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===== Homework 03 - Rigid motion as a coordinate transformation =====
=== Motivation ===
//In this task, we will study the effect of motion (rotation and translation) on a //rigid body//. 

To study this, we construct a very simple rigid body r1 consisting of only two points r1={O, X}. To describe the relative position of these two points, we introduce a coordinate system  (O, β) with the origin at one of these points - O and standard basis β. The second point X can thus be expressed as a vector X<sub>β</sub> in this coordinate system. X<sub>β</sub> = [1,2,3] in (O, β).

Now we apply the given motion on this rigid body represented by the coordinate system. That will result in a new coordinate system (O', β'). We construct a new rigid body r2 ={O', Y} where Y has the same relative pose as X in r1. Y<sub>β'</sub> = [1,2,3] in (O', β').

Finally, we apply the motion on point X denoting it as Z.  

What are the coordinates of point Z?//

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=== Task ===

Use MATLAB to solve the following problems related to rigid motion. Use different colors to display your results.

  - Simulate the rigid motion with matrix R and translation o_{\beta'} prescribed by Equation 5.4 in {{http://cmp.felk.cvut.cz/cmp/courses/PRO/Lecture/PRO-Lecture.pdf|PRO-Lecture.pdf}}. 

  % approximate rotation
  R = [0.8047   -0.5059   -0.3106
       0.3106    0.8047   -0.5059
       0.5059    0.3106    0.8047];

  % less approximate rotation
  [U,D,V] = svd(R);
  R = U*V';

  % translation o_β'
  o = [1;1;1];

  - Basis β equals the standard basis σ. O=[0; 0; 0] 
  - Find the coordinates of vectors of β' in β and vice versa.
  - Plot vectors of β and β' in the standard basis, list the numeric values.
  - Plot coordinate systems (O, β) and (O', β'). i.e. plot the basic vectors as bound vectors originating from points O and  O', respectively, list the numeric values.
  - Plot the bound vector X<sub>β</sub> = [1;2;3] representing point X in (0, β), list the numeric values.
  - Plot the position vector in (O', β') of point Y represented in (O', β') by vector Y<sub>β'</sub> = [1;2;3], list the numeric values.
  - Consider point Z, where X moves by the motion given above. Plot the bound vector representing the point Z w.r.t. (O, β), list the numeric values.
=== Upload ===

Upload  via the [[https://cw.felk.cvut.cz/upload/|course ware]] the zip archive ''hw03.zip'' containing 
  - hw03.pdf report file describing your solution with all figures 
  - hw03.m MATLAB source code, which generates the results and figures for the report
  - all your additional MATLAB files required by hw03.m