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x(t) = -cos(t)The curve follows a clockwise path as t increases, starting from the left cusp aty(t) = sin(t)sin

^{2}(t/2), 0 ≤ t < 2π

x'(t) = sin(t)so by (L1), the perimeter of the teardrop curve isy'(t) = [cos(t) - cos(2t)]/2

and its area is

By (C1), its centroid abscissa is

(Note: WolframAlpha has reversed the coefficients.)

x''(t) = sin(t)so by (X1), the teardrop curvature changes sign at (-1,0) sincey''(t) = sin(2t) - sin(t)/2,

sin(t)[sin(2t) - sin(t)/2] = cos(t)[cos(t) - cos(2t)]/2is 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

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:

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

Region Area Centroid abscissa Product Triangle 1/2 -1/3 -1/6 Remainder A = 1/3 + π/4 (1/3 + π/16)/A 1/3 + π/16 Total 5/6 + π/4 1/6 + π/16

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.

and R = |c - z| = . For verification, we have

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

x* = 0which is the endpoint of a convex hull line segment. We then havey* = 1/2,

For verification,

d/dt [x(t) - z]y(t) = d/dt 2sinhas a zero at t* = Arccos(-2/3). The corresponding coordinates are^{4}(t/2)sin(t)

from which we have

For verification,

FigureParametersPerimeter Area Centroid Incircle R = 3.864786 1.188615 (0.384900,0) Inellipse 3.913233 1.209200 (0.333333,0) Teardrop curve 5.13734 1.570796 (0.25,0) Convex hull 5.127128 1.618732 (0.224260,0) Circumellipse 5.810712 2.503567 (1.111111,0) Circumcircle R = 1 6.283185 3.141593

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:

xThese 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 x_{1}(t) = x(t) + 1 = 1 - cos(t)y

_{1}(t) = 2y(t) = 2sin(t)sin^{2}(t/2)

y^{2}= x^{3}(2 - x)

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