Cayley Sextic

Cayley Sextic 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

Metrics

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:

Region Area Centroid abscissa Product

Triangle 1/4 1/3 1/12

Remainder A₁ = (15π + 44)/16 -(315π + 1016)/(480A₁) -(315π + 1016)/480

Total 15π/16 + 3 -(315π + 976)/480

Region	Area	Centroid abscissa	Product
Triangle	1/4	1/3	1/12
Remainder	A₁ = (15π + 44)/16	-(315π + 1016)/(480A₁)	-(315π + 1016)/480
Total	15π/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

Incircle

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

Inellipse

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,

Circumellipse

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,

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

For verification,

Summary Table

Figure Parameters Perimeter Area Centroid

Incircle R = 9/8 7.068584 3.976078 (-0.875,0)

Inellipse a = 1.098193
b = 1.379682 7.809609 4.760006 (-0.848193,0)

Cayley sextic 8.881262 5.875548 (-0.702073,0)

Convex hull 8.712389 5.945243 (-0.688787,0)

Circumellipse a = 1.356954
b = 1.475361 8.901868 6.289459 (-0.643046,0)

Circumcircle R = 9.155272 6.670104 (-0.603553,0)

Figure	Parameters	Perimeter	Area	Centroid
Incircle	R = 9/8	7.068584	3.976078	(-0.875,0)
Inellipse	a = 1.098193 b = 1.379682	7.809609	4.760006	(-0.848193,0)
Cayley sextic		8.881262	5.875548	(-0.702073,0)
Convex hull		8.712389	5.945243	(-0.688787,0)
Circumellipse	a = 1.356954 b = 1.475361	8.901868	6.289459	(-0.643046,0)
Circumcircle	R =	9.155272	6.670104	(-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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