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Math Help - Identify the Surface

  1. #1
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    Identify the Surface

    Identify the surface, which is defined by the following parametric equations:

    x=2cos(u)sin(v), y=2sin(u)sin(v), z=6cos(v)

    How can you tell?
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  2. #2
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    Hello, Ideasman!

    Identify the surface, which is defined by the following parametric equations:

    . . (1) x .= .2Ěcos(u)Ěsin(v)
    . . (2) y .= .2Ěsin(u)Ěsin(v)
    . . (3) z .= .6Ěcos(v)
    We can try to eliminate the parameters and get it in terms of x, y, z.


    Square (1): . x▓ .= .4Ěcos▓(y)Ěsin▓(v)
    Square (2): . y▓ .= .4Ěsin▓(u)Ěsin▓(v)

    Add: . x▓ + y▓ .= .4Ěsin▓(v)
    [sin▓(u) + cos▓(u)]

    . . . . .x▓ + y▓ .= .4Ěsin▓(v) .(4)


    From (3), we have: . cos(v) .= .z/6 . . cos▓(v) .= .z▓/36

    . . Then: .sin▓(v) .= .1 - cos▓(v) .= .1 - z▓/36 .= .(36 - z▓)/36


    Substitute into (4): . x▓ + y▓ .= .4(36 - z▓)/36

    . . which simplifies to: . 9x▓ + 9y▓ + z▓ .= .36

    . . . . . . . . . . . x▓ . .y▓ . . z▓
    And we have: . --- + --- + --- .= .1 . . . . an ellipsoid
    . . . . . . . . . . . .4 . . 4 . . 36

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