Lune
**LUNE**

*Balmoral Software*

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).
### Metrics

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:
**Region** | Area | Centroid abscissa | Product |

Convex hull | π/2 | -4/(3π) | -2/3 |

Segment | | | |

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
### Incircle

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

### Inellipse

Using in
Lemma E,

has a zero at t* = π/6. The corresponding coordinates are:

so the inellipse dimensions are

For verification,

### Circumellipse

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,

### Circumcircle

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

**Figure** | **Parameters** | Perimeter | Area | Centroid |

Incircle | R = | 1.327793 | 0.140298 | (-0.788675,0) |

Inellipse | | 2.572074 | 0.349066 | (-0.769800,0) |

Lune | Width: 1 Height: 2 | 5.559992 | 0.751883 | (-0.628821,0) |

Convex hull | | 5.141593 | 1.570796 | (-0.424413,0) |

Circumellipse | | 5.825171 | 2.418401 | (-0.333333,0) |

Circumcircle | R = 1 | 6.283185 | 3.141593 | |

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