Hugelschaffer egg curve and surface

Abstract

In this paper we consider Hugelschaffer cubic curves which are generated using appropriate geometric constructions. The main result of this work is the mode of explicitly calculating the area of the egg-shaped part of the cubic curve using elliptic integrals. In this paper, we also analyze the Hugelschaffer surface of cubic curves for which we provide new forms of formulae for the volume and surface area of the egg-shaped part. Curves and surfaces of ovoid shape have wide applicability in aero-engineering and construction, and are also of biologic importance. With respect to this, in the final section, we consider some examples of the real applicability of this Hugelschaffer model.

Figures (4)

• Figure 1: Hügelschäffer's construction of an egg curve
• Figure 2: Hügelschäffer's construction of the egg-shaped part and the Petrović construction of the hyperbolic part of a cubic curve
• Figure 3: Areas $\mathcal{A}_1$ and $\mathcal{A}_2$ of the Hügelschäffer egg curve
• Figure 4: A Hügelschäffer surface (top, front, right and axonometric view)