Piriform Curve

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The piriform curve is a knob-shaped x-symmetric closed curve with parametric equations
x(t) = 1 + sin(t)

y(t) = [1 + sin(t)]cos(t), 0 ≤ t < 2π

The curve is traced out clockwise as t increases, starting from the point (x(0),y(0)) = (1,1). Its extreme abscissas occur at (2,0) when t = π/2 and at the origin when t = 3π/2. The maximum ordinate of the piriform curve occurs at when t = π/6, so the width x height of its bounding rectangle is


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

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

so by (L1), the perimeter of the piriform curve is
and its area is π. By (C1), its centroid abscissa is
(Note: WolframAlpha has omitted the division by π.)

Convex Hull

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

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

so by (X1), the piriform curvature changes sign at the origin since
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:
RegionAreaCentroid abscissaProduct
RemainderA = 2/3 + π/2(4/3 + 5π/8)/A4/3 + 5π/8
Total5/3 + π/22 + 5π/8
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
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.


The maximum ordinate of the piriform curve does not define its inradius since the corresponding abscissa 3/2 is too close to the right edge (2,0), so its incircle is constrained by the right edge. Using z = 2 in Lemma C, we have
and R = |c - z| = . For verification, we have


Using z = 2 in Lemma E,
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


Using z = 0 in Lemma E,
d/dt [x(t) - z]y(t) = d/dt [1 + sin(t)]2cos(t)
has a zero at The corresponding coordinates are
from which we have
For verification, we have


Using the origin as the reference point in Lemma C, we have
and R = |c - z| = c. For verification, we have

Summary Table

IncircleR = 5.0739862.048748(1.192450,0)
Piriform curve7.042493.141593(1.25,0)
Convex hull7.0189173.237463(1.224260,0)
CircumcircleR = 8.2659405.437191(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:

Relationship with the teardrop curve

See the section on the teardrop curve page.

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