Learn how to draw a sphere.
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, 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 and as shown here:
It turns out that when we use the parameters and , 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.
To show that the first description contains the second, we need to explain why every point in is actually drawn by our parametric description. Given a point on the sphere, we need to assure our most stubborn readers that there will be a and a that when plugged into our formula, will produce the given point . We could do this by solving for and in terms of , , and —though we will not do that here. Instead, we will show you how the sphere is drawn by and .
By dragging the sliders around, you should be able to convince yourself that every point on the sphere can be hit by some choice of and .
Since every point on the sphere can be obtained by letting run from to , and run from to , we have shown that the parametric formula draws the entire sphere.
Now we’ll show that every point of the parametric formulais also a point of . This result will tell us that the second description contains the first. We do this by plugging the point into and through a manipulation of symbols, conclude that it does equal . Write with me. Simplifying: So we see that every point drawn by our parametric description of the sphere is in the set .