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Math Help - Define ellipse given 3 of its points

  1. #1
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    Define ellipse given 3 of its points

    Long story short, I have a camera sensor that detects 3 points a reports them as x,y pairs. All x and y values are positive.

    It's been a long time since I've done geometry and algebra and I'm struggling with getting started. Maybe someone can help me.

    I know the points are on an ellipse, and I know the major axis is vertical. I don't know the foci, but don't need them in the end.
    Ultimately, I need to find the center of the ellipse.

    Can this be done with algebra?
    Thanks for any help!
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  2. #2
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    Quote Originally Posted by ionymous View Post
    Long story short, I have a camera sensor that detects 3 points a reports them as x,y pairs. All x and y values are positive.

    It's been a long time since I've done geometry and algebra and I'm struggling with getting started. Maybe someone can help me.

    I know the points are on an ellipse, and I know the major axis is vertical. I don't know the foci, but don't need them in the end.
    Ultimately, I need to find the center of the ellipse.

    Can this be done with algebra?
    Thanks for any help!
    Three points are not enough to determine the ellipse. You need either a fourth point or some additional information such as the eccentricity of the ellipse. There will be a whole family of ellipses, with major axis vertical, through three points, one for each value of the eccentricity from 0 to 1. One of the limiting cases is eccentricity 0, when you get a circle (it is always possible to find a circle through three points, unless they happen to be collinear). The other limiting case, eccentricity 1, is when the ellipse becomes a parabola.

    The position of the centre of the ellipse will depend on the eccentricity, so you can't pin that down either, without some extra information.
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  3. #3
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    Thank you very much for your response!
    I am struggling a bit with some of the terminology. I've googled about eccentricity and now have a very basic initial understanding.

    But I'm basically struggling with the following...
    I have convinced myself that if I draw 3 points on a paper, I can draw different ellipses that intersect these points. But I can only ever come up with a single ellipse that has a major vertical axis. That makes me think there must be a mathematical way to determine this ellipse from those 3 points.

    What am I missing?
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  4. #4
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    Here's a simple illustration of three ellipses and a circle, all going through the same three points. I have made the major axes horizontal for convenience, but you can obviously rotate the picture to make them all vertical.
    Attached Thumbnails Attached Thumbnails Define ellipse given 3 of its points-ellipse.gif  
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  5. #5
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    I see it now! Thanks!
    But I still think there might be an equation I can use.
    Looking at the picture made me think... I don't actually want the center point of the ellipse, I just want the X position of the major vertical axis.
    In the picture, every ellipse has the same X position for the major vertical axis, but is that just because of where the 3 points were chosen?
    What if the three points were in less symetrical locations?... if that makes sense
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