estimate_laser_plane.md

 # Estimation of the Laser Plane position
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-We will use pinhole camera model for finding 3D points ![alt text](http://mathurl.com/hj835tq.png) in projection coordinate ![alt text](http://mathurl.com/hw2kh9a.png) described as:
+We will use pinhole camera model for finding 3D points in camera coordinate system ![alt text](http://mathurl.com/hj835tq.png) in projection coordinate ![alt text](http://mathurl.com/hw2kh9a.png) described as:
 
 <img src="http://mathurl.com/hbtk24o.png" /> ,
 
@@ -12,4 +12,29 @@ Picture is borrowed from [there](http://docs.opencv.org/2.4/modules/calib3d/doc/
 
 
 ### Finding points on laser plane
-The equation for pinhole camera is under-determined so we have to find one more condition to estimate the position ![alt text](http://mathurl.com/hj835tq.png) using only projection coordinates ![alt text](http://mathurl.com/hw2kh9a.png).
+The equation for pinhole camera is under-determined so we have to find one more condition to estimate the position ![alt text](http://mathurl.com/hj835tq.png) using only projection coordinates ![alt text](http://mathurl.com/hw2kh9a.png). We can use chessboard plane, because OpenCV can find the transformation between those coordinates and camera coordinates. Furthermore, we can say that in chessboard coordinates all points are laying on plane described as z = 0. We are able to transform this plane to camera coordinates like:
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+<img src="http://mathurl.com/gw8dwdp.png" /> ,
+<img src="http://mathurl.com/zwt5qhc.png" />.
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+Now we have to obtain normal of this plane and its translation:
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+<img src="http://mathurl.com/h23dwzm.png" /> ,
+<img src="http://mathurl.com/hxa6ykl.png" /> ,
+<img src="http://mathurl.com/j7y8cdf.png" /> , 
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+so the <img src="http://mathurl.com/j865cst.png" />.
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+Now we can add this condition to pinhole camera model equation, where [**R**|t] matrix is identity matrix and <img src="http://mathurl.com/j7ltgkr.png" />, we end up with:
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+<img src="http://mathurl.com/jzqcmgq.png" />
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