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 Fixed length from unit square. (Posted on 2009-10-01)
Given a unit square and a fixed length, r.

Construct the set of all points which are at distance r from some point on the square.

Find the area of this set.

Note: A square is composed of 4 line segments, not the interior. For some values of r the set will have a hole in it.

Example of the construction:

 No Solution Yet Submitted by Jer Rating: 5.0000 (1 votes)

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 comparison of formulas Comment 3 of 3 |

Bractals' and my formulas for the area of the center hole for r >= sqrt(2)/2 agree numerically:

DECLARE FUNCTION arcsin# (x#)
DEFDBL A-Z

FOR r = .8 TO 1.2 STEP .1
s = (SQR(1 + 4 * (r ^ 2 - 1 / 2)) - 1) / 2
theta = 2 * arcsin(s * SQR(2) / (2 * r))
a = (SQR(1 + 4 * (r ^ 2 - 1 / 2)) - 1) ^ 2 / 2 + 2 * r ^ 2 * (theta - SIN(theta))
PRINT USING "##.## ##.########"; r; a;

t = 2 * SQR(r ^ 2 - 1 / 4)
a = 2 * r ^ 2 * (ATN(t) - ATN(1 / t)) - t + 1
PRINT USING "     ##.########"; a
NEXT

FUNCTION arcsin (x)
d = SQR(1 - x * x)
arcsin = ATN(x / d)
END FUNCTION

` r    Area(Charlie)   Area(Bractals)0.80  0.03328298      0.033282980.90  0.13956677      0.139566771.00  0.31514677      0.315146771.10  0.55792407      0.557924071.20  0.86664457      0.86664457`

 Posted by Charlie on 2009-10-01 18:30:08

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