x(t) = 16sin3(t)for t ∈ T = [0,2π). The curve follows a clockwise path as t increases, starting from the cusp (0,5) at
y(t) = 13cos(t) - 5cos(2t) - 2cos(3t) - cos(4t)
t1 = 0.908063and at -t1 in the two bulges of the heart, S is non-convex by the multiple local extrema test, and the width x height of its bounding rectangle is 32 x 28.923252.
x'(t) = 48sin2(t)cos(t)so by (L1), the perimeter of the heart curve is
y'(t) = -13sin(t) + 10sin(2t) + 6sin(3t) + 4sin(4t),
and its area is A = 180π. By (C1), its centroid ordinate is
The bounding rectangle for inconics is delimited by the cusp at (0,5).
x''(t) = -12sin(t) + 36sin(3t)so by (X1), the curvature of S changes sign at (0,-17) since
y''(t) = -13cos(t) + 20cos(2t) + 18cos(3t) + 16cos(4t),
48sin2(t)cos(t)[-13cos(t) + 20cos(2t) + 18cos(3t) + 16cos(4t)] = [-12sin(t) + 36sin(3t)][-13sin(t) + 10sin(2t) + 6sin(3t) + 4sin(4t)]is satisfied for t = π. Equation (X2) is evaluated with respect to the point (0,-17) as
which has a solution in the right half-plane at
t2 = 1.973512,so convex hull line segments extend between the points (0,-17) and
which is about 7% shorter than that of the heart curve.
The right half of the convex hull can be partitioned into four areas, shown in color in the left diagram below:
A rectangle P1 (yellow) with corners at the origin and at the maximum-ordinate point (x(t1),y(t1)).The centroid ordinate of each rectangle and triangle is the average of its vertex ordinates. By (A1), the area of the comma-shaped region P4 is
A right trapezoid containing all the Quadrant IV space between the lower-right convex hull line segment and the vertical line at x(t2). For convenience, the trapezoid is separated into a rectangle P2 (orange) with corners at the origin and at (x(t2),y(t2)), and a triangle P3 (green) containing the remainder of the trapezoid.
A comma-shaped region P4 (light blue) delimited by the curve S between t1 and t2 and the boundaries of P1 and P2.
and by (C1), its centroid ordinate is
The half-hearted convex hull component metrics can be summarized as follows:
The area of the entire convex hull then is 614.722328, about 9% larger than the area of the heart curve. The centroid ordinate of the convex hull is the weighted average
Region Area Centroid ordinate Product P1 x(t1)y(t1) = 93.456921 y(t1)/2 x(t1)y2(t1)/2 = 557.155231 P2 ∣x(t2)y(t2)∣ = 43.118171 y(t2)/2 -x(t2)y2(t2)/2 = -74.611741 P3 x(t2)[y(t2) + 17]/2 = 84.342486 [2y(t2) - 17]/3 x(t2)[y(t2) + 17][2y(t2) - 17]/6 = -672.535896 P4 A = 86.443586 428.831266/A 428.831266 Total 307.361164 238.838860
and R = |d - z| = 8.39809. For verification, we have
has a zero at t* = 2.071360. The corresponding coordinates are
x* = 10.8039We then have
y* = -5.11923
has a zero at t* = 1.355197. The corresponding coordinates are
x* = 14.9143We then have
y* = 7.8781
and R = |d - z| = 16.922074. For verification, we have
The optimal circumcircle is slightly smaller than an origin-centered circle with radius = 17 exactly.
Figure Parameters Perimeter Area Centroid Incircle R = 8.39809 52.766756 221.569982 (0,-3.39809) Inellipse a = 12.475269
b = 6.746153
61.734653 264.396668 (0,-1.746153) Heart curve Width: 32
102.168 565.486678 (0,0.833333) Convex hull 94.57115 614.722328 (0,0.777063) Circumellipse a = 17.221550
b = 16.585373
106.216983 897.320010 (0,-0.414627) Circumcircle R = 16.922074 106.324527 899.615755 (0,-0.077926)
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