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I am supposed to write the hyperboloid [tex]x^2 + y^2 - z^2=1[/tex] as a parametric funktion and find an expression for the tangent plane in an arbitary point in terms of the parameters.

I think I have figured out that the parametric funktion is

[tex]

\left\lbrace\begin{array}{ccl}

x &=& \sqrt{1+t^2}\cos\varphi \\

y &=& \sqrt{1+t^2}\sin\varphi \\

z &=& t

\end{array}\right.

[/tex]

And if [tex]z = f(x,y)[/tex], the tangent plane for the point [tex](x_0, y_0, f(x_0, y_0))[/tex] is given by

[tex]

f_t(x,y) = f(x_0, y_0) + \frac{\partial f(x_0, y_0)}{\partial x}(x-x_0) + \frac{\partial f(x_0, y_0)}{\partial y}(y - y_0)

[/tex]

which in this case is evaluates to

[tex]

f_t(x,y) = \sqrt{x_0^2 + y_0^2 - 1} + \left(\frac{x_0}{\sqrt{x_0^2+y_0^2-1}}\right)(x-x_0) + \left(\frac{y_0}{\sqrt{x_0^2+y_0^2-1}}\right)(y - y_0)

[/tex]

Hope I'm right so far...

How am I supposed to express [tex]f_t(x,y)[/tex] in terms of the parameters ([tex]t, \varphi[/tex])?

Thanks in advance!

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# Tangent Plane

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