6. Remarkable Four-Centre Oval Shapes

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.