by Mark Franz


You're probably familiar with the standard parametric equations for an ellipse, namely

x = a cos t ,    y = b sin t       (1)

where t goes from 0 to 2p, but you may have wondered if these equations are really needed for anything practical. After all, we have the good old familiar equation

x2/a2 + y2/b2 = 1       (2)

which, like Equations (1), describes an ellipse centered at the origin, with one axis on the x-axis having length 2a, and the other axis on the y-axis with length 2b. If they describe the same ellipse, what do we need another set of equations for?

René Descartes with a (digital) oval mat 

Fletcher 1100 Oval/Circle Cutter from the Fletcher-Terry Company.

There are many good answers to this question, but we're going to concentrate on one that's extremely down-to-earth and practical. In the custom picture framing trade, there's a common treatment for portraits and other types of artwork and photographs called an oval mat. A mat is essentially a cardboard window which goes over a picture to protect the edges of the picture, separate it from the glass, and - hopefully - enhance its appearance. Actually, the best mats are not made of cardboard, but of 100% cotton rag pulp, rather than wood pulp. (Rag mats are acid-free, and thus they will not damage works on paper by making them become yellow and brittle.) The term oval in "oval mat" means that the opening through which the picture shows, and perhaps the outer border of the mat itself, is an ellipse. Many, if not most, of the devices which are used to cut oval mats are based on Equations (1) rather than Equation (2).


You may recall that an ellipse can be drawn by driving two nails into a board, putting a loop of string around them, and drawing the curve with a pencil which keeps the string taught during the process (see figure below).

In fact, Equation (2) is usually derived from such a construction, by noting that the sum of the distances d1 and d2 is a constant, because the length of the string doesn't change. The constant a turns out to be equal to (d1 + d2)/2, and the constant b satisfies the equation c2 = a2 - b2, where c is the center-to focus-distance of the ellipse. (You can refer to your old algebra text for a complete derivation.) However, this drawing technique, which is related to Equation (1), does not work so well in practice. The pencil must be held perfectly straight, there will be a knot in the string which will cause a bump in the curve when the pencil goes past it, and the tension in the string must be kept constant, which is difficult to do.


In my former career as a picture framer (Honest - I'm not kidding!), the gallery I was working for did not have the space for a deluxe oval mat cutter, but we occasionally got requests for oval mats. Fortunately, a terrific framer I had worked with named Jim Cunningham had taught me that with lots of practice (I mean lots), you could use a hand-held mat cutter to cut a very smooth curve, provided you could draw a super-thin, ultra smooth curve with a sharp pencil. Thus I looked at a top-of-the-line oval mat cutter to see how it worked, and made a set of "elliptical compasses" like the one below.

The device consisted of a stick with a hard (#6) pencil lead sanded to a sharp point, holes to locate a pair of pivots, and two small blocks to anchor the pivots. One pivot was confined to slide in a horizontal groove, and the other was confined to slide in a vertical groove.

To my delight, the compasses worked beautifully! I had in a sense re-invented the wheel, because I found out later that there is a toy for executives called a "vacuum grinder," which is exactly the same thing, except the pencil lead is replaced with a knob you hold to turn the device. I can understand the appeal, because it's fascinating to watch the little pivots churn back and forth. The following diagram shows why the pencil lead (or handle) traces an ellipse.

Can you verify that x = a cos t and y = b sin t ? The trick is to draw some extra vertical lines to create right triangles, and then use trigonometry.

Now we see a practical use for the parametric equations of an ellipse. They draw (and cut) better ellipses, help to support an industry, and make our lives more beautiful! We also see what the parameter "t" is: it's the angle between the stick and the x-axis. More generally, knowing multiple mathematical formulations of a concept can make engineers and inventors more creative, and better able to assist and interact with their colleagues in the arts and other disciplines.

Now watch the animation and meditate on the value of parametric equations!