
Learn how to draw a sphere.

A key challenge in mathematics is converting formulas and equations into ideas. We want to get to the point that when you see something like you say to yourself, “Hey, that’s a sphere of radius $4$ centered at the point $(1,2,3)$.” Let’s state this generally.

As we work in 3-space it is going to be very helpful to be able to draw some of the sets we commonly encounter. For now, let me show you how to draw a sphere yourself. Get out a sheet of paper, and play along—it will be fun! Start by drawing a set of axes:

Now draw a circle in the $(y,z)$-plane: Pro-tip: If you have trouble drawing a circle, and most people do, try drawing circles on graph paper. Practice makes perfect, and if you practice enough, soon you’ll be able to impress your friends and enemies alike with your circle-drawing skills. Now draw an ellipse, dashing the part at the “back” of the sphere: And volià, we have a sphere!

Now, back to some equations! Above we gave an implicit formula for the surface of the sphere. Sometimes parametric formulas are easier to work with. We’ll be talking about parametric formulas for surfaces a lot in this course, so consider these equations your introduction. The parameters we are going to use now are $\theta$ and $\phi$ as shown here:

It turns out that when we use the parameters $\theta$ and $\phi$, the formulas below give us a sphere. (Don’t worry about where those formulas come from. You’ll get to that later in your calculus journey!)

Now let me tell you something: people who like mathematics really like asking (and answering) questions like the following.