![]() ![]() Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. We recommend using aĪuthors: Gilbert Strang, Edwin “Jed” Herman Use the information below to generate a citation. Then you must include on every digital page view the following attribution: ![]() If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses theĬreative Commons Attribution-NonCommercial-ShareAlike License This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission. We can view these surfaces as three-dimensional extensions of the conic sections we discussed earlier: the ellipse, the parabola, and the hyperbola. Some other common types of surfaces can be described by second-order equations. We have learned about surfaces in three dimensions described by first-order equations these are planes. We now explore more complex surfaces, and traces are an important tool in this investigation. Not all surfaces in three dimensions are constructed so simply, however. The trace in the xy-plane, though, is just a series of parallel lines, and the trace in the yz-plane is simply one line.Ĭylindrical surfaces are formed by a set of parallel lines. Notice, in Figure 2.80, that the trace of the graph of z = sin x z = sin x in the xz-plane is useful in constructing the graph. For a cylinder in three dimensions, though, only one set of traces is useful. Traces are useful in sketching cylindrical surfaces. The trace is simply a two-dimensional sine wave. (b) To find the trace of the graph in the xz-plane, set y = 0. In this way, any curve in one of the coordinate planes can be extended to become a surface.įigure 2.80 (a) This is one view of the graph of equation z = sin x. We can then construct a cylinder from the set of lines parallel to the z-axis passing through circle x 2 + y 2 = 9 x 2 + y 2 = 9 in the xy-plane, as shown in the figure. Imagine copies of a circle stacked on top of each other centered on the z-axis ( Figure 2.75), forming a hollow tube. In three-dimensional space, this same equation represents a surface. In the two-dimensional coordinate plane, the equation x 2 + y 2 = 9 x 2 + y 2 = 9 describes a circle centered at the origin with radius 3. A rectangular heating duct is a cylinder, as is a rolled-up yoga mat, the cross-section of which is a spiral shape. ![]() As we have seen, cylindrical surfaces don’t have to be circular. Although most people immediately think of a hollow pipe or a soda straw when they hear the word cylinder, here we use the broad mathematical meaning of the term. The first surface we’ll examine is the cylinder. In this section, we use our knowledge of planes and spheres, which are examples of three-dimensional figures called surfaces, to explore a variety of other surfaces that can be graphed in a three-dimensional coordinate system. We have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. 2.6.3 Use traces to draw the intersections of quadric surfaces with the coordinate planes.2.6.2 Recognize the main features of ellipsoids, paraboloids, and hyperboloids.2.6.1 Identify a cylinder as a type of three-dimensional surface. ![]()
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