Spheres and Ellipsoids

We now come to another important type of function. To understand what it looks like, we'll take a brief detour of what it means to measure distance in three dimensions. Recall that in two dimensions, we can use the Pythagorean theorem to calculate the distance from the point (x₀, y₀) to the origin. It's simply $\sqrt{x_0^2 + y_0^2}$. This is the same as the magnitude of the position vector that points to the points (x₀, y₀). In three dimensions, we can do the same thing. If a point has coordinates (x₀, y₀, z₀) then it has a position vector $\vec{r} = x_0 \hat{i} + y_0 \hat{j} + z_0 \hat{k}$ and is a distance $d = \vert\vert \vec{r} \vert\vert = \sqrt{x_0^2 + y_0^2 + z_0^2}$ from the origin.

Suppose we want to know what the equation of a sphere is. We can use the distance formula above to help. If the sphere is centered at the origin and has a radius of R units, then we know that the surface of the sphere is made up of every point that is exactly R units from the origin. Thus, if we let (x,y,z) be a point on the surface of the sphere, we have

$$R = \sqrt{x^2 + y^2 + z^2} \quad \Rightarrow \quad x^2 + y^2 + z^2 = R^2$$

as the equation for the surface of the sphere. This is only for a sphere centered at the origin.

For a sphere centered elsewhere, we modify the distance formula. If the center of the sphere is at (x₀, y₀, z₀) and the sphere has a radius of R, then each point (x,y,z) on the surface is R units from the center, so

$$R = \sqrt{(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2}.$$

Now, let's modify the formula for the sphere centered at the origin a little. Suppose we want to draw the surface represented by

$$\frac{x^2}{4} + y^2 + z^2 = 1.$$

A little work with drawing the contours and sections will convince you that this surface is similar to a sphere, except that it has been stretched a factor of $\sqrt{4} = 2$ along the x axis. Thus, it is more ellipsoidal in shape. In general, the graph of the surface

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$$

is an ellipsoid with intercepts at $(\pm a,0,0), (0,\pm b,0)$ and $(0,0,\pm c)$.