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mthsc206-fall-2010-notes-13.6 - 13.6 Cylinders and Quadric...

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13.6 Cylinders and Quadric Surfaces Cylinders Definition 13.6.1 A cylinder is a surface obtained by taking all of the lines parallel to a given line that intersect a given (plane) curve. Admittedly, this definition is at first confusing. Let’s try to illustrate the meaning of the definition. Consider the plane curve y = x 2 . In the xy -plane the graph is just the “standard” parabola which opens up and has vertex (0 , 0). However, if we were to graph this equation in 3-space, we would obtain much more than a plane curve. Why? Well if we look at the trace of this surface in any plane parallel to the xy -plane, we will always obtain the plane curve y = x 2 . So what does this look like? Note: If one of the 3 variables is missing in an equation, then the graph of the equation will be a cylinder. The “missing” variable is the variable that will have free range. Thus the cylinder will be the lines parallel to the axis belonging to the missing variable that intersect the plane curve with the equation in the two given variables.
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