3.10. QUADRATIC CURVES AND SURFACES 399 ξ 2 − 1 √ 2 ξ 1 + √ 2 • ( − 1 2 , 1) x y Figure 3.31: The curve x 2 − 2 xy + y 2 + 3 x − 5 y + 5 = 0 Quadric Surfaces The most general equation of degree two in x , y and z consists of three “square” terms, three “mixed product” turns, three degree one terms (multiples of a single variable), and a constant term. A procedure similar to the one we used for two variables can be applied here: combining the six quadratic terms (the three squares and the three mixed products) into a quadratic form Q ( x,y,z ), we can express the general quadratic equation in three variables as Q ( x,y,z ) + Ax + By + Cz = D. (3.41) Using the Principal Axis Theorem (Proposition 3.9.2 ) we can create a new coordinate system, a rotation of the standard one, in which the quadratic
This is the end of the preview. Sign up
access the rest of the document.
This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.