12 Vectors and the Geometry of Space Copyright Cengage Learning All rights reserved 12 6 Cylinders and Quadric Surfaces Copyright Cengage Learning All rights reserved Cylinders and Quadric Surfaces Learning 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 surfaces 3 Cylinders and Quadric Surfaces We have already looked at two special types of surfaces planes and spheres Here we investigate two other types of surfaces cylinders and quadric surfaces In 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 4 Cylinders 5 Cylinders A 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 6 Quadric Surfaces 7 Quadric Surfaces A quadric surface is the graph of a second degree equation in three variables x y and z The most general such equation is Where 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 0 Quadric surfaces are the counterparts in three dimensions of the conic sections in the plane 8 Quadric Surfaces 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 Graphs of quadric surfaces Table 1 9 Applications of Quadric Surfaces 10 Applications of Quadric Surfaces Examples 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 11 Applications of Quadric Surfaces In 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 axes 12
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