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The Elegant Ellipse

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 Basics

(Note: Click any image to enlarge)

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)

A circle only has one axis—its diameter, but an ellipse has two: a large axis called the Major Axis, and a smaller one called the Minor Axis.

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)


43 Responses to “The Elegant Ellipse”

  1. David Pugh

    Great article Mike. Thanks for the careful explanation and helpful videos. I’ll be using that suggestion for the trammel arm with my Festool Router. Thanks again.

  2. Keith Mathewson

    Nicely presented. clear and informative, any carpenter should feel confident producing one after reading this article.

  3. Yannis Tsakiris

    Can you say keep it simple stupid (KISS). Nicely done Mike. We do a lot or renovations in old historic homes here in Philadelphia. This will save me and my crew time which means adding to my bottom line. Now if only the masonry arches could be clean lined as the new window and trim.
    Thank you

    Yannis Tsakiris

  4. Garry Macdonald

    Fantastic article. Thanks so much for taking the time to share this. You make it look so easy, even I might give it a go!
    Thanks again,

  5. Mike Kennedy

    I love that router jig. Simple and effective. I also like the way that the router was turned into a cartoon on the second pass in the video. I would love to see Mike cartoonized that way ; )
    Keep up the good work.

  6. Stanley D. Jubas

    This was excellent. I am a cabinet maker and will use the article to design elliptical shaped table tops. Thank you.

  7. Gene Paletta

    Thank you for enlightening me on ellipses. Very helpful- less math is always better for me- pictures and illustrations work best!

  8. Joshua Farrand


    I saw the demonstration at JLC Live and it never gets old. The trammel arm setup is pure genius. Thanks for sharing some wisdom and breaking it all down for us. Well put together article guys!!

  9. Al

    Nice presentation.
    A permanent jig can be built to make the marking and the cutting of an elipse by using 2 T tracks screwed to a board instead of using those to square board to produce two tracks.
    I will try to build one this weekend in my shop and shot a video and post it here to complement this presentation.
    Instead of wheels I will be using T track bolts that will slide inside the T tracks and the tremmel arm will be adjustable.

    • W.A.Strahan

      Al, how do you keep the jig from binding up during use, if your using t track bolts instead of some sort of bearing or “wheel” ?

    • Gary Katz

      Of course you will, Al! But the T track bolts won’t slide smoothly or hold the same radius measurements–they’re not round. But then, you haven’t tried it yet so you probably will figure that out later! :) Mike’s approach, on the other hand, is beautiful because it’s easy to set up on a jobsite without any special tracks. Just carry some aluminum bar stock and a few shower-door wheels whenever you have to cut an elliptical shape. And you don’t have to order that stuff online. It’s easy to find within blocks of most jobs.

  10. Emanuel

    Thank you for a well written article and easy step by step videos. I can’t wait to use this on my future projects that require an ellipse.

    Thanks Again
    Emanuel Silva

  11. bruce

    thank you for the nice article. Brought back some memories.
    I installed a kitchen in an addition with a high cathedral ceiling. The plans called for a range hood that extended up to the ceiling. I used a similar router setup (think it was from Jim Chestnut’s web site) and made six large ellipses out of plywood, used them to frame a box to house the range hood (The ellipses ran vertically from the hood to the ceiling). Bending plywood formed the walls, 1/4″ drywall with a stucco like finish and it was really cool looking.
    It was deceptively simple to lay out (just as you have described) and any time I’ve seen that client in the last 6 years or so, she still thanks me as if I just completed the project. I wish I’d had your pipe illustrations when I was first figuring it out though.

  12. Ron Anderson

    I’ve done a few projects involving elliptical layout and been able to get away with sharp pencil lines and careful cutting with a jigsaw and sander. Until now.
    Using a router makes it repeatable and enjoyable . Thanks for also putting it in language we can all understand.

  13. Jesse wright

    Ok can we just all agree that MIKE is the MAN!!!!

    Oh btw Todd, that SU animation was incredible! It was hilarious because it was injected in the video so perfectly Gary. I love it!!!

