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Math Help - Parametrization of Hyperboloid

  1. #1
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    Parametrization of Hyperboloid

    a) Find a parametrization of the hyperboloid x^2 + y^2 - z^2 = 25
    b) Find an expression for a unit normal to this surface.
    c) Find an equation for the plane tangent to the surface at (x_0,y_0,0), where x_0^2 + y_0^2 = 25

    For the parametrization I got \Phi(r,\theta) = (rcos\theta, rsin\theta, \sqrt{r^2 - 25})

    is that the correct parametrization?
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  2. #2
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    Let  x = rcos(\theta) and  y = rsin(\theta) .
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  3. #3
    Junior Member bondesan's Avatar
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    Think about an auxilliary plane uz, for example, which is perpendicular to the xy plane and makes \theta degress starting from x axis. In this plane, we will make each point P = ( Pu, Pz ) rotate. But Pu and Pz will have a parametrization too thus this will nicely describe the hyperboloid. Lets see:

    Parametrization of Hyperboloid-parametr.gif

    In the uz plane, you parametrize the hyperbola with Pu  = cosh (s) and Pz = sinh (s) with -1\leq s \leq 1. Notice that this parameter s will describe the contour of the hyperbola.

    Now you should use \theta as another parameter, an this would be the "revolution parameter". It will take each contour of hyperbola and rotate it \theta degrees.

    So, in the xy plane, we should have x = 5Pu\cos\theta and y=5Pu\sin\theta with 0\leq \theta \leq 2\pi. If we make z = 0 in x^2 + y^2 - z^2 = 25, we would end with x^2 + y^2 = 25, which requires the r = 5 in the parametrization x = r\cos\theta and y = r\sin\theta.

    Putting all together, we must have: \Phi(s,\theta) = ( Pu\cos\theta, Pu\sin\theta, Pz) = ( 5\cosh(s)\cos\theta, 5\cosh(s)\sin\theta, 5sinh (s) )

    For the B) you want the expression of the normal vector?
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  4. #4
    Junior Member bondesan's Avatar
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    Only to ellucidate, here's the hyperboloid. We can see the mesh caused by this type of parametrization - there are circunferences parallel to the xy plane, in which we have some z coordinate fixed, and there are hyperbolas, when we have either x or y fixed.

    Parametrization of Hyperboloid-parametr2.gif

    For the B, I guess you should use the definition of gradient. I'll work on this later. I hope it helps!
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  5. #5
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    Quote Originally Posted by larryboi7 View Post
    a) Find a parametrization of the hyperboloid x^2 + y^2 - z^2 = 25
    b) Find an expression for a unit normal to this surface.
    c) Find an equation for the plane tangent to the surface at (x_0,y_0,0), where x_0^2 + y_0^2 = 25

    For the parametrization I got \Phi(r,\theta) = (rcos\theta, rsin\theta, \sqrt{r^2 - 25})

    is that the correct parametrization?
    Yes, if x= r cos(\theta), y= r sin(\theta), and z= \sqrt{r^2- 25}, then x^2+ y^2- z^2= r^2 sin^2(\theta)+ r^2 cos^2(\theta)- (r^2- 25)= r^- (r^2- 25)= 25
    (Although there is no such thing as "the" parameterization. A surface has many different parameterizations. Since your x, y, and z satisfy the equation x^2+ y^2- z^2= 25, it is a valid parameterization.)

    To find a vector perpendicular to the surface x^2+ y^2- z^2= 25 think of that as a "level surface" of F(x,y,z)= x^2+ y^2- z^2 and use the fact that grad F is always perpendicular to level surfaces of F.

    And, once you have a perpendicular, use the fact that if the vector A\vec{i}+ B\vec{j}+ C\vec{k} at point (x_0,y_0,z_0) in the plane, then the plane is given by A(x- x_0)+ B(y- y_0)+ C(z- z_0)= 0.
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