Cayley Sextic

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The Cayley sextic having its cusp on the right is a heart-shaped closed curve S with polar equation
r(t) = -2cos3(t/3), -pi ≤ t < pi
For a simple curve describing the outer envelope of the Cayley sextic, T is restricted to the interval [-π,π]. The scaling factor -2 is used for comparison with the cardioid. Since r(t) is an even function of t, the Cayley sextic is symmetric with respect to the x-axis.

The curve is traced out in a counterclockwise direction, starting from the cusp (1/4,0) at t = -π. Its minimum abscissa occurs at the point (-2,0) when t = 0. Its maximum abscissas occur at the points (1/2,±1/2) when t = ∓3π/4, which determine that the Cayley sextic is non-convex by the multiple local extrema test. The extreme ordinates of the Cayley sextic occur at the points when t = ∓3π/8, so the width x height of its bounding rectangle is


We have
r'(t) = 2sin(t/3)cos2(t/3),
so by (L2), the perimeter of this portion of the Cayley sextic is
By (A2), the corresponding area is
By (C2), the centroid abscissa of this portion of the Cayley sextic is
The bounding rectangle for inconics is delimited by the cusp at (1/4,0).

Convex Hull

The convex hull is created by connecting the maximum abscissa points with a vertical line segment of length 1 at x = 1/2. By (L2), the perimeter of the convex hull is
which is about 2% shorter than that of the Cayley sextic portion.

The line segment of the convex hull creates an isosceles triangle with the origin, having area (1/2)2 = 1/4, as shown in blue in the left diagram below. The centroid abscissa of this triangle is the average of its vertex abscissas, or 1/3. 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:
RegionAreaCentroid abscissaProduct
RemainderA1 = (15π + 44)/16-(315π + 1016)/(480A1)-(315π + 1016)/480
Total15π/16 + 3-(315π + 976)/480
The area of the convex hull is about 1% larger than that of the Cayley sextic. The centroid abscissa of the convex hull is the weighted average


A candidate incircle is located on the x-axis between the minimum abscissa -2 and the cusp (1/4,0), and so has radius R = 9/8 and center abscissa c = -7/8. For verification, we have


Using z = 1/4 in Lemma E,
d/dt [x(t) - z]y(t) = d/dt 4sin(t)cos(t)cos6(t/3) + sin(t)cos3(t/3)/2
has a zero at t* = -0.707452. The corresponding coordinates are
x* = -1.39729

y* = 1.19484

We then have
For verification,


Using z = -2 in Lemma E,
d/dt [x(t) - z]y(t) = d/dt 4sin(t)cos6(t/3)cos(t) - 4sin(t)cos3(t/3)
has a zero at t* = -1.598518. The corresponding coordinates are
x* = 0.035429

y* = 1.2777

We then have
For verification,


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 For verification,

Summary Table

IncircleR = 9/87.0685843.976078(-0.875,0)
Inellipsea = 1.098193
b = 1.379682
Cayley sextic8.8812625.875548(-0.702073,0)
Convex hull8.7123895.945243(-0.688787,0)
Circumellipsea = 1.356954
b = 1.475361
CircumcircleR = 9.1552726.670104(-0.603553,0)

The Cayley sextic (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 cochleoid:

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