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Math Help - Parametric points along an ellipse equation

  1. #1
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    Parametric points along an ellipse equation

    I am attempting to parametrically layout points along an ellipse in 3D space by dividing a quadrant into equally spaced radial lines from an ellipse center. I have reduced the problem to this formula:
    Code:
    (x^2/b^2) + ((mx)^2/h^2) - 1 = 0
    'b' = the ellipse semi-major axis, 'h' = the semi-minor axis, and 'm' = the tangent of the angle of the radial line. Can someone show me how to reduce this formula to be in terms of 'x'? Once this is solved, I believe I can translate the point locations into 3D space. Thanks.
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  2. #2
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    Quote Originally Posted by bvdet View Post
    I am attempting to parametrically layout points along an ellipse in 3D space by dividing a quadrant into equally spaced radial lines from an ellipse center. I have reduced the problem to this formula:
    Code:
    (x^2/b^2) + ((mx)^2/h^2) - 1 = 0
    'b' = the ellipse semi-major axis, 'h' = the semi-minor axis, and 'm' = the tangent of the angle of the radial line. Can someone show me how to reduce this formula to be in terms of 'x'? Once this is solved, I believe I can translate the point locations into 3D space. Thanks.
    I don't know what you're doing, but can't you just solve for x?

    \frac{x^2}{b^2}+\frac{(mx)^2}{h^2}-1=0

    Thus: \frac{x^2}{b^2}+\frac{m^2x^2}{h^2}=1

    Therefore: h^2x^2+b^2m^2x^2=b^2h^2

    Then: \left(h^2+b^2m^2\right)x^2=b^2h^2

    Divide: x^2=\frac{b^2h^2}{h^2+b^2m^2}

    Finally: x=\pm \sqrt{\frac{b^2h^2}{h^2+b^2m^2}}
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  3. #3
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    Quote Originally Posted by Quick View Post
    I don't know what you're doing, but can't you just solve for x?

    \frac{x^2}{b^2}+\frac{(mx)^2}{h^2}-1=0

    Thus: \frac{x^2}{b^2}+\frac{m^2x^2}{h^2}=1

    Therefore: h^2x^2+b^2m^2x^2=b^2h^2

    Then: \left(h^2+b^2m^2\right)x^2=b^2h^2

    Divide: x^2=\frac{b^2h^2}{h^2+b^2m^2}

    Finally: x=\pm \sqrt{\frac{b^2h^2}{h^2+b^2m^2}}
    Thank you! My algebra skills are rusty. The points I am laying out on the ellipse will be used as reference points for structural steel elements in buildings. Typically ellipses are approximated by a series of arcs. For each three reference points I can determine the center point and radius in 3D space for each element. I will be using a Python script in SDS/2 by Design Data.
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  4. #4
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    Below are images of the finished project. Image 1 shows a view of the ellipse generated in 3D space aligned with a structural steel beam. Image 2 is a screenshot in the plane of the ellipse. I created 3 point construction circles parametrically to approximate the ellipse.
    Attached Thumbnails Attached Thumbnails Parametric points along an ellipse equation-ellipselayout1.gif   Parametric points along an ellipse equation-ellipselayout2.gif  
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