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Thread: Question about Normal to an Ellipsoid?

  1. #1
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    Question about Normal to an Ellipsoid?

    Hi.

    I know (from many books) that normal to an ellipsoid of equation (x^2+y^2)/a^2+(z^2)/b^2=1 at any point will passes the minor axis of that ellipsoid. i just want to know how to prove it?

    All Answers are Appreciated /////

  2. #2
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    Quote Originally Posted by Omid82 View Post
    Hi.

    I know (from many books) that normal to an ellipsoid of equation (x^2+y^2)/a^2+(z^2)/b^2=1 at any point will passes the minor axis of that ellipsoid. i just want to know how to prove it?

    All Answers are Appreciated /////
    This question is beyond what I would consider to be "Geometry and Trig." To find the equation of the normal to a surface, I would find the gradient, which involves partial derivatives.

    Surface: [tex] x^2/a^2 + y^2/a^2 + z^2/b^2 - 1 = 0 [/tex]

    Normal vector: [tex]\vec{n} = (2x/a^2)\ \hat{x} + (2y/a^2)\ \hat{y} + (2z/b^2)\ \hat{z} [/tex]

    Let [tex](X, Y, Z)[/tex] be a point on the ellipsoid. The equation of the normal line through that point is

    ........[tex] (2X/a^2)\ (x - X) + (2Y/a^2)\ (y - Y) + (2Z/b^2)\ (z - Z) = 0 [/tex]

    ........[tex] (X/a^2)\ x + (Y/a^2)\ y + (Z/b^2)\ z - \left[ X^2/a^2 + Y^2/a^2 + Z^2/b^2\right] = 0 [/tex]

    ........[tex] (X/a^2)\ x + (Y/a^2)\ y + (Z/b^2)\ z - 1 = 0 [/tex]

    Can you show that that line intersects the z-axis?
    DrPhil (not the TV guy)
    If we knew what we were doing,we wouldn't have to do it

  3. #3
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    Quote Originally Posted by DrPhil View Post
    This question is beyond what I would consider to be "Geometry and Trig." To find the equation of the normal to a surface, I would find the gradient, which involves partial derivatives.

    Surface: [tex] x^2/a^2 + y^2/a^2 + z^2/b^2 - 1 = 0 [/tex]

    Normal vector: [tex]\vec{n} = (2x/a^2)\ \hat{x} + (2y/a^2)\ \hat{y} + (2z/b^2)\ \hat{z} [/tex]

    Let [tex](X, Y, Z)[/tex] be a point on the ellipsoid. The equation of the normal line through that point is

    ........[tex] (2X/a^2)\ (x - X) + (2Y/a^2)\ (y - Y) + (2Z/b^2)\ (z - Z) = 0 [/tex]

    ........[tex] (X/a^2)\ x + (Y/a^2)\ y + (Z/b^2)\ z - \left[ X^2/a^2 + Y^2/a^2 + Z^2/b^2\right] = 0 [/tex]

    ........[tex] (X/a^2)\ x + (Y/a^2)\ y + (Z/b^2)\ z - 1 = 0 [/tex]

    Can you show that that line intersects the z-axis?
    Thank you , but for showing that this line intersects the z axis , i should put x=y=0 ? Thank you again.

  4. #4
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    Quote Originally Posted by Omid82 View Post
    Thank you , but for showing that this line intersects the z axis , i should put x=y=0 ? Thank you again.
    Yes, and then you can see there is a perfectly well-defined solution for z, the point of intersection, as a function of Z, the arbitrary point on the ellipsoid where you constructed the normal. Since the limits of Z are ħb, what are the limits for z? Does that satisfy the theorem that the point of intersection is "on the minor axis" of the ellipsoid?
    DrPhil (not the TV guy)
    If we knew what we were doing,we wouldn't have to do it

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