Chapter 13

# In general the horizontal trace in the plane z k is

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Unformatted text preview: etch the quadric surface with equation x2 y2 9 z2 4 1 0, we ﬁnd that the trace in the xy-plane is x 2 y 2 9 1, which we recognize as an equation of an ellipse. In general, the horizontal trace in the plane z k is SOLUTION By substituting z y2 9 x2 k2 4 1 which is an ellipse, provided that k 2 4, that is, Similarly, the vertical traces are also ellipses: z (0, 0, 2) z 2 k k 2. y2 9 0 (1, 0, 0) (0, 3, 0) y z2 4 1 k2 x k if 1 k 1 x2 z2 4 1 k2 9 y k if 3 k 3 x FIGURE 4 The ellipsoid ≈+ [email protected] [email protected] + =1 9 4 Figure 4 shows how drawing some traces indicates the shape of the surface. It’s called an ellipsoid because all of its traces are ellipses. Notice that it is symmetric with respect to each coordinate plane; this is a reﬂection of the fact that its equation involves only even powers of x, y, and z. EXAMPLE 4 Use traces to sketch the surface z 4x 2 y 2. 0, we get z y 2, so the y z-plane intersects the surface in a parabola. If we put x k (a constant), we get z y 2 4k 2. This means that if we slice the graph with any plane parallel to the y z-plane, we obtain a parabola that opens upward. Similarly, if y k, the trace is z 4 x 2 k 2, which is again a parabola that opens upward. If we put z k, we get the horizontal traces 4 x 2 y 2 k, which we recognize as a family of ellipses. Knowing the shapes of the traces, we can sketch the graph in Figure 5. Because of the elliptical and parabolic traces, the quadric surface z 4 x 2 y 2 is called an elliptic paraboloid. SOLUTION If we put x z FIGURE 5 The surface z=4≈+¥ is an elliptic paraboloid. Horizontal traces are ellipses; vertical traces are parabolas. 0 x y 5E-13(pp 868-877) 1/18/06 11:35 AM Page 871 ❙❙❙❙ S ECTION 13.6 CYLINDERS AND QUADRIC SURFACES EXAMPLE 5 Sketch the surface z y2 871 x 2. k are the parabolas z y 2 k 2, which open upward. The traces in y k are the parabolas z x 2 k 2, which open down2 2 ward. The horizontal traces are y x k, a family of hyperbolas. We draw the families of traces in Figure 6, and we show how the traces appear when placed in their correct planes in Figure 7. SOLUTION The traces in the vertical planes x z z y...
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