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r(t) = sinThe curve is traced around and through the origin in a counterclockwise direction, starting from its maximum abscissa point (1,0) at^{3}(t) + cos^{3}(t), 0 ≤ t < π

r(π/4 + t) = r(π/4 - t),the lima bean curve is symmetric with respect to the line y = x, and is more amenable to analysis after being rotated clockwise by π/4, producing an x-symmetric curve S:

The polar function of the rotated curve is:

rand the corresponding coordinate functions are_{S}(t) = sin^{3}(t + π/4) + cos^{3}(t + π/4), 0 ≤ t < π,

x(t) = rThe minimum abscissa point of S is the origin at t = π/2; and its two maximum abscissa points are at_{S}(t)cos(t) = [sin^{3}(t + π/4) + cos^{3}(t + π/4)]cos(t)y(t) = r

_{S}(t)sin(t) = [sin^{3}(t + π/4) + cos^{3}(t + π/4)]sin(t)

The bounding rectangle for inconics is delimited by the ordinate extrema and the cusp

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

which is less than 1% shorter than that of the lima bean curve.

The line segment of the convex hull creates an isosceles triangle with the origin, having area

as shown in blue in the diagram above. The centroid abscissa of this triangle is the average of its vertex abscissas, or 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:

The area of the convex hull is 1.019853, about 4% larger than that of the lima bean curve.

RegionArea Centroid abscissa Product Triangle Remainder Total

The centroid abscissa of the convex hull of S is the weighted average

After a counterclockwise rotation by π/4, the centroid's position with respect to the lima bean curve is (0.308956,0.308956).

After a counterclockwise rotation by π/4, the incircle's center is (1/4,1/4).

d/dt [x(t) - z]y(t) = d/dt (-1/8)sinhas a zero at t* = 1.245619. The corresponding coordinates are^{3}(t)[-2cos(t) - 3cos(3t) + cos(5t)]

x* = 0.201781We then havey* = 0.598499

For verification,

d/dt [x(t) - z]y(t) = d/dt [26sin(2t) + 2sin(4t) - 6sin(6t) + sin(8t)]/64has a zero at t* = π/4. The corresponding coordinates are

We then have

For verification,

and center abscissa For verification,

After a counterclockwise rotation by π/4, the circumcircle's center is (3/8,3/8).

FigureParametersPerimeter Area Centroid Incircle 2.221442 0.392699 (0.25,0.25) Inellipse * a = 0.336884

b = 0.6910873.326054 0.731414 (0.370223,0) Lima bean curve Width: 1.282887

Height: 1.2828873.931703 0.981748 (0.3,0.3) Convex hull 3.911279 1.019853 (0.308956,0.308956) Circumellipse * 4.119019 1.209201 (0.471405,0) Circumcircle 4.891247 1.903835 (0.375,0.375) *: With respect to rotated curve S

The rotated lima bean curve (red) is a member of a group of similarly-shaped figures described on these pages, including (inside to outside) the cardioid, the cochleoid and the Cayley sextic.

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