Trammel of Archimedes

Back in the summer I posted a method for drawing the type of foreshortened ("squashed") circles one needs when drawing circular rooms on axonometric dungeon maps.

Unfortunately it's a bit math heavy and slow, and +Nate McD was quick to point out that if your axes are truly isometric (most of my maps are actually dimetric), there's an approximation with a pair of compasses that's much faster.

Recently, however, I needed to draw this:

If you squint, you can see that it's built out of several foreshortened circles.  The problem is that perspective basis isn't isometric, the foreshortening is much steeper than that.  This rules out compass approximations, which start to look wonky at steep angles.

Making foreshortened circles is trivial in Photoshop or Illustrator, but I was working on paper, old-school style.

Enter the Trammel of Archimedes!

This method is pretty simple, once you get your head around it, it's exact (not an approximation), and it can be used for any degree of foreshortening.  It's fairly quick, but you do need to be willing to draw a freehand curve through a series of dots.

Using the Trammel

First, you figure out how big you want your ellipse to be.  Assuming it's wider than it is tall, the width is the 'major axis', and the height is the 'minor axis'. Divide each of those values in half, which gives you the 'semimajor' and 'semiminor' axes.  I've illustrated as green and blue lines, respectively.

Lay these distances out on your ruler, but in a particular way.  Starting at zero (blue circle), go up your ruler the length of your semi-major axis (green), and make a mark (shown by the red circle).

Then, come back towards zero the length of your semi-minor axis, and make a second mark (shown by the green circle).

Now, here's the neat trick.  If you place the tip (blue) of your ruler on the minor axis, and your second mark (green) on the major axis, the first mark (red) will tell you where to draw the ellipse.

Now, work your trammel: the tip (blue) slides up and down the minor axis, and the middle point (green) slides back and forth along the major axis. As you move it, make tick marks at the outer mark (red). These will trace out your ellipse:

It takes a bit of practice to get used to sliding your ruler around (it can be helpful to have a second ruler to slide the tip against), but in no time you'll have traced out your

You'll need to freehand the ellipse itself, but that just adds charm. For the Zentac Dreadnought, I used this technique for two of the eight ellipses, and then used a ruler to translate them up and down by hand:

You still need to draw the rest of the owl, of course, but it's a lot easier once you have solid elliptical guides to work from.