Circular Segment
**CIRCULAR SEGMENT**

*Balmoral Software*

Let a circular segment
S of radius 1 be oriented with the origin at the midpoint of the chord
connecting the endpoints of its arc, and with the center of the circle on the
negative y-axis, so that S is symmetric with respect to that axis. If
β < π/2 is one-half the angle subtended by the segment arc,
then the extreme abscissa points of S are (±sin(β),0), the maximum
ordinate point is (0,1 - cos(β)), and the width x height of the bounding
rectangle of S is 2sin(β) x 1 - cos(β).

Let t represent the usual parameter, ranging from 0 to 2π through the
quadrants in counterclockwise order. In Quadrants I and II, the parametric
equations of the segment arc are:
x(φ) = cos(φ)
y(φ) = sin(φ) - cos(β),

where the angle φ is measured from the center of the circle and ranges from
π/2 - β to π/2 + β. We can scale t to
the angular parameter φ as follows:
Range of t | Range of φ |

[0,π] | [π/2 - β,π/2 + β] |

or equivalently,
φ = 2βt/π + π/2 - β

In Quadrants III and IV, the parametric function x(t) of the horizontal chord is
linear in t. Combining definitions, we have

### Metrics

The perimeter of the segment is the arc length of the circular arc (in radians)
plus the chord:
L = 2β + 2sin(β)

The area of S is
A = [2β - sin(2β)]/2

and its centroid ordinate is

*Specific Example*

In the remainder of this paper, we use

β = 35° = 7π/36
L = 2.368883

A = 0.141019

y_{c} = 0.072931

Bounding rectangle: 1.147153 x 0.180848

### Incircle

A candidate incircle is located on the y-axis between the ordinate extrema of S,
and so has radius and center ordinate both equal to
R = d = [1 - cos(β)]/2 = 0.090424

For verification, we have

### Inellipse

Using z = 0 in Lemma E,
d/dt x(t)[y(t) - z] = d/dt cos(2βt/π + π/2 - β)[sin(2βt/π + π/2 - β) - cos(β)]

has a zero over [0,π) at t* = 0.682719. The corresponding coordinates are
x* = 0.338539
y* = 0.121800

We then have

For verification,

### Circumellipse

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

For verification,

### Circumcircle

The smallest enclosing circle is one with the chord of S as its diameter, so
R = sin(β) = 0.573576
d = 0

Since the radius of the circumcircle is less than the unit radius of the
segment, the circumcircle has greater curvature and thus encloses S.
### Summary Table

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

Incircle | R = [1 - cos(β)]/2 | 0.568151 | 0.025687 | (0,0.090424) |

Inellipse | a = 0.390911 b = 0.081200 | 1.649400 | 0.099720 | (0,0.081200) |

Circular segment | Width: 2sin(β) Height: 1 - cos(β) | 2.368883 | 0.141019 | (0,0.072931) |

Circumellipse | a = b = | 2.765929 | 0.250860 | (0,0.060283) |

Circumcircle | R = sin(β) | 3.603887 | 1.033552 | |

Top Page

Home

Copyright © 2021 Balmoral Software (http://www.balmoralsoftware.com). All
rights reserved.