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r(t) = sin(t)cos2(t), 0 ≤ t < πThe curve is traced out in a counterclockwise direction, starting from its minimum ordinate point at the origin when
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):
r'(t) = 3cos3(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
By (C2), the associated centroid ordinate is![]()
The convex hull component metrics can be summarized as follows:![]()
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
Region Area Centroid ordinate Product Triangle 1/16 1/6 1/96 Remainder A = (4 + 3π)/192 Total (16 + 3π)/192
x(t) = r(t)cos(t) = sin(t)cos3(t)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 radiusy(t) = r(t)sin(t) = sin2(t)cos2(t) = sin2(2t)/4
d/dt x(t)[y(t) - z] = d/dt sin3(t)cos5(t)has a zero at t* = Arccos(1/4)/2. The corresponding coordinates are
We then have![]()
For verification,![]()
Since the bifolium is defined by a polar function, tR is also the clockwise rotation angle. The parametric coordinate functions of the rotated curve are:![]()
x(t) = r(t)cos(t - tR)The asymmetry can then be seen by comparing the upper half of the lobe (red) with a reflection of the lower half (blue):y(t) = r(t)sin(t - tR)
R = 0.087970
c = 0.189708
d = 0.139099
a = 0.165788To display the tilted ellipse in the left lobe for comparison with the incircle, we negate c and then rotate the ellipse clockwise by
b = 0.084029
c = 0.217709
d = 0.005909
*: With respect to rotated curve S
Figure Parameters Perimeter Area Centroid Incircle (lobe) R = 0.087970 0.552732 0.024312 (0.189708,0.139099) Inellipse (lobe) * a = 0.165788
b = 0.0840290.805983 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)
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