Elliptical Crescent Calculator

Name: Elliptical Crescent - Geometry Calculator
Author: Jürgen Kummer

Calculations at an elliptical crescent. This is a crescent, which isnt't based on two circles, but on two ellipses. Those two ellipsed have a common semi-axis of equal length. The other semi-axes, which are perpendicular to it, have different lengths. The elliptical crescent is formed by the area between each pair of semi-ellipses.
Enter the common semi-axis and the two different semi-axes of the two underlying ellipses. Choose the number of decimal places, then click Calculate.

Formulas:

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

λ = \frac{a - b}{a + b}

μ = \frac{a - c}{a + c}

A = \frac{π \cdot a \cdot (b - c)}{2}

pi:

π = 3.141592653589793...

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

Unlike a crescent, which is formed with two circles, the two ellipses of an elliptical crescent share a common center. If these ellipses were circles, they would be the same size and located at the same point; subtracting one from the other would leave nothing behind. However, the shape of an elliptical crescent results from the difference in their semi-axes. It doesn't matter whether the common semi-axis of the two ellipses is the major or minor one. In the sketch above, it is the major semi-axis. Of the other two semi-axes, the one with the longer length is called the outer semi-axis, and the one with the shorter length is called the inner semi-axis. The circumference is calculated using Ramanujan's second approximation, just like at a standard ellipse. This is the arithmetic mean of the circumferences of both underlying ellipses.

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