r(t) = -2sin(t)/t, t ∈ T,where t is measured in radians. For a simple curve describing the outer envelope of the cochleoid, T is restricted to the interval [-π,π). The scaling factor -2 is used for comparison with the cardioid.
The curve is traced out in a counterclockwise direction, starting from the cusp
at the origin when
so by (L2), the perimeter of this portion of the cochleoid is
By (A2), the corresponding area is
By (C2), the centroid abscissa of this portion of the cochleoid is
The bounding rectangle for inconics is delimited by the cusp at the origin.
which is about 5% shorter than that of the cochleoid.
The line segment of the convex hull creates an isosceles triangle with the origin, having area
(0.434467)(0.541786) = 0.235388as shown in blue in the left diagram below. The centroid abscissa of this triangle is the average of its vertex abscissas, or
By (C2), the associated centroid abscissa is
The convex hull component metrics can be summarized as follows:
The area of the convex hull is about 2% larger than that of the cochleoid. The centroid abscissa of the convex hull is the weighted average
Region Area Centroid abscissa Product Triangle 0.235388 0.289645 0.068179 Remainder A1 = 5.53872 -4.21048/A1 -4.21048 Total 5.774108 -4.142301
d/dt [x(t) - z]y(t) = d/dt 4sin3(t)cos(t)/t2has a zero at t* = -0.671886. The corresponding coordinates are
x* = -1.45016We then have
y* = 1.15335
d/dt [x(t) - z]y(t) = d/dt [-2sin(t)cos(t)/t + 2]2sin2(t)/thas a zero at t* = -π/2. The corresponding coordinates are
x* = 0We then have
y* = 4/π
Figure Parameters Perimeter Area Centroid Incircle R = 1 6.283185 3.141593 (-1,0) Inellipse a = 0.966773
b = 1.331774
7.266693 4.044873 (-0.966773,0) Cochleoid Width: 2.434467
9.056693 5.672606 (-0.736250,0) Convex hull 8.603362 5.774108 (-0.717392,0) Circumellipse 8.812839 6.158400 (-0.666667,0) Circumcircle R = 1.449223 9.105737 6.598122 (-0.621685,0)
The cochleoid (red) is a member of a group of similarly-shaped figures described on these pages, including (inside to outside) the rotated lima bean curve, the cardioid and the Cayley sextic:
Copyright © 2021 Balmoral Software (http://www.balmoralsoftware.com). All rights reserved.