Egg-Profile Calculator

Name: Egg-Profile - Geometry Calculator
Author: Jürgen Kummer

Calculations at an egg-profile. This is made of two semi-ellipses with equal semi-axes, which are joined together along these semi-axes.
Enter the semi axis and both half heights and choose the number of decimal places. Then click Calculate.

Formulas:

b = 2 \cdot a

h = h_{1} + h_{2}

p \approx \frac{π}{2} \cdot [(a + h_{1}) \cdot [1 + \frac{3 κ^{2}}{10 + \sqrt{4 - 3 κ^{2}}}] + (a + h_{2}) \cdot (1 + \frac{3 λ^{2}}{10 + \sqrt{4 - 3 λ^{2}}})]

κ = \frac{a - h_{1}}{a + h_{1}}

λ = \frac{a - h_{2}}{a + h_{2}}

A = \frac{π}{2} \cdot a \cdot h

pi:

π = 3.141592653589793 ...

Semi axis, heights, width and circumference have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

There are several different ways to draw an oval. In fact, the word derives from the Latin ovum, meaning egg. Since the name oval was already taken on these pages, the term egg profile was chosen here to designate this specific shape. If a standard chicken egg outline is drawn, the resulting shape corresponds closely to the one presented here. The three-dimensional equivalent of this egg profile is referred to here as an egg-shape. It is the solid of revolution generated by rotating the egg profile around its height line.

Two semi-ellipses with identical semi-axes can be joined together seamlessly, without a kink, because their curved sides meet the semi-axis at a right angle, and a tangent line can be drawn at the point where the two semi-ellipses meet. This creates the impression of a unified, continuous form.
The mathematical challenge regarding semi-ellipses lies in the fact that the perimeter of the curved section cannot be calculated algebraically. For this calculation, an approximation formula developed by Ramanujan is used. For further details, please refer to the ellipse calculator. Calculating the area, however, is relatively straightforward.

The egg profile is axially symmetric with respect to its vertical axis. Apart from this, it possesses no other symmetries. It has no corners and no straight sides. This shape features four distinct points along its perimeter. Two are located at the ends of the semi-axes, and two at the ends of the vertical axis. If these points are connected by straight lines, the resulting figure is a kite. Furthermore, an egg profile fits precisely within a rectangle, such that these four points lie upon its sides and the sides of the rectangle are the tangents at these points.

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