13.6Cylinders and Quadric SurfacesCylindersDefinition 13.6.1A cylinder is a surface obtained by taking all of the lines parallel to a given line thatintersect a given (plane) curve.Admittedly, this definition is at first confusing. Let’s try to illustrate the meaning of the definition.Consider the plane curvey=x2. In thexy-plane the graph is just the “standard” parabola which opensup and has vertex (0,0). However, if we were to graph this equation in 3-space, we would obtain muchmore than a plane curve. Why? Well if we look at the trace of this surface in any plane parallel to thexy-plane, we will always obtain the plane curvey=x2. 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 linesparallel to the axis belonging to the missing variable that intersect the plane curve with the equationin the two given variables.
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