Lune

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)

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

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

Top Page

Home