Lune Calculator

Name: Lune - Geometry Calculator
Author: Jürgen Kummer

Calculations with lunes. Lune is a Latin word, the english term for lunes which is also sometimes used is crescents. Lunes occur, when a circle partially eclipses another one. The circles can have different sizes. In addition to the two lunes, there is an intersection of two circular segments. The union of the two circles can also be viewed as a shape resulting from the three previous ones.
Enter the radiuses of the circles and the distance of their two centers. Choose the number of decimal places, then click Calculate. The distance must be between difference and sum of the two radiuses. The shapes will be plotted in a graphic.

Lune shapes:

Formulas:

Δ = \frac{\sqrt{(a + b + c) \cdot (b + c - a) \cdot (c + a - b) \cdot (a + b - c)}}{4}

A_{1} = 2 Δ + a^{2} \cdot \arccos (\frac{b^{2} - a^{2} - c^{2}}{2 a c}) - b^{2} \cdot \arccos (\frac{b^{2} + c^{2} - a^{2}}{2 b c})

Source: Weisstein, Eric W. From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Lune.html

A_{s} = π a^{2} - A_{1}

A_{2} = π b^{2} - A_{s}

A = A_{1} + A_{s} + A_{2}

pi:

π = 3.141592653589793...

Radiuses and distance have a one-dimensional unit (e.g. meter), the areas have this unit squared (e.g. square meter).

areas of the lunes A₁ and A₂, section A_s

radiuses and distance, here: a=2.8; b=3.6; c=4

Both the lunes, the intersection of the two circles, and the entire shape have exactly one axis of symmetry if the circles have different sizes. This axis of symmetry passes through the centers of the two circles. For circles of different sizes, there are no further symmetries. However, if both circles have the same size, then there is another axis of symmetry for the intersection and the overall shape, as well as for both lunes together, which passes through the two intersection points of the circles. Then this shape is also point-symmetric about the center, which is the intersection point of the two axes of symmetry. Such a shape is also rotationally symmetric about this point at a rotation of 180 degrees and multiples thereof.
If both circles have the same size, then the three-dimensional equivalent of this figure is the double sphere.

↑ up