Engineering Calculus Notes 419

Engineering Calculus Notes 419 - The intersection of the...

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3.10. QUADRATIC CURVES AND SURFACES 407 3. When all three variables appear to the second power but the quadratic form is not deFnite, there are three basic shapes that occur. These are illustrated by “model equations” below. In each, we assume that x 2 occurs with coe±cient 1 and z 2 with coe±cient 1. (a) If there is no constant term, then we have an equation of the form x 2 ± y 2 z 2 = 0 which boils down to one of the two equations x 2 + y 2 = z 2 or x 2 = y 2 + z 2 . Since the second of these equations results from the Frst by interchanging x with z , we will concentrate on the Frst.
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Unformatted text preview: The intersection of the surface x 2 + y 2 = z 2 (3.47) with the horizontal plane z = k is a circle, centered at the origin, of radius | k | (igure 3.42 ). x y z igure 3.42: Horizontal Sections The intersection of this surface with each of the vertical coordinate planes is a pair of lines (igure 3.43 ) and Ftting this together, we see that this is just the conical surface K of Archimedes ( 2.1 ) which we refer to in this context as simply a cone (igure 3.44 )....
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