Lima Bean Curve

Balmoral Software

The lima bean curve is a tilted closed curve with polar equation
r(t) = sin3(t) + cos3(t), 0 ≤ t < π
The curve is traced around and through the origin in a counterclockwise direction, starting from its maximum abscissa point (1,0) at t = 0, reaching its closest approach to the origin in the first quadrant at (1/2,1/2) when t = π/4, then its maximum ordinate point (0,1) at t = π/2 and through Quadrant II to the origin at t = 3π/4 before returning through Quadrant IV to its starting point. Its minimum abscissa and ordinate points are (-0.282887,0.621170) and (0.621170,-0.282887), respectively, so its bounding rectangle is a 1.282887 x 1.282887 square.


The perimeter of the lima bean curve is 3.931703 and its area is 5π/16. Its centroid is the point (3/10,3/10).

Rotated Curve

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:
rS(t) = sin3(t + π/4) + cos3(t + π/4), 0 ≤ t < π,
and the corresponding coordinate functions are
x(t) = rS(t)cos(t) = [sin3(t + π/4) + cos3(t + π/4)]cos(t)

y(t) = rS(t)sin(t) = [sin3(t + π/4) + cos3(t + π/4)]sin(t)

The minimum abscissa point of S is the origin at t = π/2; and its two maximum abscissa points are at t = π/6 and t = 5π/6, so S and the lima bean curve are non-convex by the multiple local extrema test. The maximum ordinate point of S is
The bounding rectangle for inconics is delimited by the ordinate extrema and the cusp

Convex Hull

The convex hull is created by connecting the maximum abscissa points with a vertical line segment of length We have
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:
RegionAreaCentroid abscissaProduct
The area of the convex hull is 1.019853, about 4% larger than that of the lima bean curve.

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).


A candidate incircle is located on the x-axis between the origin and the cusp, and so has radius R and center abscissa c both equal to For verification, we have
After a counterclockwise rotation by π/4, the incircle's center is (1/4,1/4).


Using in Lemma E,
d/dt [x(t) - z]y(t) = d/dt (-1/8)sin3(t)[-2cos(t) - 3cos(3t) + cos(5t)]
has a zero at t* = 1.245619. The corresponding coordinates are
x* = 0.201781

y* = 0.598499

We then have
For verification,


Using z = 0 in Lemma E,
d/dt [x(t) - z]y(t) = d/dt [26sin(2t) + 2sin(4t) - 6sin(6t) + sin(8t)]/64
has a zero at t* = π/4. The corresponding coordinates are
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,
After a counterclockwise rotation by π/4, the circumcircle's center is (3/8,3/8).

Summary Table

Inellipse *a = 0.336884
b = 0.691087
Lima bean curveWidth: 1.282887
Height: 1.282887
Convex hull3.9112791.019853(0.308956,0.308956)
Circumellipse *4.1190191.209201(0.471405,0)

*: 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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