Teardrop Curve

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A second-order teardrop curve opening to the right is a non-convex x-symmetric closed curve with parametric equations
x(t) = -cos(t)

y(t) = sin(t)sin2(t/2), 0 ≤ t < 2π

The curve follows a clockwise path as t increases, starting from the left cusp at (x(0),y(0)) = (-1,0). Its extreme abscissas occur at the cusp and at (1,0) when t = π. The maximum ordinate of the teardrop curve occurs at when t = 2π/3, so the width x height of its bounding rectangle is


We have
x'(t) = sin(t)

y'(t) = [cos(t) - cos(2t)]/2

so by (L1), the perimeter of the teardrop curve is
and its area is
By (C1), its centroid abscissa is
(Note: WolframAlpha has reversed the coefficients.)

Convex Hull

We have
x''(t) = sin(t)

y''(t) = sin(2t) - sin(t)/2,

so by (X1), the teardrop curvature changes sign at (-1,0) since
sin(t)[sin(2t) - sin(t)/2] = cos(t)[cos(t) - cos(2t)]/2
is satisfied for t = 0. Equation (X2) is evaluated with respect to the point (-1,0) as
which has a solution in the upper half-plane at t = π/2, so the convex hull line segments extend between the points (-1,0) and (x(π/2),±y(π/2)) = (0,±1/2). These line segments are shown in blue in the left diagram below, and each has length By (L1), the perimeter of the convex hull is:
which is less than 1% shorter than that of the teardrop curve.

The line segments of the convex hull create an isosceles triangle with its base on the y-axis, having area 1/2. The centroid abscissa of this triangle is the average of its vertex abscissas, or -1/3. By (A1), the remainder of the convex hull has area

By (C1), the associated centroid abscissa is
The convex hull component metrics can be summarized as follows:
RegionAreaCentroid abscissaProduct
RemainderA = 1/3 + π/4(1/3 + π/16)/A1/3 + π/16
Total5/6 + π/41/6 + π/16
The area of the convex hull is 1.618732, about 3% larger than that of the teardrop curve. The centroid abscissa of the convex hull is the weighted average
which is 1/(20 + 6π) to the left of the centroid of the teardrop curve itself, exactly the same distance as the corresponding value for the piriform curve.


The maximum ordinate of the teardrop curve does not define its inradius since the corresponding abscissa 1/2 is too close to the right edge (1,0), so its incircle is constrained by the right edge. Using z = 1 in Lemma C, we have
and R = |c - z| = . For verification, we have


Using z = 1 in Lemma E,
d/dt [x(t) - z]y(t) = d/dt -sin3(t)/2
has a zero at t* = π/2. The corresponding coordinates are
x* = 0

y* = 1/2,

which is the endpoint of a convex hull line segment. We then have
For verification,


Using z = -1 in Lemma E,
d/dt [x(t) - z]y(t) = d/dt 2sin4(t/2)sin(t)
has a zero at t* = Arccos(-2/3). The corresponding coordinates are
from which we have
For verification,


Since the abscissa extrema are on the x-axis and the bounding rectangle is wider than it is tall, a candidate for the circumcircle has radius R = 1 and center at the origin. For verification, we have

Summary Table

IncircleR = 3.8647861.188615(0.384900,0)
Teardrop curve5.137341.570796(0.25,0)
Convex hull5.1271281.618732(0.224260,0)
CircumcircleR = 16.2831853.141593

The teardrop curve (red) is a member of a group of similarly-shaped figures described on these pages, including (inside to outside) the Tschirnhausen cubic, the right strophoid, the Trisectrix of Maclaurin and the piriform curve:

Relationship with the piriform curve

The teardrop curve coincides with the piriform curve after translating its abscissas to the right by 1 and doubling its ordinates. The transformed coordinate functions are
x1(t) = x(t) + 1 = 1 - cos(t)

y1(t) = 2y(t) = 2sin(t)sin2(t/2)

These coordinate functions cannot be directly compared with the corresponding ones for the piriform curve since the rates of travel along the path T (if t is considered a time variable) vary between the two curves. However, the path itself is the same for both curves since x1(t), y1(t) satisfy the cartesian form of the piriform curve:
y2 = x3(2 - x)

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