Definition:Paraboloid/Hyperbolic

Definition

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Definition 1

Let $\PP$ be a paraboloid.

Let $P_1$ and $P_2$ be two plane sections of $\PP$ such that both $P_1$ and $P_2$ are parabolas.

Let $P_3$ be a plane section of $\PP$ perpendicular to both $P_1$ and $P_2$.

Then $\PP$ is an hyperbolic paraboloid if and only if $P_3$ is a hyperbola.

Definition 2

A hyperbolic paraboloid is a paraboloid which can be embedded in a Cartesian $3$-space and described by the equation:

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

Also see

• Equivalence of Definitions of Hyperbolic Paraboloid

• Results about hyperbolic paraboloids can be found here.

Sources

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