In the last part I hopefully made it clear why you wouldn’t want to use an actual ellipse when making a real object, curves constructed from multiple fixed radius arcs are much more useful and look just the same.
So there is a traditional draughting technique for drawing a five centred arch that makes quite a good imitation of an ellipse, here is the diagram again:
This is fine, but it is a solution to an old problem of drawing an ellipse on a drawing board with calipers and compass. These days we don’t have this problem, every CAD tool imaginable will let you easily draw ellipses — the problem we have is how to convert that ellipse into fixed radius curves that suit our practical building purposes — for this a bit more control would be useful.
Following is my technique, this starts with an ellipse drawn in CAD drawn to the required height and width:
The next step is to decide how we want to split the curve, this is done by dividing up a circle to one side, in this case I am splitting a quadrant into a 45 degree segment, a 30 degree segment and a 15 degree segment, the final ellipse will be made up from segments with the same angles. While you are there, connect the ends of the radial lines with straight chords and you should have something like this:
The next step is to copy these chords to the ellipse and fit them as best you can without changing the angles, the lengths will be different of course:
Note that I say “as best you can”, not all of these intersections can be exactly on the curve of the ellipse.
Then similarly, copy the radial lines, without changing any angles, these give you the centres for the circular arcs:
That’s it, if you set out a stadium or an arena using this a geometry like this then all the angles between stands will be multiples of 15 degrees and everybody on site will be very very happy (maybe).
Of course you don’t need to use these particular angles, here is the circle quadrant divided into nine equal 10 degree segments:
This is what they look like mapped onto the ellipse:
This is a seventeen centre arch, if you build this probably no one can ever tell that it isn’t actually an ellipse:
Just to complete the set, the simplest ellipse like curve is the three centre arch, here I’m dividing the quadrant into a 60 degree segment and a 30 degree segment:
Here it is with the chords and radial lines mapped onto the same ellipse:
Note that the two chords don’t actually meet on the line of the ellipse, you can’t have everything.
The result is a basic three centre arch. If you care to look, this is obviously not an ellipse, but I still think it looks fine — in some ways it may be more useful than the five centre arch as you get a bit more headroom to the sides:
Three centre arches really are fine, this one is split with the segments at about 67/23 degrees, if it’s good enough for Christopher Wren then it will be good enough for you:
That’s it, there may be another piece in the series showing how to decompose any curve into arcs, the principle is basically the same for pointed arches, ogees, parabolas etc…