The hyperboloid as a level surface
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The picture below shows the level surface \(f^{-1}(k)\) of the function \(f : \mathbb{R}^3 \rightarrow \mathbb{R}\) defined by \(f(x, y, z) = x^2 + y^2 - z^2\).
You can move the slider on the left to change the value of \(k\). Note that at \(k=0\) the surface becomes singular (it is a cone). When \(k > 0\) the surface is connected, and is called the Hyperboloid of one sheet. When \(k < 0\) the surface has two components, and is called the Hyperboloid of two sheets
