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Classification of Quadric Surfaces

  • Thomas Banchoff
  • John Wermer
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

A quadric surface is the 3-dimensional generalization of a conic section. Such a surface is determined by an equation in the variables x, y, z so that each term is of second degree; for example,
$$ {x^2}\;{\rm{ + }}\;{\rm{2}}xy\;{\rm{ + }}\;3{z^2}\; = \;{\rm{1}}{\rm{.}} $$
The general form of the equation of a quadric surface is
$$ a{x^2}\;{\rm{ + }}\;2bxy\;{\rm{ + }}\;2cxz\;{\rm{ + }}\;d{y^2}\;{\rm{ + }}\;2eyz\;{\rm{ + }}\;f{z^2}\;{\rm{ = }}\;{\rm{1,}} $$
(1)
where the coefficients a, b, c, d, e, and f are constants. We would like to predict the shape of the quadric surface in terms of the coefficients, much in the same way that we described a conic section in terms of the coefficients of an equation
$$ a{x^2}\;{\rm{ + }}\;2bxy\;{\rm{ + }}\;c{y^2}\;{\rm{ = }}\;{\rm{1}} $$
in two variables.

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Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • Thomas Banchoff
    • 1
  • John Wermer
    • 1
  1. 1.Department of MathematicsBrown UniversityProvidenceUSA

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