 - mx2 1st and 2nd rows are orthogonal axay + bxby+ mxmy = 0 3rd column is cross product of 1st two columns mz = axby - bxay

given: find: m and 

r11 + r22 + r33 = (1-cos) (mx2+my2+mz2) + 3 cos r11 + r22 + r33 =1 - cos + 3 cos r21 - r12 = 2 mz sin r13 - r31 = 2 my sin r32 - r23 = 2 mx sin