Hügelschäffer Egg

The Hügelschäffer egg is the closed oval component of the cubic curve

(1)

where , , and are positive real numbers, with . The closed oval lies in . The cubic curve also has an unbounded real component with (Petrović and Malešević 2023).

The name honors the German engineer Fritz Hügelschäffer, who introduced the circle construction in 1944 (Petrović and Malešević 2023). It generalizes the construction of an ellipse using two concentric circles. Moving the center of the second circle along the common horizontal axis breaks the left-right symmetry and produces an egg-shaped oval. Thus the construction gives a family of shapes, rather than a single fixed egg profile. Petrović and Malešević (2023) describe its origin in aeronautical design and its use for smooth fuselage cross-sections. The name therefore describes the outline rather than a requirement to reproduce a particular bird's egg.

The circle construction illustrated above starts with and , with the second center labeled . For a point on the first circle, take on the second circle where the ray from through meets it. The point traces the closed component (Schmidbauer 1948, Petrović and Malešević 2023). The gray horizontal and vertical line segments transfer the two coordinates from and to . Equivalently, that component has the parametric equations

			(2)
			(3)

for .

Writing

(4)

makes both real components explicit. At the equation factors as , a line together with a parabola.

In the usual egg-shaped range , the endpoints are and , and the maximum width occurs at . Consequently, controls the width-to-length ratio, while controls the asymmetry. A small value of gives a flattened profile, but this is a parameter choice rather than an intrinsic feature of the curve. At , the two circles are concentric and the profile is an ellipse. The model has also been applied to measured chicken-egg profiles (Narushin et al. 2020).

The area enclosed by the oval is

(5)

where , and and are the complete elliptic integral of the first kind and complete elliptic integral of the second kind, respectively, in the modulus convention. This follows by integrating the two branches over (Petrović and Malešević 2023).