  14. Mike S

    Thanks for all the Nice Comments, This Article was really a collaboration, without Gary’s’ skillful editing and Todd’s superb Sketchup drawings and animations it would not have been intelligible.
    I’ve always loved curved work, I’ll just ask you to promise never to show your clients just how easy this is to do :)


  15. aaron pinkins

    thanks so much for ellipse layot and cutting class. really shows that careful planning and forethought can either make or break a project, especially if its to be truly admired. thnx again for the video truly helpful

  16. Doug Simmons

    Thanks Mike!! I carry a 48″ aluminum ruler/straightedge in my rig which I can drill and use for those crazy curved router jobs.

  17. chris olson

    Saw him do this at the roadshow yesterday! I can’t wait to use this method on-site.!

  18. Karl Gage

    Thank you Mike and the whole Katz Roadshow crew for a great presentation yesterday at Hayward Lumber in San Luis Obispo, CA
    Seeing this in person is very inspiring.

  19. Morton

    This is a fantastic “jig” and extremely well explained in the text and video. Really awesome job. I’m planning a piece of furniture with many concentric, routed, ellipses and this may be the right method for me to get the job done. Sweet.

  20. tamsin

    really helpful page, really clearly explained. thank you so much for sharing

  21. Jim Lehman

    New to your site, I can really appreciate the simplicity of your instruction. I’ve done several arcs in my career but just recently did an elliptical. Expanding an opening between a kitchen and dinng room I framed an elliptical for drywall. The clients decided they wanted it cased. I see the math after years doing carpentry but this article will save me a lot of time in the future. I had to match moldings from the 1860s with routed detail in the field of the moldings. Any jigs for helping do this? I got it done but it was a pain. Tx! Jim

  22. Facebook-carpenter Scott hunt

    I like the string method. The way to calculate an elliptical arch is, in this example, find center @ 20″ and square or plumb up to the spring line @14″ from the same point plumb or scribe a line straight down. Add half the major axis @ 20″ and the minor axis (spring line). This example is 34″, measure from the spring line straight down and mark 34″. Set a nail or screw at that point and hook your chalk line on it, stretch it out to the spring line mark and wrap it around your pencil then strike your arch. This is a little more stable than the string method, and it’s quicker to layout. Also don’t try the string method without using all 3 screw or nail points. It’s like taking a wire and bending it to fit a rudimentary arch and trying to look like a professional carpenter. Too many variables in that example

  23. Facebook-carpenter Scott hunt

    I meant to say pull half the major axis @ 20″ from top of spring line. Then strike an arch. Disregard the 34″ comment as I was working on something else

  24. Peter Siersma

    How does one cut a small ellipse with a router? I want to make an elliptical reservoir on a bamboo cutting board, with major/minor axes of 4″/2″ respectively. Any suggestions?

    In anticipation, thanks.

  25. don

    I just played this out in 1″ scale on my drawing board. Fantastic and easy, thank you. Now I can show clients what it really should look like.

  26. Benj Lawrenz

    Question about offset. I want to make a 4″ wide trim board. The jig produces a true ellipse for my inside cut but when I use the same jig to make the second cut 4″ greater than the first, the trim piece is narrower on the shoulders than on the major and minor axis. The intersection of the ellipse and the major and minor axis is truely 4″ but the shoulders are not. Is this just the nature of ellipses? . . . that one ellipse inside another cannot have a true consistent offset? Am I missing something? I’ve even drawn the ellipses in AutoCad and get the same narrow shoulder. I do however, get a consistent offset in AutoCad using the “offset” tool but the arc is no longer an ellipse.

    • Wm. Todd Murdock


      Yes. Unfortunately, this is the nature of the ellipse… The offset won’t be consistent around the perimeter—only along the Major and Minor axes. One option, since you are using ACAD, is to use the “Offset” tool and create full size templates to use as a pattern for the trim pieces. Another, more traditional approach, is to use an elliptical approximation that uses simple tangent curves–which are easy to offset. For more information on this, take a look at the TiC article on Three-Centered Arches.

      Three-centered arches are more common than you might think, and they can be difficult to distinguish from a true elliptical arch. Attached below is a composite image from the Grove Arcade in Asheville, NC. While the arch looks elliptical, you can see the architect’s drawings on the right, which calls out the two radii for the arch.


  27. Mike S

    The jig does narrow down – as it rounds the curve- That has to do with Trammel – The thickness of the Guide point effects it. I had this conversation with Jed Dixon a long time ago — He explained it and at the time I thought I understood it, But I sure can’t remember the whole thing — It has something to do with Orthographic projection and how the trammel makes the swing . He said in his line of work, to make an elliptical staircase , he only used the trammel for one of the layouts , the other – he would manually offset – this way all of the stair treads would have the same width. Much like using the offset tool – I suppose a CNC could cut the Ellipse by using the offset. I think its cool the way it looks —
    I like doing splayed ellipses ( is that the Plural ?) since it hides all of that.

