All Sides to an Oval

When writing about ovals the first thing to do is to make sure that the reader knows what you are talking about. The word oval has, both in common and in technical language, an ambiguous meaning. It may be any shape resembling a circle stretched from two opposite sides, sometimes even more to one side than to the other. When it comes to mathematics you have to be precise if you don’t want to talk about ellipses, or about non-convex shapes, or about forms with a single symmetry axis. Polycentric ovals are convex, with two symmetry axes, and are made of arcs of circle connected in a way that allows for a common tangent at every connection point. This form doesn’t have an elegant equation as do the ellipse, Cassini’s Oval, or Cartesian Ovals. But it has been used probably more than any other similar shape to build arches, bridges, amphitheatres, churches and windows whenever the circle was considered not convenient or simply uninteresting. The ellipse is nature, it is how the planets move, while the oval is human, it is imperfect. It has often been an artist’s attempt to approximate the ellipse, to come close to perfection. But the oval allows for freedom, because choices of properties and shapes to inscribe or circumscribe can be made by the creator. The fusion between the predictiveness of the circle and the arbitrariness of how and when this changes into another circle is described in the biography of the violin-maker Martin Schleske: “Ovals describe neither a mathematical function (as the ellipse does) nor an arbitrary shape. [...] Two elements mesh here in a fantastic dialectic: familiarity and surprise. They form a harmonic contrast. […] In this shape the one cannot exist without the other.” (our translation from the German, [15], pp. 47–48).

In this chapter we sum up the well-known properties of an oval and add new ones, in order to have the tools for the various constructions illustrated in Chap. 3 and for the formulas linking the different parameters, derived in Chap. 4. All properties are derived by means of mathematical proofs based on elementary geometry and illustrated with drawings.

We can now show how most ovals can be drawn with ruler and compass if enough parameters are known. And after having mastered the basic ones, one is ready to tackle more complicated problems, such as the two Frame Problems or the Stadium Problem, which we will discuss in the second part of this chapter. All the following constructions have been made with freeware Geogebra and most of them are linked through the website www.mazzottiangelo.eu/en/pcc.asp, as described further. The much used Connection Locus—the CL—has just been defined in Chap. 2. All constructions in this chapter are general purpose constructions, in the sense that any combination of parameters, constrained within some values, can be chosen. Further combinations of parameters then those illustrated here are listed in the Appendix, for the reader to try out and find the corresponding constructions. Selected oval forms will then be presented in Chap. 6.

Many tools for drawing a polycentric oval subject to geometrical or aesthetical constraints have been presented in Chap. 3. In this chapter we derive formulas which on the other hand allow to calculate all important parameters when the value of three independent ones is known, as well as the limitations the given parameters are subject to. These formulas allow for a deeper insight in the properties of any oval, as will be shown on the chosen forms of Chap. 6. Cases included are numbered consistently with the construction numbers of Chap. 3. Also included is the proof of the formulas yielding the solution to the frame problem presented in Sect. 3.3 as well as formulas for the length of an oval and for the area surrounded by it. The final section is devoted to concentric ovals and their properties.

Once formulas describing parameters or other characteristics of an oval have been derived (in Chap. 4), calculus can be used to consider them as functions of a certain quantity and see how they vary. In this short chapter as an example we present here the solution to the problems of minimising the difference and the ratio of the radii of a simple oval with given axes a and b. The first of these two problems being suggested by Edoardo Dotto.

There are ∞³ simple ovals of given axis lines, and since any of them has infinite versions with the same shape (you just need to multiply all parameters except p and β by the same positive number), we can say that there are ∞² different shapes of four-centre ovals. What we have dealt with in Chaps. 3–6 are general purpose constructions and formulas, where you can end up with any oval varying the parameter values. This chapter is dedicated to particular oval shapes, chosen either for their use in architecture and/or for their geometric elegance. Parameter formulas in Chap. 4 help in investigating the properties of these specific oval shapes. If, on the other hand, one fixes some kind of relation between two parameters, or fixes a value for p or for β, then one gets an oval shape family of ∞¹ shapes with something in common. Cases in the literature include Serlio’s first construction (see [13]), the oval shape families by Gridgeman and Franchi (used to fit a given ellipse—see [9] for references and comments), those by Bianchi, Kitao and the third study by Hewitt (see [10] for references and comments) as well as studies by Zerlenga (in [16]) of variations of single parameters and borderline cases of Bosse’e constructions. In this book both the minimal radius ratio and the minimal radius difference can be considered examples of oval shape families.

We have always been surprised, fascinated and intrigued by Borromini’s architecture, especially if we consider how carefully his drawings were made, whether they were perspective sketches made with a pen or orthogonal projections made with a pencil, where shapes, dimensions and proportions were displayed. Graphite, “which had been used since the last thirty years of the sixteenth century” (Joseph Connors) is what enabled Borromini to make clear, precise and detailed representations, increasing his control over the project. That is why it makes sense to ask which drawing was the basis for the building of the dome, or at least which elements or data he needed for the purpose. His precision and his obsessive control of the project and of its realisation, as well as his marginal notes on the drawings where he illustrates his lines of reasoning, allow, justify and legitimate this study of ours. Aware that we cannot be totally sure of our results, we can suggest a method which can be the basis for further investigations and new hypotheses.

Ovals must have a number of centres which are a multiple of 4. Ovals with more than four centres have been used to align important points in a building and/or to make the oval look more like an ellipse. In this chapter we will extend a property presented in Chap. 2 which allows solving some construction problems for ovals with more than four centres, and illustrate a procedure to draw eight-centre ovals once two of the three radii are known. We will then illustrate the procedure that, according to Trevisan, may have been adopted for the ground plan of the Colosseum, where one four-centre oval and more eight-centre ovals were used.