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Curves. A good part of math deals with curves. Approximating them using Taylor Polynomials, finding their properties, creating more curves. Maybe you are a little bored of drawing the same 3-degree polynomial on your graph paper and need something a little more exciting. That is where envelopes come in!

Let’s take the basic parabola to start. Most people know the basic formula (x-h)^2 +k =y. But there are a couple of other properties of a parabola. The first is called a focus point. (See below):

The coordinate for this focus point is (h, {k + a/4}). Next, the directrix is a line at y = k- a/4, or the same distance from the vertex that the focus is. The most important property about these two features is that every point on the parabola is equal distance from the focus to the directrix. (see above)

Now, this is where the fun starts. Draw a dot (focus) and a straight line (directrix) on a sheet of paper. Draw another line connecting the dot and one place on the line. Next, in a different color, mark out where the mid point of that line segment is. Now draw a line perpendicular to the line that you just drew. Continue this until you feel like your piece of paper is properly covered. You should notice a familiar shape beginning to form…

A parabola!

The set of all of these tangent lines that form a parabola is called an envelope. Because the envelope is the family of all of these tangent lines, it can be further classified down to parametric equations.

Let’s look at another common example: the astroid.

First, chose two points on the x and y-axis. The picture below uses (0, 5) and (4, 0). Now draw a straight line from the point on the x-axis to a few points on the y axis that are below the chosen point. Repeat for the y-axis. You should begin to see a start forming:

Now, the above image is by no means perfect, but as more and more lines are drawn, a more distinct astroid should form. What is most interesting about this shape is that it is actually connected to the ellipse, similarly to the parabola in the first example.

First, take the equation of an ellipse and the equation of its partial derivative with respect to c:

Its derivative is useful since the envelope is just a fancy way of saying the collection of tangent lines.

Now that we have the two equations for the ellipse and derivative, we can put them both into matrix form:

We can them move to solve this matrix by rearranging the equation and finding the inverse of the matrix:

Finally, we can rearrange this equality to eliminate a, resulting in the following equation:

Which… when plugged into a graphing calculator, results in the asteroid we graphed above:

Also notice, that the original ellipse graph has its focus points exactly where the asteroid we derived is graphed. This is a really cool connection to how all of the properties of conic graphs and their envelope, or set of derivatives are connected. And whenever you are bored in class, try out drawing your own conic shape and finding the properties that its envelope reveals about it.