Arbelos
**ARBELOS**

*Balmoral Software*

Consider the arbelos S
with left radius r and right radius 1 - r, 0 < r < 1,
and therefore with enclosing radius 1, oriented so that the center of the
enclosing semicircle is at the origin. Then the centers of the left and right
semicircles are (r - 1,0) and (r,0) respectively. S is asymmetric
if r ≠ 1/2.

So that S is traversed in a counterclockwise direction starting from its right
edge (1,0) at t = 0, its parametric equations can be written:

### Metrics

The perimeter of the arbelos is the sum of the semicircumferences of circles of
radius 1, r and 1 - r, or 2π, the perimeter of a unit circle
(the answer from
WolframAlpha
is in error by a factor of 2). Its area is that of a unit semicircle, less the
areas of the two smaller semicircles, or r(1 - r)π, which ranges
from 0 to π/4.
Using the formula for
the centroid of a semicircle, the arbelos metrics can be summarized as follows:

**Region** | Area | Centroid | Product |

Unit semicircle | π/2 | (0,4/(3π)) | (0,2/3) |

Left semicircle | r^{2}π/2 | (r - 1,4r/(3π)) | (r^{2}(r - 1)π/2,2r^{3}/3) |

Right semicircle | (1 - r)^{2}π/2 | (r,4(1 - r)/(3π)) | (r(r - 1)^{2}π/2,2(1 - r)^{3}/3) |

Arbelos (by subtraction) | r(1 - r)π | | (r(1 - r)(2r - 1)π/2,2r(1 - r)) |

The centroid of the arbelos is the
weighted
average
(r(1 - r)(2r - 1)π/2,2r(1 - r))/[r(1 - r)π] = (r - 1/2,2/π)

Since the arbelos has three cusps, it is non-convex by the
multiple local extrema test. The
width x height of its bounding rectangle is that of the unit semicircle, 2 x 1.
### Convex Hull

The convex hull of the arbelos is created simply by connecting its three cusps
with a horizontal 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, which is about 18%
shorter than the perimeter of the arbelos. The area of the convex hull is that
of a unit semicircle, or π/2, which is always at least twice the area of the
arbelos itself. The centroid of the convex hull is that of a unit semicircle
centered at the origin, as shown above: (0,4/(3π)). When r is greater than
8/(9π^{2}) + 1/2 = 0.59, the centroid of the convex hull
is outside of the arbelos.
### Circumellipse

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

The parametric equations of the convex hull of S are:

For verification,

### Circumcircle

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

Denote the radius and center of the largest circle inscribed in the arbelos as R
and (c,d), as shown in the diagram above. Since the points of tangency of the
incircle with the left and right semicircles lie along lines connecting their
centers, we have
[c - (r - 1)]^{2} + (d - 0)^{2} = (r + R)^{2}
(c - r)^{2} + (d - 0)^{2} = (1 - r + R)^{2}

By "Surprise No. 7" in [10], a
vertical line from (c,d) to the horizontal base of the arbelos is bisected by
the lower arc of the incircle; or equivalently,
d = 2R

Solving the preceding three equations simultaneously for R, c and d yields

*Specific Example*

In the remainder of this paper, we use r = 2/3 as illustrated in the diagram
above. Then the metrics of the incircle are:

R = 2/7
(c,d) = (3/7,4/7)

and the centroid of the arbelos is (1/6,2/π).
### Inellipse

Since it's evident that an optimal inscribed ellipse will be tilted downwards
toward the smallest semicircle, we can formulate an equivalent problem by
rotating the arbelos itself by a counterclockwise angle -φ to orient its
inellipse in a rectilinear position. This ellipse will be tangent to each of
the smaller semicircles and twice tangent to the enclosing semicircle, and is
y-symmetric. A numerical search can be used to determine the optimal tilted
inellipse:
φ = -25.514°
a = 0.592932

b = 0.236724

c = 0.318006

d = 0.666294

Area = 0.440957

### Summary Table

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

Incircle | R = 2/7 | 1.795196 | 0.256457 | (0.428571,0.571429) |

Inellipse | a = 0.592932 b = 0.236724 | 2.728001 | 0.440957 | (0.318006,0.666294) |

Arbelos | Width: 2 Height: 1 | 6.283185 | 0.698132 | (0.166667,0.636620) |

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

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

Circumcircle | R = 1 | 6.283185 | 3.141593 | |

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