Equation of Hyperboloid of One Sheet

Theorem

Let $\HH$ be a hyperboloid of one sheet.

Let $\HH$ be embedded in a cartesian $3$-space such that the conjugate axes of the hyperbolas forming its okane sections coincide with the $z$-axis.

The equation for $\HH$ can be expressed in the form:

$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} - \dfrac {z^2} {c^2} = 1$

where $a, b, c \in \R_{\ne 0}$.

Proof

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Also see

• Equation of Hyperboloid of Two Sheets

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hyperboloid
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hyperboloid