Bicorn
**BICORN**

*Balmoral Software*

The bicorn is a
y-symmetric closed curve S with a shape reminiscent of the
bicorne hat worn by Napoleon and
others. It is defined by the parametric equations

The curve follows a clockwise path as t increases, starting from its maximum
ordinate at (x(0),y(0)) = (0,1), reaching its maximum abscissa
(1,0) at t = π/2, then the local height maximum (0,1/3) at
t = π and the minimum abscissa (-1,0) at
t = 3π/2 before returning to the y-axis. Since two ordinate
minima occur on the x-axis, the bicorn is non-convex by the
multiple local extrema test,
and the width x height of its bounding rectangle is 2 x 1.
### Metrics

The perimeter of the bicorn is approximately 5.056530 and its area is

Its
centroid
ordinate is

The bounding rectangle for inconics is delimited by the lower y-intercept
y(t) = 1/3 occurring when t = ±Arccos(2/3) and
the corresponding outer abscissas
### Convex Hull

The convex hull is created simply by connecting the two cusps of the bicorn with
a horizontal line segment of length 2. We have
x'(t) = cos(t)
y'(t) = sin(t){1 - 4/[cos(t) - 2]^{2}},

so by (L1), the perimeter of the convex
hull is

which is about 3% shorter than that of the bicorn.
By (A1), the area A of the convex hull is
determined by integrating over the same subinterval of T:

which is about 62% larger than the area of the bicorn.
By (C1), the centroid ordinate of the
convex hull is

### Incircle

A candidate incircle is located between the ordinate extrema of S on the y-axis,
and so has radius R = 1/3 and center ordinate d = 2/3.
For verification, we have

### Inellipse

Using z = 1/3 in Lemma E,
d/dt x(t)[y(t) - z] = d/dt sin(t){cos^{2}(t)/[2 - cos(t)] - 1/3}

has a zero at

The corresponding coordinates are

We then have

For verification,

### Circumellipse

Using z = 1 in Lemma E,
d/dt x(t)[y(t) - z] = d/dt sin(t){cos^{2}(t)/[2 - cos(t)] - 1}

has a zero at t* = π/2. The corresponding coordinates are
x* = 1
y* = 0,

from which we have

For verification,

### Circumcircle

Since the abscissa extrema are on the x-axis and the bounding rectangle is wider
than it is tall, a candidate for the circumcircle has radius 1 and center at the
origin. For verification, we have

### Summary Table

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

Incircle | R = 1/3 | 2.094395 | 0.349066 | (0,0.666667) |

Inellipse | a = 0.470553 b = 0.289332 | 2.421315 | 0.427715 | (0,0.622665) |

Bicorn | Width: 2 Height: 1 | 5.056530 | 0.746456 | (0,0.542011) |

Convex hull | | 4.90528 | 1.210026 | (0,0.386245) |

Circumellipse | | 5.825171 | 2.418399 | (0,0.333333) |

Circumcircle | R = 1 | 6.283185 | 3.141593 | |

| |

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