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Math Help - How to Determine a Hyperbola From Points

  1. #1
    Junior Member masoug's Avatar
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    Question How to Determine a Hyperbola From Points

    Hi,
    Is there a way to determine a hyperbola (equation) from a few points (three maybe)?

    I thought that there is only one way, which is by graphing and finding the vertex, transverse axes, etc... but I hope there are other ways to determine and equation.

    Thanks!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by masoug View Post
    Hi,
    Is there a way to determine a hyperbola (equation) from a few points (three maybe)?

    I thought that there is only one way, which is by graphing and finding the vertex, transverse axes, etc... but I hope there are other ways to determine and equation.

    Thanks!
    If you have the general equation of a hyperbola a set of points defines a set of simultaneous equations in the parameters which can then be solved.

    CB
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  3. #3
    Junior Member masoug's Avatar
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    Question

    Quote Originally Posted by CaptainBlack View Post
    If you have the general equation of a hyperbola a set of points defines a set of simultaneous equations in the parameters which can then be solved.

    CB
    Um, can you explain what you mean by that?
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  4. #4
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    If you know that a hyperbolas axes are parallel to the x and y axes, then a hyperbola can be written as either
    \frac{(x- x_0)^2}{a^2}- \frac{y- y_0)^2}{b^2}= 1
    or
    \frac{(y- y_0)^2}{b^2}- \frac{(x- x_0)^2}{a^2}=

    That depends on 4 parameters, x_0, y_0, a, and b. If you know 4 points you can put the x,y values of those points into the equation, getting 4 equations to solve for the parameters.

    If it is possible that the axes of the hyperbola are tilted with respect to the x and y axes the problem becomes much more complicated. However, any conic section (hyperbola, parabola, ellipse, circle and some special cases) can be written in the form Ax^2+ Bxy+ Cy^2+ Dx+ Ey+ F= 0. That has 6 parameters but we could always divide the entire equation by one of the so there are 5 independent parameters. That means that 5 points are sufficient to determine any conic section.
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  5. #5
    Junior Member masoug's Avatar
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    Smile Aha!

    Quote Originally Posted by HallsofIvy View Post
    If it is possible that the axes of the hyperbola are tilted with respect to the x and y axes the problem becomes much more complicated. However, any conic section (hyperbola, parabola, ellipse, circle and some special cases) can be written in the form Ax^2+ Bxy+ Cy^2+ Dx+ Ey+ F= 0. That has 6 parameters but we could always divide the entire equation by one of the so there are 5 independent parameters. That means that 5 points are sufficient to determine any conic section.
    I see...
    So for most conic sections, just use the Ax^2+ Bxy+ Cy^2+ Dx+ Ey+ F= 0 relation and pick six points to satisfy each unknown.

    Thanks!
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  6. #6
    MHF Contributor

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    For all conic sections and 5 points will be sufficient.
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