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Math Help - Equation of Parabola from two points with additional contrainsts...

  1. #1
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    Equation of Parabola from two points with additional contrainsts...

    Hi, is it possible to find the (best fit) equation of a/any Parabola given *two* points and the following additional constraints:

    i) The Axis is parallel to the y-axis ( vertical )
    ii) The Axis lies between the two points
    iii) The vertex is higher than both points
    iv) The vertex is a maxima, or the Parabola opens downwards.
    v) One point can be the origin.

    This is basically for the path of a projectile under gravity that originates from a start point and must hit a target point, following a parabolic path.

    So for the equation: y = a(x - b)^2 + c, I understand that I need three equations to solve all constants, which normally requires three points.

    Is it possible to find ( or approximate ) the third equation given the above constraints somehow?

    Is there possibly a (simple'ish) iterative technique? ( Unfortunately speed is a factor as this is part of program's run-time algorithm ).

    Many thanks for any help!
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  2. #2
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    What do you mean by axis? Do you mean the directrix?
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Choose b=0.
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    Sorry, I should have said the axis of symmetry. Thanks!
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  5. #5
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    If your two points are your source and target, then there are an infinite number of solutions from the constraints you have given.

    If you just want to find an arbitrary solution.
    Draw the line between your two points. Pick any point in between them that lies above the line segment. Use this as your 3rd constraint.
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  6. #6
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    I see!

    I've just been trying some example calculations, and it seems that I don't get the constants that describe the parabola that I want - multiple parabolas through the same points?
    This might possibly just be the root selection of the equations, I'll try to nut it out and ask for more help if I can't get it.

    Thanks for the speedy replies.
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