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Eelipsoid and line intersection

  1. Jan 18, 2009 #1
    The problem statement, all variables and given/known data

    I have an ellipsoid with center (000). There is a point A inside the ellipsoid with known coordinates(1,2,3) I draw a line from center to point A and extend it to cut the ellipsoid on on point p(x,y,z).

    2. Relevant equations

    I want to find the coordinates of point P(x,y,z)

    3. The attempt at a solution

    The equation of ellipsoid for p is x^2/a^2+y^2/b^2+z^2/c^2=1
    i have the values of a,b and c
    i want to know if the ellipsoid equation is applicable to coordinates of A and coordinates of P

    and how can i create equation using the coordinates of p with the coordinates of A
    by equation of line method as both the points lie on a straight line with one end on (000) as the third point.
  2. jcsd
  3. Jan 18, 2009 #2


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    Express the line through (0,0,0) and (1,2,3) in parametric form, i.e. x=t, y=2t, z=3t. Put that into the ellipsoid equation. Solve for t.
  4. Jan 21, 2009 #3
    Thank you very much . I tried that method but the problem I am facing is say my point A (1,2,3) is inside the ellipsoid and center(000) and the point on the ellipsoid i solve using
    x^2/a2+y^2/b2+Z^2/c^2=1 by substituting x=t,y=2t,z=3t and solving for t. But the distance between center and point (x,y,z) should be equal to the sum of the dist between center and A and A and point(x,y,z) . that is not matching .i am using the formula for dist between two pints say(x1,y1,z1) and (x2,y2,z2) as sqrt(x2-x1)^2+(y2-y1)^2+(z2-z1)^2.
    can you tell me where i am going wrong
  5. Jan 21, 2009 #4


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    What did you use for a, b and c, what did you get for t and hence for (x,y,z)? Of course you should find that the distance from O to A plus the distance from A to (x,y,z) should equal the distance from O to (x,y,z). But it's impossible to say what you are doing wrong until you tell us what you did.
  6. Jan 21, 2009 #5
    Thank you for your responce.
    I used a=1
    for coordinates of A(1,2,3) and center(0,0,0)

    now my equaation becomes for point p(x,y,z) on ellipsoid

    put in ellipsoid equation



    now pa=sqrt[(x-1)^2+(y-2)^2+(z-3)^2]=1.603
    point A and center(000)=3.74
    whereas my p to center is 3.30
    i cannoot understand where i am going wrong
  7. Jan 21, 2009 #6


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    I don't think your distance from p to the center is right. But more importantly, if you choose a=1, b=2, and c=3 then your selected point A=(1,2,3) is OUTSIDE of the ellipsoid. I thought you were going to put it inside.
  8. Jan 21, 2009 #7
    i need the point A to be inside . I guess I should have given the arbitraty values for the axis as a=3 b=2 and c=1 . I hope a,b,c are the semimajor axis,semiminor axis and the z axis repectively in the ellipsoid equation . and the coordinates of A(1,2,3) are separate.
    the point on the ellipsoid will have x^2/a^2+y^2/b^2+z^2/3^3=1.where the x=t,y=2t,z=3t as the line touching the point on ellipsoid pwill pass through the center(000) and A(1,2,3) located inside the ellipsoid.
    I tried that too but still it is not matching.
  9. Jan 21, 2009 #8


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    Sure, a, b and c are the axes of the ellipse. But if you want A(1,2,3) to be inside the ellipse, you need to make a, b and c larger than 1, 2 and 3.
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