Definition:Paraboloid

Definition

A paraboloid is a figure whose sections parallel to at least one plane are parabolas.

Elliptical Paraboloid

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 elliptical paraboloid if and only if $P_3$ is an ellipse.

Hyperbolic Paraboloid

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.

Circular Paraboloid

A circular paraboloid is a surface of revolution obtained by revolving a parabola around its axis.

Also see

• Volume of Paraboloid

• Definition:Ellipsoid
• Definition:Hyperboloid

• Results about paraboloids can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): paraboloid
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): conicoid (conoid)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): paraboloid
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conicoid (conoid)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): paraboloid
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): paraboloid