Part 3: Cylinders

A cylinder is a surface traced out by translation of a plane curve along a straight line in space. For example, the right circular cylinder shown below is the translation of a circle in the xy-plane along a straight line parallel to the z-axis. (The "right" in the name means that the translation is along a line perpendicular to the plane of the "generating circle.")

We can use that description to write an equation for the surface: The "generating circle" in the plane is x² + y² =16. If we interpret that equation as having three variables (with 0 coefficient for z), the equation describes every point on the surface.

This idea works for all cylinders that are parallel to one of the coordinate axes. For example, if we translate the parabola y² = z in the yz-plane along the x-axis, we get the parabolic cylinder defined by the same equation. (Like the previous example, this is a "right" cylinder, but of course it's not a circular cylinder.) The following figure is a graph of this parabolic cylinder.

The equations for both the circular and parabolic cylinders are quadratic, so technically these are quadric surfaces. However they are "degenerate" quadrics because each of the equations has a variable with 0 coefficient. As we will see in step 4, not all cylinders are quadric surfaces.

1. Make your own plots of the circular and parabolic cylinders in your worksheet. Rotate and use various views to be sure you understand what these surfaces look like.
2. Define and plot an elliptic cylinder that is not circular.
3. Define and plot a hyperbolic cylinder.
4. On paper, sketch what you think the graph of the cylinder y = sin(x) would look like. Then verify the shape in your worksheet.
5. Choose another equation in two variables, and use it to plot a cylinder different from any of those plotted previously.

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Copyright CCP and the author(s), 2001-2002