Construction of an Oval - the Roman Way

We shall never know how the Romans planned and constructed many of their buildings. For example, this is what A. Hahn says about construction of the Colosseum [Mathematical Excursions, p. 37],

How exactly the Romans laid out the parallel sequence of ovals for the Colosseum has been a matter of scholarly debate. However, it is likely that they were laid out , one by one, as a combination of circular arcs. What is known is that the oval structures of some earlier and smaller Roman amphitheaters were laid out with circular arcs according to the following scheme.

The construction starts with two equal isosceles triangles ABC and ABD. Let's agree that C(O, r) denotes the circle centered a point O with radius r. The oval displayed by the applet consists of four circular arcs FH, HI, IJ, JF. These are arcs in the circles C(B, BF), C(C, CH), C(A, BF), and C(D, CH), respectively.

The applet has 2 controls: 1) point C is draggable, so that you can change the shape of the isosceles triangles; 2) the slider in the lower right corner allows you to change the distance d between B and F.

The resulting shape - oval - is a smooth curve, in the sense that it has neither cusps nor corners. The only places where corners might be suspected are points F, H, I, J. At point P, for example, there is a transition from C(C, BF) to C(D, CH) = C(D, DF). A tangent to a circle at its circumference is perpendicular to the radius at the point of tangency. So that the tangent to C(C, BF) at F is perpendicular to BF. On the other hand, the tangent to C(D, DF) at F is perpendicular to DF. But DF and BF are one and the same line, so that the tangents to C(C, BF) and C(D, DF) at F really coincide.

The curve so obtained is not an ellipse - it consists of four circular arcs. You can observe the difference with an ellipse by checking the box "Conic". Five points on the oval will appear and the unique conic (an ellipse, in fact) passing through these five points. For some positions of C and the value of d, the oval is very close to an ellipse for other positions it is not. (The shape of the ellipse depends also on the positions of the five points on the four arcs.)

References

  1. A. J. Hahn's, Mathematical Excursions to the World's Great Buildings, Princeton University Press, 2012

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