Bifolium
**BIFOLIUM**

*Balmoral Software*

The bifolium is a two-lobed
y-symmetric closed curve with polar equation
r(t) = sin(t)cos^{2}(t), 0 ≤ t < π

The curve is traced out in a counterclockwise direction, starting from its
minimum ordinate point at the origin when t = 0 and reaching its
maximum abscissa point at
t = π/6. It reaches the first of its maximum ordinate points at
t = π/4 before returning to the origin at
t = π/2 and repeating the pattern symmetrically in the left
half-plane. Since the bifolium has two maximum ordinate points at
(±1/4,1/4), it is non-convex by the
multiple local extrema test. The
width x height of its bounding rectangle is
which has an aspect ratio of
about 2.6.
The lobes of the bifolium (red) are spread farther apart than those in the upper
half-plane of the comparably-sized four-leaved
rose
(black):

### Metrics

The area of the
bifolium is π/32 and its perimeter is
1.788887. Its
centroid
is the point (0,1/8).
### Convex Hull

The convex hull is created by connecting the maximum ordinate points with a
horizontal line segment of length 1/2 at y = 1/4. We have
r'(t) = 3cos^{3}(t) - 2cos(t),

so by (L2), the perimeter of the convex
hull is

which is about 14% shorter than that of the bifolium.
The line segment of the convex hull creates an isosceles triangle with the
origin, having an area of (1/4)^{2} = 1/16, as shown in
blue in the left diagram below. The centroid ordinate of this triangle is the
average of its vertex ordinates, or 1/6. By
(A2), the remainder of the convex hull has
area

By (C2), the associated centroid
ordinate is

The convex hull component metrics can be summarized as follows:
Region | Area | Centroid ordinate | Product |

Triangle | 1/16 | 1/6 | 1/96 |

Remainder | A = (4 + 3π)/192 | | |

Total | (16 + 3π)/192 | | |

The area of the convex hull is 0.132421, about 1/3 larger than that of the
bifolium. The centroid ordinate of the convex hull is the
weighted
average

### Circumcircle

The parametric coordinate functions of the bifolium are
x(t) = r(t)cos(t) = sin(t)cos^{3}(t)
y(t) = r(t)sin(t) = sin^{2}(t)cos^{2}(t) = sin^{2}(2t)/4

The radius of a circle centered on the y-axis and circumscribing the bifolium
is at least its maximum abscissa, so a candidate for the circumcircle has radius
and center ordinate
d = 3/16. For verification,

### Circumellipse

Using z = 0 in Lemma E,
d/dt x(t)[y(t) - z] = d/dt sin^{3}(t)cos^{5}(t)

has a zero at t* = Arccos(1/4)/2. The corresponding coordinates are

We then have

For verification,

### Lobe

Unlike the lobe of a
rose, the bifolium
lobe is asymmetric with respect to its maximum-radius axis (which is also the
case for the lobes in the tail of the
trefoil). To see this, rotate the bifolium
clockwise so that the spine of its right lobe is oriented along the x-axis. To
determine the angle of rotation, we begin by finding the maximum radius point
of the bifolium in Quadrant I, which occurs at
when

Since the bifolium is defined by a polar function, t_{R} is also the
clockwise rotation angle. The parametric coordinate functions of the rotated
curve are:
x(t) = r(t)cos(t - t_{R})
y(t) = r(t)sin(t - t_{R})

The asymmetry can then be seen by comparing the upper half of the lobe (red)
with a reflection of the lower half (blue):

### Incircle (lobe)

Since the lobe is asymmetric, we resort to an unconstrained numerical search
over the lobe parameter range 0 ≤ t < π/2 to determine the optimal lobe
incircle:
R = 0.087970

c = 0.189708

d = 0.139099

### Inellipse (lobe)

An inellipse can be found by first using an unconstrained numerical search of
the rotated lobe defined by the equations above over
the parameter range 0 ≤ t < π/2 to determine an optimal axis-aligned
inellipse with an off-axis center:
a = 0.165788

b = 0.084029

c = 0.217709

d = 0.005909

To display the tilted ellipse in the left lobe for comparison with the incircle,
we negate c and then rotate the ellipse clockwise by
t_{R} = 35.26439° to produce the result in the
left-hand diagram below. This approach presupposes that the major axis of the
ellipse is along the spine of the lobe, which produces a slightly suboptimal
inellipse.
### Summary Table

**Figure** | **Parameters** | Perimeter | Area | Centroid |

Incircle (lobe) | R = 0.087970 | 0.552732 | 0.024312 | (0.189708,0.139099) |

Inellipse (lobe) * | a = 0.165788 b = 0.084029 | 0.805983 | 0.043766 | (0.217709,0.005909) |

Bifolium | | 1.788887 | 0.098175 | (0,0.125) |

Convex hull | | 1.530382 | 0.132421 | (0,0.199639) |

Circumellipse | | 1.646991 | 0.171504 | (0,0.15625) |

Circumcircle | R = | 2.040524 | 0.331340 | (0,0.1875) |

*: With respect to rotated curve S

| |

*Visitor from the planet Bifolium* | |

Top Page

Home

Copyright © 2021 Balmoral Software (http://www.balmoralsoftware.com). All
rights reserved.