Cochleoid
**COCHLEOID**

*Balmoral Software*

A cochleoid having its cusp on
the right is a heart-shaped x-symmetric closed curve S with even polar function
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 t = -π. Its minimum abscissa occurs at the
limit point (-2,0) as t → 0. Its maximum abscissas occur at
the points (0.434467,∓0.541786) when t = ±2.246705,
which determine that the cochleoid is non-convex by the
multiple local extrema test. The
extreme ordinates of the cochleoid occur at the points
(-0.621685,∓1.449223) when t = ±1.165561, so the
width x height of its bounding rectangle is 2.434467 x 2.898446.

### Metrics

We have

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.
### Convex Hull

The convex hull is created by connecting the maximum abscissa points with a
vertical line segment of length 1.083572 at x = 0.434467. By
(L2), the perimeter of the convex hull is

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) = 0.235388

as shown in blue in the left diagram below. The centroid abscissa of this
triangle is the average of its vertex abscissas, or
2(0.434467)/3 = 0.289645. By
(A2), the remainder of the convex hull has
area

By (C2), the associated centroid
abscissa is

The convex hull component metrics can be summarized as follows:
**Region** | Area | Centroid abscissa | Product |

Triangle | 0.235388 | 0.289645 | 0.068179 |

Remainder | A_{1} = 5.53872 | -4.21048/A_{1} | -4.21048 |

Total | 5.774108 | | -4.142301 |

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 -4.142301/5.774108 = -0.717392.
### Incircle

A candidate incircle is located on the x-axis between the minimum abscissa -2
and the cusp at the origin, and so has radius R = 1 and center
abscissa c = -1. For verification, we have

### Inellipse

Using z = 0 in Lemma E,
d/dt [x(t) - z]y(t) = d/dt 4sin^{3}(t)cos(t)/t^{2}

has a zero at t* = -0.671886. The corresponding coordinates are
x* = -1.45016
y* = 1.15335

We then have

For verification,

### Circumellipse

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

has a zero at t* = -π/2. The corresponding coordinates are
x* = 0
y* = 4/π

We then have

For verification,

### Circumcircle

The radius of a circle centered on the x-axis and circumscribing S is at least
its maximum ordinate, so a candidate for the circumcircle has radius
R = 1.449223 and center abscissa c = -0.621685. For
verification,

### Summary Table

**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 Height: 2.898446 | 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:

Top Page

Home

Copyright © 2021 Balmoral Software (http://www.balmoralsoftware.com). All
rights reserved.