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Math Help - Projection, perspective distortion of circle geometry

  1. #1
    Junior Member
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    Arrow Projection, perspective distortion of circle geometry

    Hello all!

    I'd like to determine positions in a physical reality by using measurements in a photo. It's tougher than I thought! Maybe there is some whole different kind of approach for taking care of perspective transpformations out there some where, but I sure as hell can't find anything useful about it. And my geometry/trigonometry knowledges are strained here, so I seek guidence!

    I've made some illustrations in Sketchup free demo. (Side question: Sketchup is very easy to use, but not at all designed for this purpose, does anyone know about other geometry drawing software?)

    Question (refering to text and illustrations below):
    What is "delta Z"? I.e. what is the distance between the center of the circle and the center of the ellipse? Physical distance OR the angle CenterCircle-Camera-CenterEllipse will do.

    Situation:
    -- The physical radius of a circle is known beforehand.

    -- A camera takes a digital photo of the circle from unknown angle and distance.

    -- In the photo, one pixel corresponds to a certain fraction of an angular degree. Hence measurements of pixel distances between any two marked points in the photo gives us the angle between these two points, with vertex at the position of the camera.

    -- The center of the circle is NOT MARKED on the photo, we cannot make direct measurements of angles relating to that point.

    -- In the photo, the circle is depicted as an ellipse due to both the angle of view and perspective distortions. Hence I talk about ELLIPS (in photo) and CIRCLE (out there).

    -- 2*small axis of the ellipse (North to South) corresponds to the diameter of the circle. So we know the side North-South and the angle South-Camera-North in that triangle.

    -- Perspective distortions makes the center of the ellipse offset from the center of the circle by a distance which I call "delta Z" (offset in deapth, Z-dimension). The great axis of the ellipse is on a CHORD of the circle.

    -- We do not know the length of the chord because we do not know its physical position. We can however measure the angle ChordEndOne-Camera-ChordEndTwo and we ofcourse know that the distance between a ChordEnd and the circle center is one radius.

    -- The only two points marked on the circle are points A and B, both on the perimeter of the circle. We know that there is a right angle between them and the center of the circle. We could use points A and B to measure angles to other points, but again, we CANNOT measure angles directly from the center of the circle because it is unmarked (which I think is the whole case of all this headache, and so I want to find the center of the circle by finding "delta Z", its offset from the center of the ellipse).

    Illustrations:
    View from above the center of the circle:


    View from the camera:
    NOTE HOW IT IS NOW AN ELLIPSE AND THAT A CHORD IS ITS MAIN AXIS!


    View from diagonal side:


    Phui, I've tried to explain a complicated geometrical problem. I will sort out any remaining unclarities swiftly!

    Solutions, suggestions, questions... ANYTHING is very welcome!
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  2. #2
    Junior Member
    Joined
    Sep 2006
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    Lightbulb

    I've just found something about prespective distortion of circles inte ellipses. It has to do with two concentric circles, but the concept might be useful in my problem with only one circle, if advantage can be taken from the information about points A and B to create, for example, two chords through A and B respectively, and relate the points where those chords cross the North-South-diameter line???

    Please help me THINK!

    I here cite the (most) relevant section (3.2) out of a paper by Vincent Fremont and Ryad Chellali of Ecole des Mines de Nantes: "Direct Camera Calibration using Two Concentric Circles from a Single View" available at: link to paper.

    [IMG][/IMG]

    [IMG][/IMG]

    PS
    At least my intuition that: "the center of projected concentric circles always lies on a line definied by the ellipses centers under ANY projective transformation"
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