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x(t) = 1 + sin(t)The curve is traced out clockwise as t increases, starting from the pointy(t) = [1 + sin(t)]cos(t), 0 ≤ t < 2π

x'(t) = cos(t)so by (L1), the perimeter of the piriform curve isy'(t) = cos(2t) - sin(t),

and its area is π. By (C1), its centroid abscissa is

(Note: WolframAlpha has omitted the division by π.)

x''(t) = -sin(t)so by (X1), the piriform curvature changes sign at the origin sincey''(t) = -2sin(2t) - cos(t),

cos(t)[-2sin(2t) - cos(t)] = -sin(t)[cos(2t) - sin(t)]is satisfied for t = 3π/2. Equation (X2) is evaluated with respect to the origin as

which has a solution in the upper half-plane at t = 0, so the convex hull line segments extend between the origin and the points (1,±1). 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 piriform curve.

The line segments of the convex hull create an isosceles triangle with area 1. The centroid abscissa of this triangle is the average of its vertex abscissas, or 2/3. 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 3.237463, about 3% larger than that of the piriform curve. The centroid abscissa of the convex hull is the weighted average

Region Area Centroid abscissa Product Triangle 1 2/3 2/3 Remainder A = 2/3 + π/2 (4/3 + 5π/8)/A 4/3 + 5π/8 Total 5/3 + π/2 2 + 5π/8

which is 1/(20 + 6π) to the left of the centroid of the piriform curve itself, exactly the same distance as the corresponding value for the teardrop curve.

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

d/dt [x(t) - z]y(t) = d/dt -coshas a zero at t* = 0. The corresponding coordinates are^{3}(t)

x* = y* = 1,from which we have

For verification, we have

d/dt [x(t) - z]y(t) = d/dt [1 + sin(t)]has a zero at The corresponding coordinates are^{2}cos(t)

from which we have

For verification, we have

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

FigureParametersPerimeter Area Centroid Incircle R = 5.073986 2.048748 (1.192450,0) Inellipse 5.825171 2.418399 (1.333333,0) Piriform curve 7.04249 3.141593 (1.25,0) Convex hull 7.018917 3.237463 (1.224260,0) Circumellipse 8.029365 5.007134 (1.111111,0) Circumcircle R = 8.265940 5.437191 (1.315565,0)

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

The piriform curve adjoins and is the same height as the cardioid:

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