Michael Corral: Vector Calculus

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44 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE For the hyperboloid of one sheet, the trace in any plane parallel to the xy-plane is an ellipse. The traces in the planes parallel to the xz- or yz-planes are hyperbolas (see Figure 1.6.5), except for the special cases x=±a and y=±b; in those planes the traces are pairs of intersecting lines (see Exercise 8). For the hyperboloid of two sheets, the trace in any plane parallel to the xy- or xzplane is a hyperbola (see Figure 1.6.6). There is no trace in the yz-plane. In any plane parallel to the yz-plane for which|x|>|a|, the trace is an ellipse. The elliptic paraboloid is another type of quadric surface, whose equation has the form: z x 2 a 2+y2 b 2=z c (1.37) The traces in planes parallel to the xy-plane are ellipses, though in the xy-plane itself the trace is a single point. The traces in planes parallel to the xz- or yz-planes are parabolas. Figure 1.6.7 shows the case where c>0. When c
1.6 Surfaces 45 The hyperbolic paraboloid can be tricky to draw; using graphing software on a computercanmakeiteasier. Forexample,Figure1.6.8wascreatedusingthefreeGnuplot package (see Appendix C). It shows the graph of the hyperbolic paraboloid z=y 2 − x 2 , whichisthespecial casewherea=b=1andc=−1inequation(1.38). Themeshlines on the surface are the traces in planes parallel to the coordinate planes. So we see thatthetracesinplanesparalleltothe xz-planeareparabolaspointingupward,while the traces in planes parallel to the yz-plane are parabolas pointing downward. Also, notice that the traces in planes parallel to the xy-plane are hyperbolas, though in the xy-plane itself the trace is a pair of intersecting lines through the origin. This is true in general when c0, the surface would be similar to that in Figure 1.6.8, only rotated 90 ◦ around the z-axis and the nature of the traces in planes parallel to the xz- or yz-planes would be reversed. The last type of quadric surface that we will consider is the elliptic cone, which has an equation of the form: x 2 a 2+y2 b 2−z2 c2= 0 (1.39) The traces in planes parallel to the xy-plane are ellipses, y except in the xy-plane itself where the trace is a single 0 point. The traces in planes parallel to the xz- or yz-planes are hyperbolas, except in the xz- and yz-planes themselves where the traces are pairs of intersecting lines. x Notice that every point on the elliptic cone is on a line which lies entirely on the surface; in Figure 1.6.9 these Figure 1.6.9 Elliptic cone lines all go through the origin. This makes the elliptic cone an example of a ruled surface. The cylinder is also a ruled surface. What may not be as obvious is that both the hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces. In fact, on both surfaces there are two lines through each point on the surface (see Exercises 11-12). Such surfaces are called doubly ruled surfaces, and the pairs of lines are called a regulus. It is clear that for each of the six types of quadric surfaces that we discussed, the surface can be translated away from the origin (e.g. by replacing x 2 by (x− x 0 ) 2 in its equation). It can be proved 11 that every quadric surface can be translated and/or rotated so that its equation matches one of the six types that we described. For example, z=2xy is a case of equation (1.33) with “mixed” variables, e.g. with D0so that we get an xy term. This equation does not match any of the types we considered. However, by rotating the x- and y-axes by 45 ◦ in the xy-plane by means of the coordinate transformation x=(x ′ − y ′ )/ √ 2, y=(x ′ + y ′ )/ √ 2, z=z ′ , then z=2xy becomes the hyperbolic paraboloid z ′ = (x ′ ) 2 − (y ′ ) 2 in the (x ′ ,y ′ ,z ′ ) coordinate system. That is, z=2xy is a hyperbolic paraboloid as in equation (1.38), but rotated 45 ◦ in the xy-plane. 11 See Ch. 7 in POGORELOV. z
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188 Bibliography Pogorelov, A.V., A
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Appendix B We will prove the right-
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Appendix C 3D Graphing with Gnuplot
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202 GNU Free Documentation License
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Index Symbols D....................
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Michael Corral: Vector Calculus

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