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Help: Perpendicular project is regular surface?

  1. Apr 21, 2009 #1
    Help: Perpendicular project is regular surface??

    (edit found a better way of showing this)

    1. The problem statement, all variables and given/known data

    Hi

    I have this problem here which is causing me trouble.

    Show that the perpendicular projection of the center (0,0,0) of the ellipsoid
    \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1

    onto its tangent planes constitutes a regular surface given by

    {(x,y,z) \in R^3; (x^2+y^2+z^2)/2 = a^2x^2 + b^2y^2 + c^2z^2}-{(0,0,0)}

    What can do here is arrive at tangent plane


    3. The attempt at a solution

    First I find the tangent

    gradF(x0,y0,z0) = <2x0/a^2, 2y0/b^2, 2z0/c^2>

    Which gives us a tangentplane

    2x0/a^2*(x-x0)+2y0/b^2(y-y0)+2z0/c^2(z-z0) = 0

    which by rearrangement gives

    xx0/a^2 + yy0/b^2 + zz0/c^2 = 1


    the normal is x0/a^2*x = y0/b^2*y = z0/c^2*z


    But how do I continue from here???


    Best Regards
    Alphaboy

    2. Relevant equations
     
    Last edited: Apr 21, 2009
  2. jcsd
  3. Apr 21, 2009 #2
    Re: Help: Perpendicular project is regular surface??

    You have a normal n=<x0/a^2, y0/b^2, z0/c^2>.

    Can you consider tn and determine what value of t makes tn lie in the plane? Isn't that the projection of the origin into the plane?

    Then verify that this point satisfies your desired equation:

    which unfortunately I tried three times, getting two different results, neither of which was exactly this. Are you sure it's divided by 2?
     
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