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Math Help - Identify the Parametric Surface

  1. #1
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    Identify the Parametric Surface

    Identify the following parametric surface.

    1.) x = sinh(v), y = cos(u)*cosh(v), z = sin(u)*cosh(v)
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  2. #2
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    Hello, Ideasman!

    You're expected to know: .cosh²(x) - sinh²(x) .= .1


    Identify the following parametric surface.

    . . [1] .x .= .sinh(v)
    . . [2] .y .= .cosh(u)·cosh(v)
    . . [3] .z .= .sinh(u)·cosh(v)
    We will eliminate the parameters . . .

    Square [2]: . .= .cosh²(u)·cosh²(v)
    Square [3]: . .= .sinh²(u)·cosh²(v)

    Subtract: .y² - z² .= .cos²(u)·cos²(v) - sinh²(u)·cosh²(v)
    . . . . . . . . . . . . . = .[cosh²(u) - sin²(u)]·cosh²(v)
    . . . . . . . . . . . . . = .cosh²(v)

    We have: .y² - z² .= .cosh²(v) .= .sinh²(v) + 1 .[4]


    Square [1]: . .= .sinh²(v)

    Substitute into [4]: .y² - z² .= .x² + 1


    We have: . y² - x² - z² .= .1 . . . a hyperboloid of two sheet.

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  3. #3
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    Hi Soroban,

    Thanks for the help. The solution, however, says that its a hyperboloid of ONE sheet. Thus, one of the signs is wrong, but I can't see why. I see how you used the identity from cosh^2(x) - sinh^2(x) = 1...

    Hmm
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  4. #4
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    Quote Originally Posted by Ideasman View Post
    Thanks for the help. The solution, however, says that its a hyperboloid of ONE sheet.
    I believe Soroban is correct. If you multiply through by a negative,

    x^2-y^2+z^2=-1
    Which is the form for two sheets because, two squared variables are positive and the third is negative and there is a negative on the RHS. (Unless he made a mistake arriving at this conclusion).
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