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Published in:
2011 | OriginalPaper | Chapter
8. Affine Classification of Real Quadrics
Author : Agustí Reventós Tarrida
Published in: Affine Maps, Euclidean Motions and Quadrics
Publisher: Springer London
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Abstract
In this chapter we study quadrics considering equivalent two quadrics when there is an affinity that takes one onto the other. Quadrics are the zeros of quadratic polynomials, and therefore they are the most natural objects to consider after straight lines (zeros of linear polynomials). From this point of view, there are essentially only three quadrics (conics) in the plane: ellipse, hyperbola and parabola. We also give the complete list of affine quadrics in dimension three.
We study some properties of a given quadric, such as the tangent cone, the polar hyperplane, the center, etc.
The subsections are
8.1
Introduction
8.2
Quadratic polynomials
8.3
Definition of quadric
8.4
Change of affine frame
8.5
Image of a quadric under an affinity
8.6
Equivalent quadratic polynomials
8.7
Invariants
8.8
Canonical representatives without linear part
8.9
Canonical representatives with linear part
8.10
Number of equivalence classes of quadratic polynomials
8.11
Regular zeros
8.12
Affine classification of quadrics
8.13
Affine classification of conics
8.14
Affine classification of quadrics in dimension three
8.15
Quadrics without regular points
8.16
Quadrics with center
8.17
Tangent cone
Exercises