The Quadric Surfaces: three squares yield 6 curves

This page describes the Quadric Surfaces, there are 6.

The surfaces defined by the set of equations generated from the meta equation:

      (+/-) x^2 (+/-) y^2 (+/-) z^[1,2] = c

These are "nice" surfaces because they are easy to create, easy for the programmer and processor alike.

The sphere, an ellipsoid.

The paraboloid.

The cone and 2 hyperboloids

The hyperbolic paraboloid

The cylinder (not a quadric, but related)

Quadric Surface	Equation
sphere	x^2 + y^2 + z^2 =   1
cone	x^2 + y^2 - z^2 =   0
hyperboloid of one sheet	x^2 + y^2 - z^2 =   1
hyperboloid of two sheets	x^2 + y^2 - z^2 = -1
paraboloid	x^2 + y^2 - z     =   0
hyperboloic paraboloid	x^2 - y^2 - z     =   0
cylinder	x^2 + y^2         =   1

Note: the cylinder is not a quadric

Since we're on the topic of the Quadric Surfaces, we may wish to recall our old friend the Quadradic Equation.

            -b (+/-) square_root(  b^2 - 4 * a * c )
t  =     -------------------------------------------------
                               2 * a

The quadradic equation can be used to find the roots of a second degree equation such as

      y = a*x^2 + b*x + c

The paraboloid can be defined by the area swept out by rotating a parabola.

Conversely, a parabola can be defined by the intersection of a plane and a paraboloid.

Which might come in handy...

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Last modified: 01/10/97 Revision #1 First Created: 11/06/95

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