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Engineering Calculus Notes 417

Engineering Calculus Notes 417 - z = k we substitute this...

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3.10. QUADRATIC CURVES AND SURFACES 405 if k = 0 this is just the origin, and if k< 0 this gives an empty locus; if k> 0, then we can divide by k and modify the divisors on the left to get an equation of the form x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 . (3.46) We study the locus of this equation by slicing : that is, by looking at how it intersects various planes parallel to the coordinate planes. This is an elaboration of the idea of looking at level curves of a function. The xy -plane ( z = 0) intersects the surface in the ellipse x 2 a 2 + y 2 b 2 = 1 . To find the intersection with another horizontal plane,
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Unformatted text preview: z = k , we substitute this into the equation of the surface, getting x 2 a 2 + y 2 b 2 = 1 − k 2 c 2 ; to get a nonempty locus, we must have the right side nonnegative, or | k | ≤ c. When we have equality, the intersection is a single point, and otherwise is an ellipse similar to that in ±igure 3.39 , but scaled down: we have superimposed a few of these “sections” of the surface in ±igure 3.39 . To see how these Ft together, we can look at where x y z ±igure 3.39: Horizontal sections...
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