Cardioid
**CARDIOID**

*Balmoral Software*

A cardioid having its cusp on
the right is a heart-shaped x-symmetric closed curve S with polar equation
r(t) = 1 - cos(t), 0 ≤ t < 2π

The curve is traced out in a counterclockwise direction, starting from the cusp
at the origin. Its minimum abscissa occurs at the point (-2,0) when
t = π. Its maximum abscissas occur at the points
when
t = ±π/3, which determine that the cardioid is non-convex
by the multiple local extrema
test. The maximum ordinate of the cardioid occurs at
when t = 2π/3,
so the width x height of its bounding rectangle is
### Metrics

The perimeter of the
cardioid is 8 and its
area is 3π/2. Its
centroid
is the point (-5/6,0). 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
at x = 1/4. We have
r'(t) = sin(t),

so by (L2), the perimeter of the convex
hull is

which is about 3% shorter than that of the cardioid.
The line segment of the convex hull creates an isosceles triangle with the
origin, having area

as shown in blue in the left diagram below. The centroid abscissa of this
triangle is the average of its vertex abscissas, or 1/6. 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 | | 1/6 | |

Remainder | | | |

Total | | | |

The area of the convex hull is 4.765390, about 1% larger than that of the
cardioid. The centroid abscissa of the convex hull is the
weighted
average

### 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 [1 - cos(t)]^{2}sin(2t)/2

has a zero at t* = 4π/5. The corresponding coordinates are

We then have

For verification,

### Circumellipse

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

has a zero at

The corresponding coordinates are

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
and center abscissa
c = -3/4. For verification,

### Summary Table

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

Incircle | R = 1 | 6.283185 | 3.141593 | (-1,0) |

Inellipse | | 6.945153 | 3.763482 | (-0.975684,0) |

Cardioid | | 8 | 4.712389 | (-0.833333,0) |

Convex hull | | 7.794229 | 4.765390 | (-0.821975,0) |

Circumellipse | | 7.955280 | 5.025080 | (-0.782871,0) |

Circumcircle | R = | 8.162097 | 5.301438 | (-0.75,0) |

The cardioid (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
cochleoid and the
Cayley sextic:

The cardioid adjoins and is the same height as the
piriform curve:

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