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Thread: Parametrization of Hyperboloid

  1. #1
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    Parametrization of Hyperboloid

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

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

    is that the correct parametrization?
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  2. #2
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    Let $\displaystyle x = rcos(\theta) $ and $\displaystyle 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 $\displaystyle xy$ plane and makes $\displaystyle \theta$ degress starting from x axis. In this plane, we will make each point $\displaystyle P = ( Pu, Pz )$ rotate. But $\displaystyle Pu $ and $\displaystyle Pz $ will have a parametrization too thus this will nicely describe the hyperboloid. Lets see:

    Parametrization of Hyperboloid-parametr.gif

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

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

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

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

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

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

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

    To find a vector perpendicular to the surface $\displaystyle x^2+ y^2- z^2= 25$ think of that as a "level surface" of $\displaystyle F(x,y,z)= x^2+ y^2- z^2$ 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 $\displaystyle A\vec{i}+ B\vec{j}+ C\vec{k}$ at point $\displaystyle (x_0,y_0,z_0)$ in the plane, then the plane is given by $\displaystyle A(x- x_0)+ B(y- y_0)+ C(z- z_0)= 0$.
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