Fish Curve

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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


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:

RegionAreaCentroid abscissaProduct
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


Using z = 1 in Lemma C, we have
and For verification, we have


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,


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

Incircle (tail)0.7747710.047768(-0.876691,0)
Inellipse (tail)a = 0.091683
b = 0.201920
Incircle (main lobe)R = 1/23.1415930.785398(0.353553,0)
Inellipse (main lobe)a = 0.722306
b = 0.477908
Fish curve6.4640821.333333(0.141421,0)
Convex hull5.6462551.927596(-0.129731,0)
Circumellipsea = 1.290478
b = 0.640445

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

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