Return of the plane evolute

Below we consider the evolutes of plane real-algebraic curves and discuss some of their complex and real-algebraic properties. In particular, for a given degree d ≥ 2, we provide lower bounds for the following four numerical invariants: 1) the maximal number of times a real line can intersect the evolute of a real-algebraic curve of degree d; 2) the maximal number of real cusps which can occur on the evolute of a real-algebraic curve of degree d; 3) the maximal number of (cru)nodes which can occur on the dual curve to the evolute of a real-algebraic curve of degree d; 4) the maximal number of (cru)nodes which can occur on the evolute of a real-algebraic curve of degree d. 1. Short historical account As we usually tell our students in calculus classes, the evolute of a curve in the Euclidean plane is the locus of its centers of curvature. The following intriguing information about evolutes can be found on Wikipedia [35]: “Apollonius (c. 200 BC) discussed evolutes in Book V of his treatise Conics. However, Huygens is sometimes credited with being the first to study them, see [17]. Huygens formulated his theory of evolutes sometime around 1659 to help solve the problem of finding the tautochrone curve, which in turn helped him construct an isochronous pendulum. This was because the tautochrone curve is a cycloid, and cycloids have the unique property that their evolute is a cycloid of the same type. The theory of evolutes, in fact, allowed Huygens to achieve many results that would later be found using calculus, see [1, 32].” Notice that [16], originally published in 1673 and freely available on the internet, contains a large number of beautiful illustrations including those of evolutes. Further exciting pictures of evolutes can be found in the small book [31] written about hundred years ago for high-school teachers. Among several dozens of books on (plane) algebraic curves available now, only very few [6,15,26,31] mention evolutes at all, the best of them being [26], first published more than one and half century ago. Some properties of evolutes have been studied in connection with the so-called 4-vertex theorem of Mukhopadhyaya–Kneser as well as its generalizations, see e.g. [9, 30]. Their definition has been generalized from the case of plane curves to that of plane fronts and also from the case of Euclidean plane to that of the Poincaré disk, see e.g. [10]. Singularities of evolutes and involutes have been discussed in details by V. Arnold and his school, see e.g. [1, 2] and more recently in [27,28]. In recent years the notion of Euclidean distance degree of an algebraic variety studied in e.g. [8] and earlier in [5, 18] has attracted substantial attention of the algebraic geometry community. In the case when the variety under consideration is a plane curve, the ED-discriminant in this theory is exactly the standard evolute. In our opinion, this connection calls for more studies of the classical evolute since in spite of more than three hundred years passed since its mathematical debut, the evolute of a plane algebraic curve is still far from being well-understood. Below we attempt to develop some real algebraic geometry around the evolutes of real-algebraic curves and their duals hoping to attract the attention of the fellow mathematicians to this beautiful and classical topic. 2. Initial facts about evolutes and problems under consideration From the computational point of view the most useful presentation of the evolute of a plane curve is as follows. Using a local parametrization of a curve Γ in R, one can parameterize its evolute EΓ as (2.1) EΓ(t) = Γ(t) + ρ(t)n̄(t), Date: October 25, 2021. 2000 Mathematics Subject Classification. Primary 14H50, Secondary 51A50, 51M05.