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Math Help - Line Intersection Point on Ellipsoid

  1. #1
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    Line Intersection Point on Ellipsoid

    Hi all,

    Hopefully someone might be able to give me some insight into this problem.

    I'am trying to find the point in 3D space where a line intersects with an ellipsoid. Shown below



    Now i have points A (xyz), B(x1,y1,z1) and C (x2,y2,z2) already.

    The ellipsoid is a scaled sphere with a known radius, i,e 10mm for each axis and scaled by say 1.1, 1.6, 2 for the given xyz coordinates.

    I just wanted to know whether it was actually possible to calculate the position of Tau i.e its actual 3D coordinates.

    Any help is appreciated.

    Regards Wolfe
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by cdrwolfe View Post
    Hi all,

    Hopefully someone might be able to give me some insight into this problem.

    I'am trying to find the point in 3D space where a line intersects with an ellipsoid. Shown below



    Now i have points A (xyz), B(x1,y1,z1) and C (x2,y2,z2) already.

    The ellipsoid is a scaled sphere with a known radius, i,e 10mm for each axis and scaled by say 1.1, 1.6, 2 for the given xyz coordinates.

    I just wanted to know whether it was actually possible to calculate the position of Tau i.e its actual 3D coordinates.

    Any help is appreciated.

    Regards Wolfe
    Your ellipsoid is of the form:

    \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=\r  ho^2

    The line segment BC is of the form:

    (u,v,w)=\lambda (x1,y1,z1)+ (1-\lambda)(x2,y2,z2), \ \lambda \in [0,1]

    So we are looking for \lambda such that:

    \frac{ (\lambda x_1+(1-\lambda)x_2)^2}{a^2}+\frac{(\lambda y_1+(1-\lambda)y_2)^2}{b^2}+\frac{(\lambda z_1+(1-\lambda)z_2)^2}{c^2}=\rho^2

    Expand and simplify this to get a quadratic in \lambda and solve, the root in [0,1] will give the point required.

    CB
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  3. #3
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    Jun 2007
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    Great Thanks!

    Looks complicated . It will probably take me a month to catch up on the maths required to solve it lol.

    Better get cracking.

    Regards Wolfe
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