From the early part of my career I’ve been dealing with a lot of curved work. The neighborhood I specialize in was built in the early 1900s, and many of the homes are graced with both simple and complex arches. When I started in the business, I relied on millwork shops whenever I needed to restore or remodel projects. But all that changed on one single job.
A client on a tight budget sent me a picture of an arch he wanted built in his family room. It looked a little complicated; it wasn’t a simple radius. That was my first encounter with an ellipse. He had found a millwork shop that would make the arch for a competitive price. As usual for that time, I was happy doing just the rough framing and installing the owner-supplied trim…at least until the piece showed up on the jobsite.
That old saying—if the price is too good, there is something wrong—proved true. What was supposed to be an elliptical arch looked more like someone traced a large garbage can lid and two coffee cans—an unsuccessful attempt at a three-centered arch.
I wasn’t pleased, and neither was my client. While it’s usually against my nature to criticize another craftsman’s work, I couldn’t tolerate that trim. It had to go. That is the moment I decided to learn more about the ellipse—to learn not only how to draw one, but how to make one.
I turned to George Collings’ Circular Work in Carpentry and Joinery and fell deeply into the mystery of circular work and its use in millwork. Along the way, I discovered that the ellipse can be the perfect form for arches on homes. It can be used in places where a segmental arch won’t look right, or when a semicircular arch won’t fit, like in flanking arched openings with different spans, or an arch with an extremely low rise. No matter what the height or width of an opening, the shape of an elliptical arch is always pleasing and consistent.
The Ellipse Defined
Even with Collings’ great book in my hands, understanding how to layout an ellipse wasn’t easy. Just look at this online definition: A curved line forming a closed loop, where the sum of the distances from two points (foci) to every point on the line is constant (source).
And here’s another description, with a formula, that I found online: A closed conic section shaped like a flattened circle and formed by an inclined plane that does not cut the base of the cone. Standard equation x2/a2 + y2/b2 = 1, where 2a and 2b are the lengths of the major and minor axes. Area: πab
Well, regardless of what Gary Katz says about my abilities with math, I’m not very good at understanding advanced equations. Like most carpenters, I need to get a handle on things—I need to get my hands on something tangible, something physical, in order to understand it.
The easiest way for me to explain what an ellipse looks like is to share a simple illustration of something that any carpenter can visualize. If you take a 4″ PVC pipe and cut it on your miter saw at a 90˚ angle, (zero on most miter saws!), the cut end of the pipe forms a circle with a 2 in. radius, and a diameter of 4 in.
If you cut that same pipe at an angle, by swinging the saw to 22 ½˚ or 45˚, the cut end of the pipe will form an ellipse. And the size and shape of the ellipse is mathematically predictable.
Drawing an Elliptical Arch (the string method)
If we look at that piece of pipe we cut on the miter saw, the minor axis would be the diameter of the pipe.
If we are going to use this shape to create an arch, there are a few important features we need to identify in order to really understand an ellipse. Those features are:
• The Rise and the Run of the arch (the Rise is one half of the minor axis, and the Run is equal to the major axis; since we are only using half of the ellipse)
• The focal points
In the illustration of the ellipse we can see a few elements that define the shape. For a carpenter Rise and Run are more familiar, so I’ll use those terms instead of major axis and ½ the minor axis. The ellipse we will draw for this article will have a Run of 40 in. and a Rise of 14 in.
Starting with a horizontal baseline (the spring line of the arch), mark off the 40 in. in addition to the midpoint at 20 in. (see below).
Using a square, draw a perpendicular line from the midpoint of the arch’s Run to define the Rise of the arch. In this example the rise is 14 in. (see below).
Next, find the ellipse’s focal points by using a measurement of ½ the Run length (20 in. in this example) to strike a mark on the Run line measuring from the top of the rise line. I usually make a small story pole for this to make it easier.
Set two screws at each end of the Run, and then connect a non-stretch line or cable between the two points. This gives us a string with a measurement equal to the Run, the major axis of the ellipse.
NOW, move the cable connections to the focal points without changing the length of the string.
All that’s left to do is to stretch out the string and draw your ellipse. The shape that is drawn can be cut with a jig saw with a reasonable degree of accuracy for rough framing. I use this technique for drywall arches and for framing barrel ceilings and porches.
Note: An alternate method for setting the string length is to set a third screw at the top of the Rise line. Tightly stretch the string from a screw set at one focal point, over the height line screw, and secure it to a screw set at the opposite focal point. Next, remove the Rise line screw and use the string as a guide to trace the arch’s shape.
|Download the Quick Reference Guide for Drawing an Elliptical Arch (the string method)|
Cutting an Elliptical Arch with a Router
The string method is normally not accurate enough for molding and case work, at least not in my hands. I suppose there are carpenters with the skills and patience to make it work—but I prefer to use a router to cut my trim. The technique and layout may be different, but now that we understand the shape it’s really not hard at all.
First determine the layout of the arch—the Rise and the Run. These dimensions will determine how to set up the router’s trammel arm. The trammel that I use consists of a piece of 1/8 in. thick aluminum stock, and two sliding shower door rollers to act as pivots.
To set up the trammel, mount your router at one end of the trammel arm and drill a hole to allow the cutting bit to drop through. Measuring from the appropriate cutting side of the bit, mount the rollers along the trammel arm as shown below. To make life easy, I run a score line down the center of my trammel to help in layout. This allows me to locate the pivots on the center of the arm very quickly. Both rollers must be placed accurately in order to create a predetermined shape.
Now that the trammel is set up, we need to create a T-slot for the trammel’s pivot rollers to ride in. The T-slot is set along the Run line (the spring line of the arch) with its perpendicular slot centered on the Rise line. The width of the slots corresponds to the width of the pivot rollers being used. I create this T-slot by cutting two rectangles out of whatever scrap I happen to have onsite, and use the roller wheels, or spacers of the same width, to set the slot width. When everything is aligned and positioned correctly, I screw the pieces down to a backer board, including the piece I’m going to cut, which forms the top of the ‘Run’ track.
With the trammel rollers dropped into the slots, this jig cuts an almost perfect elliptical arch. There is no need to locate the focal points, the Rise and Run dimensions are constrained by the T-slot, and the geometry is automatically created.
You’ll find that once you make of a few of these, there are a lot of things you can do with the ellipse. Exterior ornaments, arched trim heads for bookcases, arched passage ways, and a host of other cool projects ….
The real trick is never to let your client see just how easy it is!
(SketchUp drawings by Wm. Todd Murdock)