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x(t) = 16sinfor t ∈ T = [0,2π). The curve follows a clockwise path as t increases, starting from the cusp (0,5) at^{3}(t)y(t) = 13cos(t) - 5cos(2t) - 2cos(3t) - cos(4t)

tand at -t_{1}= 0.908063

x'(t) = 48sinso by (L1), the perimeter of the heart curve is^{2}(t)cos(t)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) sincey''(t) = -13cos(t) + 20cos(2t) + 18cos(3t) + 16cos(4t),

48sinis satisfied for t = π. Equation (X2) is evaluated with respect to the point (0,-17) as^{2}(t)cos(t)[-13cos(t) + 20cos(2t) + 18cos(3t) + 16cos(4t)] = [-12sin(t) + 36sin(3t)][-13sin(t) + 10sin(2t) + 6sin(3t) + 4sin(4t)]

which has a solution in the right half-plane at

tso convex hull line segments extend between the points (0,-17) and_{2}= 1.973512,

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 PThe centroid ordinate of each rectangle and triangle is the average of its vertex ordinates. By (A1), the area of the comma-shaped region P_{1}(yellow) with corners at the origin and at the maximum-ordinate point (x(t_{1}),y(t_{1})).A right trapezoid containing all the Quadrant IV space between the lower-right convex hull line segment and the vertical line at x(t

_{2}). For convenience, the trapezoid is separated into a rectangle P_{2}(orange) with corners at the origin and at (x(t_{2}),y(t_{2})), and a triangle P_{3}(green) containing the remainder of the trapezoid.A comma-shaped region P

_{4}(light blue) delimited by the curve S between t_{1}and t_{2}and the boundaries of P_{1}and P_{2}.

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 P _{1}x(t _{1})y(t_{1}) = 93.456921y(t _{1})/2x(t _{1})y^{2}(t_{1})/2 = 557.155231P _{2}∣x(t _{2})y(t_{2})∣ = 43.118171y(t _{2})/2-x(t _{2})y^{2}(t_{2})/2 = -74.611741P _{3}x(t _{2})[y(t_{2}) + 17]/2 = 84.342486[2y(t _{2}) - 17]/3x(t _{2})[y(t_{2}) + 17][2y(t_{2}) - 17]/6 = -672.535896P _{4}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 havey* = -5.11923

For verification,

has a zero at t* = 1.355197. The corresponding coordinates are

x* = 14.9143We then havey* = 7.8781

For verification,

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.

FigureParametersPerimeter Area Centroid Incircle R = 8.39809 52.766756 221.569982 (0,-3.39809) Inellipse a = 12.475269

b = 6.74615361.734653 264.396668 (0,-1.746153) Heart curve Width: 32

Height: 28.923252102.168 565.486678 (0,0.833333) Convex hull 94.57115 614.722328 (0,0.777063) Circumellipse a = 17.221550

b = 16.585373106.216983 897.320010 (0,-0.414627) Circumcircle R = 16.922074 106.324527 899.615755 (0,-0.077926)

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