Hugelschaffer egg curve and surface

TL;DR

This paper explores the geometric properties of Hugelschaffer cubic curves and surfaces, providing explicit formulas for their egg-shaped areas and volumes using elliptic integrals, with applications in engineering and biology.

Contribution

It introduces new formulas for calculating areas and volumes of Hugelschaffer curves and surfaces, enhancing understanding of their geometric properties.

Findings

Explicit area formulas for egg-shaped parts using elliptic integrals

New expressions for volume and surface area of Hugelschaffer surfaces

Demonstrations of practical applications in engineering and biology

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.