PowerPoint PresentationSlide 2Cylinders and Quadric SurfacesSlide 4Slide 5CylindersSlide 7Quadric SurfacesSlide 9Slide 10Applications of Quadric SurfacesSlide 12Copyright © Cengage Learning. All rights reserved. 12Vectors and the Geometry of SpaceCopyright © Cengage Learning. All rights reserved. 12.6Cylinders and Quadric Surfaces3Cylinders and Quadric SurfacesLearning objectives: Investigate on your own: cylinders and quadric surfaces• Section 12.6 in the Stewart text• via the Media tab within Enhanced WebAssign The goal is geometric intuition for our study of surfaces, and the ability to visualize a number of surfaces that will repeatedly serve as examples You do not have to memorize the various formulas for quadric surfaces4Cylinders and Quadric SurfacesWe have already looked at two special types of surfaces: planes and spheres. Here we investigate two other types of surfaces: cylinders and quadric surfacesIn order to sketch the graph of a surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes.These curves are called traces (or cross-sections) of the surface.5Cylinders6CylindersA cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and pass through a given plane curve.7Quadric Surfaces8Quadric SurfacesA quadric surface is the graph of a second-degree equation in three variables x, y, and z. The most general such equation isWhere A,B,C,…, J are constants, but by translation and rotation it can be brought into one of the two standard forms Ax2 + By2 + Cz2 + J = 0 or Ax2 + By2 + Iz = 0Quadric surfaces are the counterparts in three dimensions of the conic sections in the plane.9Quadric SurfacesTable 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.Table 1Graphs of quadric surfaces10Applications of Quadric Surfaces11Applications of Quadric SurfacesExamples of quadric surfaces can be found in the world around us. In fact, the world itself is a good example.Although the earth is commonly modeled as a sphere, a more accurate model is an ellipsoid because the earth’s rotation has caused a flattening at the poles.Circular paraboloids, obtained by rotating a parabola about its axis, are used to collect and reflect light, sound, and radio and television signals.12Applications of Quadric SurfacesIn a radio telescope, for instance, signals from distant stars that strike the bowl are all reflected to the receiver at the focus and are therefore amplified.The same principle applies to microphones and satellite dishes in the shape of paraboloids.Cooling towers for nuclear reactors are usually designed in the shape of hyperboloids of one sheet for reasons of structural stability.Pairs of hyperboloids are used to transmit rotational motion between skew
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