All traces are labeled with the value of k z z z 1 0 y

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Unformatted text preview: 1 2 0 _1 1 _1 0 y 1 FIGURE 6 Vertical traces are parabolas; horizontal traces are hyperbolas. All traces are labeled with the value of k. x x 0 2 1 Traces in x=k are z=¥-k @ Traces in y=k are z=_≈+k @ Traces in z=k are ¥-≈=k z z z 1 0 y 0 FIGURE 7 1 Traces moved to their correct planes x _1 x x _1 _1 0 1 Traces in y=k Traces in x=k In Module 13.6A you can investigate how traces determine the shape of a surface. y y Traces in z=k In Figure 8 we fit together the traces from Figure 7 to form the surface z y 2 x 2, a hyperbolic paraboloid. Notice that the shape of the surface near the origin resembles that of a saddle. This surface will be investigated further in Section 15.7 when we discuss saddle points. z 0 y x FIGURE 8 The surface z=¥-≈ is a hyperbolic paraboloid. EXAMPLE 6 Sketch the surface x2 4 y2 z2 4 SOLUTION The trace in any horizontal plane z x2 4 y2 1 1. k is the ellipse k2 4 z k 5E-13(pp 868-877) 872 ❙❙❙❙ 1/18/06 11:35 AM Page 872 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE z but the traces in the x z- and y z-planes are the hyperbolas x2 4 1 y 0 y2 and z2 4 1 x 0 This surface is called a hyperboloid of one sheet and is sketched in Figure 9. (0, 1, 0) (2, 0, 0) z2 4 y x The idea of using traces to draw a surface is employed in three-dimensional graphing software for computers. In most such software, traces in the vertical planes x k and y k are drawn for equally spaced values of k, and parts of the graph are eliminated using hidden line removal. Table 1 shows computer-drawn graphs of the six basic types of quadric surfaces in standard form. All surfaces are symmetric with respect to the z-axis. If a quadric surface is symmetric about a different axis, its equation changes accordingly. FIGURE 9 TABLE 1 Graphs of quadric surfaces Surface Equation x2 a2 Ellipsoid z y2 b2 z2 c2 Surface 1 z x z c Elliptic Paraboloid z x2 a2 y2 b2 x2 a2 y2 b2 Horizontal traces are ellipses. c, the ellipsoid is y x z2 c2 Cone All traces are ellipses. If a b a sphere. Equation y Vertical traces in the p...
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