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The lune is a crescent-shaped, x-symmetric closed curve S bounded by the arcs of two circles of different radii. Here we have taken the cusps of S at the endpoints of a diameter of a unit circle, and 2π/3 as the angle subtended in a larger circle by that line segment:
To find the radius R and center abscissa c of the larger circle, we have
which has solution So that S is traversed in a clockwise direction starting from its left edge (-1,0) at t = 0, its parametric equations are:
The abscissa maxima of the lune are at the two cusps (0,±1), so it is non-convex by the multiple local extrema test. The ordinate extrema are also at the cusps, so the width x height of the lune's bounding rectangle is 1 x 2. The lune crosses the x-axis at -1 and

Convex Hull

The convex hull of the lune is created simply by connecting its two cusps with a vertical line segment of length 2, as shown in blue in the left diagram below. The perimeter of the convex hull is half the circumference of a unit circle plus the line segment, or π + 2. The area of the convex hull is that of a unit semicircle, or π/2. The centroid abscissa of the convex hull is that of a unit semicircle centered at the origin and in the left half-plane, or -4/(3π).


The perimeter of the lune is the semicircumference of a unit circle plus the arc length subtending 2π/3 on a larger circle of radius
so the perimeter of the convex hull is about about 8% shorter than that of the lune itself.

The portion of the unit semicircle that is not part of the lune is the segment of the larger circle, which has area

The centroid abscissa of this segment is
The lune metrics can be summarized as follows:
RegionAreaCentroid abscissaProduct
Convex hullπ/2-4/(3π)-2/3
Lune (by subtraction)
The convex hull has a little more than twice the area of the lune. The centroid abscissa of the lune is is the weighted average
The bounding rectangle for inconics is delimited by the upper x-intercept occurring when and the corresponding outer ordinates


A candidate incircle is centered on the x-axis between the axis crossings of S, and so has radius and center abscissa For verification, we have


Using in Lemma E,
has a zero at t* = π/6. The corresponding coordinates are:
so the inellipse dimensions are
For verification,


The extreme points of the lune create an x-symmetric isosceles triangle with vertices (-1,0) and (0,±1). Using h = -1 and w = 2 in Lemma TE, a candidate for the circumellipse has parameters
For verification,


Since the lune is a portion of a unit circle centered at the origin, its circumcircle is that circle.

Summary Table

IncircleR = 1.3277930.140298(-0.788675,0)
LuneWidth: 1
Height: 2
Convex hull5.1415931.570796(-0.424413,0)
CircumcircleR = 16.2831853.141593

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