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Math 232 – Scientific Calculus II Geometry in 3D – Part 1 Planes and Cylinders Recall that, back in our “2D-calculus” days, we used our knowledge of things like circles, lines, parabolas, etc., to help us understand the graphs of certain functions. (For example, knowing that y x 2 is the equation of a parabola tells us about the graph of the function f ( x) x 2 .) So, knowing the formulas of some geometric shapes is a good thing. In 3D, this is even more important, so, in the next couple of readings, we will learn some basics about geometry in 3D. 3D Planes. We have already learned (in Section 9.5) how to recognize an equation of a plane when we see one. Any equation of the form ax by cz d , where x, y, and z are the variables, and a, b, c, and d are constants, represents a plane. 3D Cylinders. Before we proceed, take a moment and get that old notion of “cylinder = can” out of your head! Thank you. In 3D, any equation in which exactly one of the three variables is missing is called a cylindrical surface, or just a cylinder. Example 1. In 2D, the equation x 2 y 2 4 represents a circle of radius 2, but, in 3D, it is something different. It represents the set of all points (x,y,z) in 3-space, which satisfy the equation x 2 y 2 4 . Since z does not appear in this equation, there is no restriction on z-values. The result is that, in 3D, this equation represents a circular cylinder (radius 2) that runs parallel to the z-axis, infinitely far in both z-directions. (See plot below.) 2 z 0 -2 -2 -1 0 -2 1 -1 0 y 1 2 2 x Example 2. In 2D, z y 2 is a parabola in the yz-plane. Following the same reasoning as in the previous example, we determine that, in 3D, this represents a surface that is parallel to the x-axis (the missing variable). It can be thought of as the surface obtained by starting with the 2D parabola z y 2 , drawn in the yz-plane, then extending this parabola, parallel to the x-axis, infinitely-far in both directions. The result is the plot below. -2 x 0 2 8 6 z 4 2 0 -2 0 2 y In General. Whenever exactly one variable in a 3D equation is missing, it represents a cylindrical surface, which is simply a surface that is parallel to one of the coordinate axes (the one that belongs to the missing variable), and runs infinitely-far in both parallel directions. To draw such a surface, first draw (in the appropriate coordinate plane) the 2D rendition of the equation. Then, extend this graph in the direction of the missing variable. The resulting surface is the one that belongs to the equation. Example 3. In 2D, the equation 2 y z 1 is a line. In fact, if we rewrite it as z 2 y 1 , then we see that it is a line that lives in the yz-plane, with slope 2 and z-intercept 1. (See the plot, below left.) The 3D surface comes from extending this line, parallel to the x-axis, infinitely far in the two x-directions. The result is the plane pictured in the plot below right. (Note where the red line (the line from the left plot) sits in the plot of the plane.) -2 -1 x z 0 1 2 4 4 z 2 2 0 y -2 -1 1 -2 2 -2 -2 -1 0 y 1 2