I'll try to get a hold of a picture of a truss with an arch. I also refined the theoretical elliptic arch and will try to post sketches and calcs tomorrow.
With modern calculators (especially the programmable graphing calculators) there's no inaccuracy due to rounding. And the risk of error is decreased.
Log Valley Roof : Off topic as this has nothing to do with arches, but it's a good example of why it's good to understand the math.
There was no time to test fit the parts and no room for error. As soon as the roof components were cut, the building was loaded and along with the crew and crane went on a journey by road and barge to the erection site.
The layout and cutting was done on horses sitting safely on the ground. The calcs were executed to five decimal places ... way more accuracy than needed. And the work was done quickly: the lengths and mortises on Valleys, jack rafters, header and ridge were done from a standing start in three days with a $15.00 Sharp scientific calculator.
I don't think this roof could have been done correctly and in only three days without trigonometry. Having said that, I am by no means knocking scale layout as an approach to solving joinery problems. The mortises on the irregular surfaces of logs defy calculation; they were done using jigs and projective geometry.
The same goes for this arch. I think that attempting a scale layout to "fit" an ellipse given the arch dimensions would involve a lot of time consuming hit-and-miss guesswork. The math allows us to find our working points accurately. Fitting the arch to timbers of a given width and length can also be done on paper but calculating the intersections can be onerous ... see my "Arch Layout Notes" link. This part of the job is best achieved with a full scale layout.