1. CSCE 452: Question Set 1 Spatial Descriptions
Homogeneous Transformations: Mapping and Operator
Three Angle Rotation Representations
CPSC 452 Dezhen Song

2. Last lecture: Homogeneous Transform Interpretations

3. Homogeneous Transformation for Mapping
CPSC 452 Dezhen Song

4. Transformation Operator

5. Rotation Representations
Rotation Matrix
Fixed Angle Rotation
Euler Angle Rotation
Angle-Axis Representation
Euler Parameters

6. Q1
A vector 𝐴 𝑃 is rotated about 𝑍 𝐴 by θ degrees and is subsequently rotate about 𝑋 𝐴 by φ degrees. Given the rotation matrix that accomplishes these rotations in the given order.

7. Q1 Answer
A vector 𝐴 𝑃 is rotated about 𝑍 𝐴 by θ degrees and is subsequently rotate about 𝑋 𝐴 by φ degrees. Given the rotation matrix that accomplishes these rotations in the given order.
𝑅 𝑋 𝜑 𝑅 𝑍 (𝜃) = 𝑐𝜑 −𝑠𝜑 0 𝑠𝜑 𝑐𝜑 𝑐𝜃 −𝑠𝜃 0 𝑠𝜃 𝑐𝜃

8. Q2
What is the corresponding operator T for Q1 in matrix format?

9. Q2-Answer
What is the corresponding operator T for Q1 in matrix format?
𝑇= 𝑅 𝑋 𝜑 𝑅 𝑍 (𝜃) 0 3× ×3 1

10. Q3
A frame {B} is initially coincident with a frame {A}. We rotate {B} about 𝑍 𝐵 by θ degrees, and then we rotate the resulting frame about 𝑋 𝐵 by φ degrees. Given the rotation matrix that changes the description of the vectors from 𝐵 𝑃 to 𝐴 𝑃 .

11. Q3 -Answer
A frame {B} is initially coincident with a frame {A}. We rotate {B} about 𝑍 𝐵 by θ degrees, and then we rotate the resulting frame about 𝑋 𝐵 by φ degrees. Given the rotation matrix that changes the description of the vectors from 𝐵 𝑃 to 𝐴 𝑃 .
𝐵 𝐴 𝑅= 𝑅 𝑍 (𝜃) 𝑅 𝑋 𝜑 = 𝑐𝜃 −𝑠𝜃 0 𝑠𝜃 𝑐𝜃 𝑐𝜑 −𝑠𝜑 0 𝑠𝜑 𝑐𝜑

12. Q4
What is the transformation matrix 𝐵 𝐴 𝑇 for the that of Q2? How to compute 𝐴 𝐵 𝑇

13. Q4 –Answer:
What is the transformation matrix 𝐵 𝐴 𝑇 for the that of Q2? How to compute 𝐴 𝐵 𝑇
𝐵 𝐴 𝑇 = 𝑅 𝑍 (𝜃) 𝑅 𝑋 𝜑 0 3× ×3 1
𝐴 𝐵 𝑇 = ( 𝐵 𝐴 𝑇) −1 = 𝑅 𝑋 −𝜑 𝑅 𝑍 (−𝜃) 0 3× ×3 1

14. Q5
A vector 𝐴 𝑃 is undergoing the following transformation in sequence:
Translate by vector 𝑄 1
Rotate about 𝑍 𝐴 by θ degrees
Translate by another vector 𝑄 2
Rotate about about 𝑋 𝐴 by φ degrees
Please compute transform operator T for each step and a single transform operator matrix that can perform the above sequence. What is new 𝐴 𝑃 ?

15. Q5 - Answer
A vector 𝐴 𝑃 = 1, 2, 3 𝑇 is undergoing the following transformation in sequence:
Translate by vector 𝑄 1
Rotate about 𝑍 𝐴 by θ degrees
Translate by another vector 𝑄 2
Rotate about about 𝑋 𝐴 by φ degrees
Please compute transform operate T for each step and a single transform operator matrix that can perform the above sequence. What is new 𝐴 𝑃 ?
𝑇 1 = 𝐼 3×3 𝑄 ×3 1 , 𝑇 2 = 𝑅 𝑍 (𝜃) 0 3× ×3 1 ,
𝑇 3 = 𝐼 3×3 𝑄 ×3 1 , 𝑇 4 = 𝑅 𝑋 (𝜑) 0 3× ×3 1 ,
𝑇= 𝑇 4 𝑇 3 𝑇 2 𝑇 1
𝐴 𝑃 1 =𝑇

16. Q6
A frame {B} is initially coincident with a frame {A}. We transform frame {B} according to the following sequence
Translate by vector 𝐴 𝑄1 to form frame {B’}
Rotate about 𝑍 𝐵′ by θ degrees to form frame {B’’}
Translate by another vector 𝐵′′ 𝑄2 to from frame {B’’’}
Rotate about about 𝑋 𝐵′′′ by φ degrees to form final frame {B}
What is frame mapping matrices for each step. What is the final matrix 𝐵 𝐴 𝑇 ?

17. Q6 - Answer
A frame {B} is initially coincident with a frame {A}. We transform frame {B} according to the following sequence
Translate by vector 𝐴 𝑄1 to form frame {B’}
Rotate about 𝑍 𝐵′ by θ degrees to form frame {B’’}
Translate by another vector 𝐵 ′′ 𝑄2 to from frame {B’’’}
Rotate about about 𝑋 𝐵′′′ by φ degrees to form final frame {B}
What is frame mapping matrices for each step. What is the final matrix 𝐵 𝐴 𝑇 ?
𝐵 ′ 𝐴 𝑇 = 𝐼 3×3 𝐴 𝑄 ×3 1 , 𝐵 ′′ 𝐵 ′ 𝑇 = 𝑅 𝑍 (𝜃) 0 3× ×3 1 ,
𝐵 ′′′ 𝐵 ′′ 𝑇 = 𝐼 3×3 𝐵 ′′ 𝑄 ×3 1 , 𝐵 𝐵 ′′′ 𝑇 = 𝑅 𝑋 (𝜑) 0 3× ×3 1 ,
𝐵 𝐴 𝑇 = 𝐵 ′ 𝐴 𝑇 ∙ 𝐵 ′′ 𝐵 ′ 𝑇 ∙ 𝐵 ′′′ 𝐵 ′′ 𝑇 ∙ 𝐵 𝐵 ′′′ 𝑇

18. Q7
We have the following frames {U}, {1}, {2}, {3}, {4} with known frame mapping matrices 1 𝑈 𝑇 , 2 1 𝑇 , 3 2 𝑇 , 4 3 𝑇 , how to obtain 3 𝑈 𝑇 and 𝑈 4 𝑇 ?

19. Q7 - Answer
We have the following frames {U}, {1}, {2}, {3}, {4} with known frame mapping matrices 1 𝑈 𝑇 , 2 1 𝑇 , 3 2 𝑇 , 4 3 𝑇 , how to obtain 3 𝑈 𝑇 and 𝑈 4 𝑇 ?
3 𝑈 𝑇 = 1 𝑈 𝑇 ∙ 2 1 𝑇 ∙ 3 2 𝑇
𝑈 4 𝑇 = ( 4 𝑈 𝑇 ) −1 =( 1 𝑈 𝑇 ∙ 2 1 𝑇 ∙ 3 2 𝑇 ∙ 4 3 𝑇 ) −1