Piriform Curve

Piriform Curve PIRIFORM CURVE

Balmoral Software

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

Metrics

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:

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

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

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.

Incircle

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

Inellipse

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

Circumellipse

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

Circumcircle

Using the origin as the reference point in Lemma C, we have

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

Summary Table

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)

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:

Relationship with the teardrop curve

See the section on the teardrop curve page.

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