CALCULUS Understanding Its Concepts and Methods

Quadric surfaces

A quadric surface is the set of all points
(*x*,*y*,*z*)
that satisfy an equation of the
form

*Ax*^{2} + *By*^{2} + *Cz*^{2} +
*Dxy* + *Exz* + *Fyz* +
*Gx* + *Hy* + *Iz* + *J *= 0

In order to sketch the graph of a quadric surface (or any surface), it is useful to look at the curves that intersect the surface with planes parallel to the axes of the surface. These curves are called traces (or cross-sections) of the surface.

The following equations and plots are in standard form and symmetric about the
*z*-axis.
Thus the traces are the intersections with the
*xy*-planes,
*xz*-planes,
and
*yz*-planes

A quadric surface with equation
is an *ellipsoid*. Its traces are ellipses.

Quadric surfaces with equation are hyperbolic paraboloids. The traces are parabolas and hyperbolas.

Quadric surfaces with equationare elliptic paraboloids. The traces are parabolas and ellipses.

Quadric surfaces with equations
are
hyperboloids of one sheet. The traces in the
*xz*-planes
and
*yz*-planes
are hyperbolas, and the trace in any horizontal plane is an ellipse.

Quadric surfaces with equations
are
hyperboloids of two sheets. The traces in the
*xz*-planes
and
*yz*-planes
are hyperbolas, and the trace in any horizontal plane is an ellipse or misses
the surface completely.

Quadric surfaces of the form
are
elliptic cones. The traces in the
*xz*-planes
and
*yz*-planes
are an intersecting pair of straight lines, and the trace in any horizontal
plane is an ellipse.

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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.