Surfaces of revolution

Radius in polar coordinates

Dropping back to two-dimensions for a moment, remember polar coordinates (r,θ),

PIC

where r is the radius from the origin to the point P, and θ is the angle from the positive x-axis. You may recall that we can express the radius r in terms of rectangular (Cartesian) coordinates by r = ∘ -2----2- x + y. The set of all points where r is constant is a circle of radius r centered around the origin.

Functions that depend only on radius

Let’s say we had a function f(x,y) of a special form so that it depended only on the radius r, i.e., depended on x and y only via the expression ∘ -------- x2 + y2. In this case, if we changed x and y in such a way that r = ∘ -2----2- x + y didn’t change, then the value of the function f(x,y) would not change. Combining this observation with our knowledge about r, we conclude that f(x,y) is constant along any circle centered around the origin.

What I’ve just described is a function of the form f(x,y) = g(∘x2--+--y2-), where g(r) is some one-variable function. We know, for example, that the function f(x,y) is constant on the circle of radius 2 centered at the origin, because for any point (x,y) where ∘ -------- x2 + y2 = 2, the value of the function f(x,y) is g(2).

This property makes it simple to graph the surface z = f(x,y) because it follows directly from the graph the curve z = g(r). And you already know how to graph a one-variable function.

To graph f(x,y), we take advantage of the fact that it doesn’t change as we rotate around the origin in the xy-plane. In three dimensions, the z-axis would be pointing out of the screen in the above figure. Hence, this rotation corresponds to rotation around the z-axis. The graph of f(x,y) is the graph of g(r) rotated around the z-axis. For this reason, the resulting surface is a called a surface of revolution.

To illustrate, we’ll show how the plot of

√ -2----2- z = f(x,y) = √sin---x--+-y- x2 + y2 + 1
is a surface of revolution. Since f(x,y) depends on x and y only via the combination r = ∘ -------- x2 + y2, we can rewrite f(x,y) as f(x,y) = g(∘ -------- x2 + y2), where
sinr g(r) = -----. r + 1
Here’s a plot of g(r).

PIC

In the following CVT, you can transform between the plot of g(r) and the plot of f(x,y) by changing the rotation angle θ. (Drag the blue point along the slider.) When θ = 0, you have the plot of g(r). When θ = 2π, you have the plot of f(x,y).

Can you recognize that each of these is a surface of revolution?

  • z = -----1----- 1 + x2 + y2
  • z = e(x2+y2)3
  • z = sin(∘ -2----2- x + y) - x2 - y2