Warning

## Homework 05 - Motion Axis

1. Consider the same motion as in HW-04. Draw figures from previous homework to the pdf report.
2. Find the axis of motion $a_0$ for ($R$,$\vec{o}_{\beta'} = [0; 0; 0]$) and $a_1$ for ($R$,$\vec{o}_{\beta'}=[1; 1; 1]$). (By $\vec{o}_{\beta'}$ we understand $\overrightarrow{O'O}_{\beta'}$).
3. Draw the coordinate systems and motion axes.
4. Find and draw the rotation axis $r$. What is the relationship beween $r$, $a_0$, and $a_1$?
5. Find the generators of the plane $\sigma$ which is perpendicular to the rotation axis $r$ and draw them into the figure.
6. Consider next only motion defined by ($R$, $\vec{o}_{\beta'}=[1; 1; 1]$) and the corresponding $a_1$.
7. What is the relationship between the generators of $\sigma$ and the matrix $R-I$?
8. Find the point $P$ where the motion axis $a_1$ intersects the plane $\sigma$, and draw it.
9. Find and draw the point $P'$, which is obtained by rotating $P$ by rotation $R$.
10. Find and draw the point $P''$, which is obtained by translating $P'$ along $\vec{o}'_{\beta}$.
11. What is the relationship between $P$, $P'$, $P''$ and the axis $a_1$?
12. What is the relationship between the rotation axis $r$ and the motion axis $a_1$ when:
1. $R = I$
2. $\vec{o}'_{\beta} = \vec{0}$
3. $\vec{o}'_{\beta}$ is an eigenvector of $R$

Create an empty dictionary in Python:

solution = {}

The keys for this dictionary will be:

• “a_0_dir” (3×1) and “a_0_point” (3×1) containing the description of the motion axis a_0: a_0 = x * a_0_dir + a_0_point.
• “a_1_dir” (3×1) and “a_1_point” (3×1) containing the description of the motion axis a_1: a_1 = x * a_1_dir + a_1_point.
• “sigma1” (3×1) and “sigma2” (3×1) containing the generators of the plane $\sigma$.
• “p” (3×1), “p_pr” (3×1) and “p_pr_pr” (3×1) containing the points $P$, $P'$ and $P''$.

Finally, save solution to hw05.json:

import json
with open("hw05.json", "w") as outfile:
json.dump(solution, outfile)
Upload a zip archive hw05.zip (via the course ware) containing the following files:

1. hw05.py - python script used for computation
2. hw05.pdf - report file describing your solution with all the figures and explanatory text. Please mark the discussion of 4, 7, 11, 12 so it is easy to find
3. hw05.json