#### Thesis Title

#### Date of Graduation

Spring 2013

#### Degree

Master of Science in Mathematics

#### Department

Mathematics

#### Committee Chair

Kishor Shah

#### Keywords

quadric surface, Principal Axis Theorem, eigenvalues, projective space, normal form, Segre map, ruled surface, Plücker coordinates, Mathematica

#### Subject Categories

Mathematics

#### Abstract

We begin our thesis with the study of quadric surfaces in R^n. We provide a detailed proof of the well-known Principal Axis Theorem. We apply this theorem to reduce a general degree two equation to the form Ax^2+By^2+Cz^2+J=0 or the form Ax^2+By^2+Iz=0. We construct a detailed table that complements the classification of quadrics outlined in Leonhard Euler's book Introduction to Analysis of the Infinite, James Stewart's textbook Essential Calculus, as well as John B. Fraleigh and Raymond A. Beauregard's textbook Linear Algebra. We then study quadric surfaces in the projective space P^n over the field k. We provide a proof of the well-known theorem on Normal Form for Quadrics specifically in the case of P^3. We illustrate this theorem with detailed examples of quadric surfaces in P^4 (Z_5 ), P^2 (Q(i)), and P^2 (Q). We conclude our thesis by studying the Segre map, Segre varieties, ruled surfaces, and Plü;cker coordinates.

#### Copyright

© Michael Robert Finnegan

#### Recommended Citation

Finnegan, Michael Robert, "Quadric Surfaces" (2013). *MSU Graduate Theses*. 1648.

http://bearworks.missouristate.edu/theses/1648