    Hope this helps
    Mike S

  28. john

    I want to put a small angle across the top on my 5 inch pipe entry. Its 20 feet long therefore center is at ten ft. How much of a “wedge” should I cut out of the center for a 4 to 6 inch drop at each end? thanks

  29. Jack Herring

    Great article and yet I didn’t get the information I need from it. I am not trying to make an ellipse but am trying to find the exact location of the major and minor axes.
    We are just finishing our Mbath remodel and have purchased two elliptical shaped mirrors with bent aluminum glued to the back for mounting. I have cut two pieces of masonite to frame them from the back side after finishing them and need to find the precise location of the axes so I can allow for the mounting. Any suggestions on the best method for finding them?
    I am seeing there are many articles that I can learn from but this one on The Elegant Ellipse is the closest one to try to find the answer to my dilemma.
    Best Regards,

    • Mike S

      Jack. Not sure what you are looking for. . If you watch the string method. That will show you how to find the foci & the major & minor axis. The mirrors you bought may not be a true ellipse . I have seen a lot of modern millwork that resembles the shape but cannot be plotted . Send me a photo & describe what you are looking for. Maybe we can figure it out.


        I finally measured the longest dimension and the shortest dimension several times and then drew in the lines for the two axes. I added some slop to my cuts to give me a little wiggle room for the mount clearance. I am hoping to finish this project this week as long as I get some decent painting weather.

        In Him,

  30. Mike Dorsam


    I’m working on a coffee table that will require both an external and internal ellipse – the two ellipses need to be complimentary, if not identical… in the same sense as an interior and external circle arcs have different radii, and create an identical subtended angle.

    Is there a way to calculate the associated focal points?

    Thanks, again.
    Mike Dorsam

    • mike Sloggatt

      Concentric Ellipse
      Layout your First ellipse either the inner or the outer –
      Calculate your foci using the method in this article –
      For example if the coffee table is 48″ x 24″ Layout your Minor & Major axis 24 & 48 focal points mathematically are 26 3/16 from the center point If the inner elliptical shape was 6″ inside that , reduce both the Major & Minor axis by 12″ ( 6″ per ‘side’ )
      New Major axis is – 48-12 = 36″
      New Minor Axis is 24-12 = 12″
      Mathematically Foci are 17″ from center –
      But you don’t need to calculate it – just use the method in the video.

      If using the trammel method , I suggest using very small pivot points – When done with a trammel , the ellipse is not perfectly doesn’t bother me that it is e not perfect , some folks might pick it up , I think its part of the craft – Have fun –


  31. Ockham

    Hi Mike,
    Thanks for the GREAT video. I’ve used your trammel method to make several ellipses, including one that was 80″ x 6″ in a double 2×12 beam! It came out great and using your method was super easy to calculate. I used a sheet of 4×8 plywood for the jig-base, some nice straight 2x4s for the guides, and spare Hammond Leslie 1″ roller bearings I had laying around for the trammel rollers. For the trammel arm itself, I used a left over 48″ aluminum ruler from an old gyp board T-square. I used a scrap of flat aluminum square plate I had hanging around as a router base to which the trammel could easily mount with a couple tapped holes. I also drilled bearing mounting points in the trammel itself and simply tapped those holes for 1/4-28 bolts. (The old gyp board T-square has dimensional markings on it which made the distances really easy to calculate.) I used a typical 1/4″ carbide bit and cut the 2×12 separately, each in several passes. They came out so exactly that it was impossible to detect that they had been cut separately. Thanks to your great explanation, it went totally smooth on every set of ellipses I made – now I’m the hero on the jobsite (my home) and I have you to thank!

  32. Mike Sloggatt

    Paul – – But a 12 inch cylinder goiing thru an object at 45 degrees will produce a Major axis of 17″ ( 16.97″ exactly) and a minor axis of 12″ therefor the center of the boring would be 8.5″ to the center of the major axis and 6″ to the center of the minor axis – Not sure ? go buy a 12″ sonotube and cut it at a 45 degree angle – run two strings to represent the 2 axis & you will see the center point



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