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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![]()
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| =![]()
d/dt [x(t) - z]y(t) = d/dt -cos3(t)has a zero at t* = 0. The corresponding coordinates are
x* = y* = 1,from which we have
For verification, we have![]()
d/dt [x(t) - z]y(t) = d/dt [1 + sin(t)]2cos(t)has a zero at
from which we have![]()
For verification, we have![]()
and R = |c - z| = c. For verification, we have![]()
Figure Parameters Perimeter 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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