Cardioid

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

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)]²sin(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 sin³(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)

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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