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Engineering Calculus Notes 411

# Engineering Calculus Notes 411 - 399 3.10 QUADRATIC CURVES...

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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
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