# How to get rotation angles of Image Plane relative to the World Plane?

So we have such situation:

In this illustration, the first quadrilateral is shown on the Image Plane and the second quadrilateral is shown on the World Plane. [1]

In my particular case the Image Plane has 3 quadrilaterals - projections of real world squares, which, as we know, have same size, lying on the same plane, with same rotation relative to the plane they are lying on, and are not situated on same line on plane.

I wonder if we can get rotation angles of Image Plane to World Plane knowing stuff described?

In my case as input I have such data structures: original image (RGB pixels), objects (squares) with angles points in pixels (x,y) on Image Plane.

Problem I am describing is generally known as pose estimation - determining the 3D orientation and position of an object relative to a camera from a 2D view.

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English is not my native language so could you please point me on to my errors. BTW google docs does not see any. –  Ole J Sep 14 '10 at 22:37
I believe this is the same question you asked earlier (22 hours ago), and I think it is asking for exactly the same answer. mathoverflow.net/questions/38627/…. Didn't Joseph O'Rourke's answer about projective geometry and his pointer to a page about projective transform on stackflow help you? stackoverflow.com/questions/169902/projective-transformation –  sleepless in beantown Sep 14 '10 at 23:35
I believe this is not. That question was about 2d transformations - of quadrilateral into quadrilateral. now my question is about 3d projections. WhatI need is angles of one plane relativly to another. –  Ole J Sep 15 '10 at 11:07

I believe this is the same question you asked earlier (22 hours ago), and I think it is asking for exactly the same answer.

You may also attempt to solve it yourself as a mathematical exercise. Using your own senses, you can see that the illustration is a 7x7 grid with three red squares on it it the positions (3,4), (5,2), and (5,5) depending on how you define your coordinate system. If this drawing were on the $xy$ plane and your camera is on the $z$-axis at a height $h$ and pointing at the origin point(0,0,0), with the $x$-axis horizontal and $y$-axis vertical in your projective image, then what would the camera see?

Now think about how the camera views the image looking at it from different positions. You will have to find the six coordinates describing the camera:

its position in space, $x, y, z$

its orientation in space, however you choose to define it

Try to work out the problem. If you have trouble with it, look at stackoverflow again. Or search for more pages about projective transforms, coordinate transforms, 3-d rendering, ray-tracing, or even the basics of OpenGL (open graphics language) which will help you understand the basics of visual raytracing and projective transformations. A lot of the pages will present matrix representations of the coordinate transforms, which may help you if you understand matrices. But if you don't understand matrices, try to solve it with separate linear transformation equations.

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So Joseph O'Rourke answered the same non-question with 3000pts rep yesterday, and gets voted up? And your vote down doesn't even come with a comment about it? Please note that I also commented just below the question about the repetition of this question and provided a different approach asking the questioner to work it out. I thinks it's an appropriate answer to an inappropriate (for mathverflow) question. –  sleepless in beantown Sep 14 '10 at 23:56
I believe this is not. That question was about 2d transformations - of quadrilateral into quadrilateral. now my question is about 3d projections. What I need is angles of one plane relativly to another. Not I downvoted - my rep is 2 low. problem I am describing is generally known as pose estimation - determining the 3D orientation and position of an object relative to a camera from a 2D view. –  Ole J Sep 15 '10 at 12:05
@Ole J: you may attempt to do pose estimation using the heuristics I provide in this answer. Set up a camera location, set up an oriented rectangle to view through form the camera location, and calculate the viewn image. The location of the camera and the orientation of the camera together form the pose which you are looking for. Calculating that is not a research level mathematical problem. It is a problem which has been solved in many ways. Follow the pointers to the solutions; or, alternately. follow the heuristics to calculate the answer yourself. Knowledge is worth doing the work. –  sleepless in beantown Sep 16 '10 at 2:51