~~NOTOC~~

===== Homework 05 - Motion Axis =====

=== Task ===

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

=== Upload ===

Create an empty dictionary in Python:

<code>
solution = {}
</code>

The keys for this dictionary will be:
  * ''"a_0_dir"'' (3x1) and ''"a_0_point"'' (3x1) containing the description of the motion axis a_0:  a_0 = x * a_0_dir + a_0_point. 
  * ''"a_1_dir"'' (3x1) and ''"a_1_point"'' (3x1) containing the description of the motion axis a_1:  a_1 = x * a_1_dir + a_1_point. 
  * ''"sigma1"'' (3x1) and ''"sigma2"'' (3x1) containing the generators of the plane $\sigma$. 
  * ''"p"'' (3x1), ''"p_pr"'' (3x1) and ''"p_pr_pr"'' (3x1) containing the points $P$, $P'$ and $P''$. 

Finally, save ''solution'' to ''hw05.json'':

<code>
import json
with open("hw05.json", "w") as outfile:
    json.dump(solution, outfile)
</code>
Upload a zip archive ''hw05.zip'' (via the [[https://cw.felk.cvut.cz/upload/|course ware]]) containing the following files:
  - ''hw05.py'' - python script used for computation
  - ''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
  - ''hw05.json''