All of the references give a method of drawing an ellipse using the two foci and some string. I have found this method suitable only for drawing jelly bean or kidney shapes. Keeping the tension and hence the length of the string constant is problematic.
This method, based on the principle of the trammel, is IMHO much preferable. It requires a little more prep than the kidney method; but the results are uniformly true, exactly and rapidly repeatable, and the apparatus (constructed from scrap) can be used to draw any number of exactly concentric ellipses.
There are two parts; the ellipstick and the track block.
Since I was doing this fairly often; I kept a stock of ellipstick blanks on hand. 8”- 24” lengths of scrap, with a scant 1/4” hole centered on the stock at 1/2” from one end, with the center lines chased around the stick; and the mid-lines down its length. Best from the drill press. You’ll be holding the pencil 3/4” away from the mark and out of plumb affects precision.
You can do the same thing with a notch in the end that will let you position the center of the pencil line at the exact end of the ellipstick. The object is to make the transfer of the critical measurements as simple and reliable as possible.
Next, a little ‘rithmatic (ambition, distraction, uglification and derision). An ellipse is defined by its greatest extents, its major and minor axes (1 axis, 2 axes). An ellipse also has two foci (1 focus, 2 foci) Foci are the result of a mystical union between ½ the major axis; as an arc, struck from either terminus of the minor axis. Luckily we don’t have to foc with either of them.
All we need to know is:
“what is ½ the major axis” and “what is ½ the minor axis.” If you’ve got numbers that are easy to measure, cool. If the numbers are wild; use the bisector gag to obtain compass settings. (see trammel points)
Transfer these to the ellipstick (each measured from the center of the pencil hole.)
Drive a 4d finish nail straight into the ellipstick at each of those marks and clip off the heads leaving the pins just scant of 3/8” proud. Sweeten the ends with a kiss of a mill file.
Mark out the major axis on the work or layout. Bisect that distance with a perpendicular on which you locate the minor axis.
Next, you need to construct the track block.
This wants to begin as a true square piece of ¾ scrap, with no voids, that is twice the difference between ½ the major and ½ the minor axes +1/8” on a side.
This done, lower the tablesaw blade so it only takes a 3/8”deep kerf and re-set the fence so the blade splits the two mid-lines of the square. You should end up with something that looks like this.
With your best try square mark the centers of the kerfs at the bottom of the block. Line up the marks, under the kerf exits, with the major and minor axis lines. When all four are aligned; you’re positioned perfectly. Fasten the block in place. With one pin in each kerf and a pencil in the hole; strike the ellipse using the saw kerfs to guide the nails.
This simple square block will work just fine where the ratio between the major and minor axes is 2:1 or less. (You can see it on the ellipstick or divide the lesser into the greater to get the exact ratio.)
As this ratio increases (the ellipse get “flatter”) the corners of the block begin to intrude on your intended line. It is a simple matter to trim these to suit.
When the ratio climbs beyond 3:1 the difference becomes greater than ½ the minor axis; and more drastic action is required.
This can be readily faired with some pins and a flexible strip.
Or, you can make the offending bits of track removable/replaceable.
Fit them back when needed and remove them when they’re in the way.
Another trick with an ellipstick
You can also use the ellipstick to strike ¼ of an ellipse against a flat square; and by repetition obtain a full ellipse. There are situations where this is simpler.
Shorten the nail heads to just scant the thickness of the square (no scratches)
Lay the outside edges along one of the four right angles produced by the intersection of Maj and Min. The ellipstick with the nail-heads riding against the square will describe ¼ of an ellipse.
The orbits of concentric ellipses are parallel. The same block in the same position will guide any concentric ellipse.
With this device, as you increase the distance between the guide pins and the marking point you also change the ratio; making the ellipse less flat.
This technique can be scaled up very effectively using trammel points, or a custom built track and stick for very large radii.
You can also trammel a router or bayonet saw. Due to the forces involved it is best to add some sort of shoe to the pins as things get larger and electrified. There are a number of commercially made router jigs that work on this same principle.
I built this whiz-bang out of basswood as a kid’s toy to sell on eBay. The shoes and track are loose sliding dovetails.
Saw this in “Fine Woodworking On Proven Shop Tips” (in the previous millennium – this link is kindling?); but have not been able to find it on their website. It was sent in by Thomas Baird of Woodland, CA; attributed only to a Danish carpenter of his acquaintance. I’ve snootied the language up a bit.
Method to draw a circular arc of known displacement and chord length; where there is no access to the center.
Place a nail at the beginning, the end, and the point of maximum displacement of the curve.
Erect a parallel to the chord at the point of maximum displacement. Place a fourth nail on that line.
Fix together two straight pieces of rigid scrap so that one lies against the point of maximum displacement and the end of the curve, and the other lies parallel to the chord of the curve at the point of maximum displacement.
Two fasteners to keep the sticks from scissoring.
Remove the fourth nail. A pencil held at the crook between the two pieces of scrap will describe a uniform curve (circular arc segment) when the edges are guided against the nails.
Flip the sticks over to mark the other half of the curve.
If the displacement is greater than the radius of the circle. . .
the technique draws a bubble (and may require automatic self-lengthening sticks to fit on this page.)