rotation_table.md

@@ -19,4 +19,27 @@ regulator :
 
 ## Finding the rotation axis of table
 The rotation axis can be found when we put the calibration chessboard on table and collect some images with different rotations. Thus, we can find transformations between these boards and camera and determine the rotation axis of table.
+
+We are looking for axis described as direction vector ![alt text](http://mathurl.com/jm7yc4a.png) and translation vector ![alt text](http://mathurl.com/hrfjerk.png). 
+
+On following image we can see some basic relations between our coordinate systems. All we know are [R|t] transformations between camera coordinates and both board coordinates (green and red). 
+
 ![table_axis](http://gitlab.ciirc.cvut.cz/cmirajak/laser_plane_scanner/uploads/9c2cc52d05549e2ddbb5d11cb7f14b24/table_axis.png)
+
+### Finding transformation between boards
+
+First, we have to find a transformation between both chessboards as:
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+![alt text](http://mathurl.com/j4b8k94.png).
+
+As you can see, we will work in the coordinate system of o2.
+
+To simplify our equation we will use following identities:
+
+![alt text](http://mathurl.com/jpjc2n7.png)
+
+Thus, we end up with:
+
+![alt text](http://mathurl.com/jgluop4.png)
+
+