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This graph illustrates the transition from a hyperboloid of one sheet to a hyperboloid of two sheets. Consider the equation \[x^2 + y^2 - z^2 = C\] In case if \(C > 0\), the level curves \[x^2 + y^2 = C + k^2\] are circles at any level \(z = k\) Therefore, the surface continues from negative \(z\) to positive \(z\).
On the other hand, if \(C = -|C| < 0\), then the level curves \[x^2 + y^2 = -|C| + k^2\] exist only when \(z = k \ge \sqrt{|C|}\) or \(z = k \le -\sqrt{|C|}\). Therefore, the surface consists of an upper and a lower piece.
The transition between a hyperboloid of one sheet and a hyperboloid of two sheets can be illustrated by varying \(C\).
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Joseph Lo