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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
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:
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
Region Area Centroid abscissa Product Rectangle -1/2 Remainder Total
and For verification, we have
a = 1.290478b = 0.640445
c = -0.254244
d = 0
has a zero when t* is a complicated expression approximated by 1.083243 radians. The corresponding coordinates are
x* = -0.083459from which we havey* = 0.413881,
For verification,
has a zero when t* is a complicated expression approximated by 1.749441 radians. The corresponding coordinates are
x* = -0.862475from which we havey* = -0.174868,
For verification, we have
Figure Parameters Perimeter Area Centroid Incircle (tail) 0.774771 0.047768 (-0.876691,0) Inellipse (tail) a = 0.091683
b = 0.2019200.955186 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 = 0.4779083.809772 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 = 0.6404456.239295 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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