3.10. QUADRATIC CURVES AND SURFACES399ξ2−1√2ξ1+√2•(−12,1)xyFigure 3.31: The curvex2−2xy+y2+ 3x−5y+ 5 = 0Quadric SurfacesThe most general equation of degree two inx,yandzconsists of three“square” terms, three “mixed product” turns, three degree one terms(multiples of a single variable), and a constant term. A procedure similarto the one we used for two variables can be applied here: combining the sixquadratic terms (the three squares and the three mixed products) into aquadratic formQ(x,y,z), we can express the general quadratic equation inthree variables asQ(x,y,z) +Ax+By+Cz=D.(3.41)Using the Principal Axis Theorem (Proposition3.9.2) we can create a newcoordinate system, a rotation of the standard one, in which the quadratic
This is the end of the preview.
access the rest of the document.
Quadratic equation, Elementary algebra, general quadratic equation, PRINCIPAL AXIS THEOREM, QUADRATIC CURVES