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Re: Heart inscribed in a circle
Posted:
Aug 25, 2010 5:00 AM


> Have you or has anyone done the opposite? I.e., > circumscribed a given heart shape with a circle?
I found Stuart's problem of inscribing a heart inside an existing circle more difficult than the problem Mary proposed of circumscribing a circle around a given heart. So I tackled them in reverse order.
First I solved the problem analytically:
Consider a square of side 'a' with vertices on the coordinate axes, as A(c,0), B(0,c),C(c,0),D(0,c) where c = a/sqrt(2), with semicircles of diam 'a' on sides AB and BC resp., with E the midpoint of AB and I the midpoint of DA. Let radius of circumcircle = 'r', and circumcenter J at (0,b), and let T be the common tangent point of circles (E,a/2) and (J,r).
Then r = a/sqrt(2) + b
Let L and K be the projections on Oy of E and T resp. Then in tJEL we have JE = ra/2, EL = a/(2*sqrt(2)), L = a/(2*sqrt(2))  b = 3*a/(2*sqrt(2))  r.
Apply Pythagoras to give a/r = (3*sqrt(2)2)/2 or r/a = (3*sqrt(2)+2)/7
.: b/a = (4sqrt(2))/14
In tOTK let <OTK = theta = <AOT.
Triangles JTK and JEL are similar .: KT = a/(2*sqrt(2))*r/(ra/2) and JK = 3a/((2*sqrt(2))r)*r/(ra/2)
Then tan(theta) = (b + JK)/KT = 1/sqrt(2) .: theta = 35.264 deg approx.
Numerically: If a=1, then b=0.1847 and r= 0.8918
Next I looked for a simple graphical construction for the circumcircle of a given heart:
Note the T,E and J are collinear. With centre D swing arcDI = a/2 to meet Oy at G. Let HJ with J on Oy be the perpendicular bisector of GE, so that GJ = JE = r  a/2 Then J is the required circumcentre.
To tackle a heart within a given circle I proceeded indirectly as follows: Inside the given circle draw a square with vertices at (+r,0) and (0,+r).[The size of the square is arbitrary, but the inscribed square is convenient.] Complete the heart on that square and draw the outer circumcircle, radius R. Then complete the inner heart by proportion. One simple way is: Find point M= (r,R). Join AD and MD. Draw line y=r to meet MD at N. Drop a vertical from N to meet AD at P. Then P is the required square vertex of the inscribed heart.
Regards, Peter Scales.



