Fish Curve
**FISH CURVE**

*Balmoral Software*

The fish curve is an
x-symmetric ichthyic closed curve S with parametric equations

for t ∈ T = [0,2π). As t increases from 0, S starts from its right-hand
edge at (1,0) and moves through Quadrants I and II, crossing the x-axis at
when t = π/2
and entering Quadrant III before reversing direction at the fin tip and
returning to Quadrant II. It then reverses direction again and continues
counterclockwise through Quadrants II, III and IV back to its starting point.
The ordinate maxima of S occur at the top of the main lobe
when t = π/4
and at the fin tip when
t = 5π/4. Its minimum abscissas also occur at each fin tip, so
S is non-convex by the multiple
local extrema test. The width x height of its bounding rectangle is
### Metrics

The perimeter of the fish curve is
and its area is 4/3. Its
centroid
is
### Convex Hull

The convex hull is created by connecting the extreme ordinate and minimum
abscissa points with two horizontal line segments of length
and one vertical line segment
of length 1. We have

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

which is about 13% shorter than that of the fish curve.
The line segments of the convex hull create a rectangle of area
as shown in blue in the left
diagram below. The centroid abscissa of this rectangle is the average of its
vertex abscissas, or By
(A1) and symmetry, the remainder of the
convex hull has area

By (C1), the associated centroid
abscissa is found by integrating

Since S is defined parametrically and
is traced out in a counterclockwise direction, the usual limits of integration
are reversed.
The convex hull component metrics can be summarized as follows:

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

Rectangle | | | -1/2 |

Remainder | | | |

Total | | | |

The area of the convex hull is 1.927596, about 45% larger than that of the fish
curve. The centroid abscissa of the convex hull is the
weighted
average

### Circumcircle

Using z = 1 in Lemma C, we have

and For verification, we have

### Circumellipse

Lemma E does not produce an ellipse
that fully contains S, so we resort to a numerical search to determine the
optimal circumellipse:
a = 1.290478
b = 0.640445

c = -0.254244

d = 0

### Incircle (main lobe)

The maximum radius of a circle centered on the x-axis and inscribed in S is its
maximum ordinate, so a candidate for the main lobe incircle has radius
R = 1/2 and center abscissa
The candidate circle must be
contained within S, so we require that c - R and c + R
both be within the abscissa range
of the main lobe of S, which is
true. For verification, we have

### Inellipse (main lobe)

Using z = 1 in Lemma E,

has a zero when t* is a complicated expression approximated by 1.083243 radians.
The corresponding coordinates are
x* = -0.083459
y* = 0.413881,

from which we have

For verification,

### Tail

Inconics can also be found for the tail portion of the fish curve, occurring
when t ∈ T = [π/2,3π/2). The left edge of the tail at
(-1,0) occurs at t = π when S revisits the x-axis for the third
time.
### Incircle (tail)

Using z = -1 in Lemma C, we have

### Inellipse (tail)

Using z = -1 in Lemma E,

has a zero when t* is a complicated expression approximated by 1.749441 radians.
The corresponding coordinates are
x* = -0.862475
y* = -0.174868,

from which we have

For verification, we have

### Summary Table

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

Incircle (tail) | | 0.774771 | 0.047768 | (-0.876691,0) |

Inellipse (tail) | a = 0.091683 b = 0.201920 | 0.955186 | 0.058159 | (-0.908317,0) |

Incircle (main lobe) | R = 1/2 | 3.141593 | 0.785398 | (0.353553,0) |

Inellipse (main lobe) | a = 0.722306 b = 0.477908 | 3.809772 | 1.084465 | (0.277694,0) |

Fish curve | | 6.464082 | 1.333333 | (0.141421,0) |

Convex hull | | 5.646255 | 1.927596 | (-0.129731,0) |

Circumellipse | a = 1.290478 b = 0.640445 | 6.239295 | 2.596464 | (-0.254244,0) |

Circumcircle | | 6.854894 | 3.739311 | (-0.090990,0) |

The fish curve (red) is similarly shaped to the
trefoil:

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