88 lines
5.7 KiB
HTML
88 lines
5.7 KiB
HTML
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<h2>Enmeshed unit circle</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 392</h3><span style="width:300px;color:#666;">Published on Saturday, 1st September 2012, 02:00 pm; Solved by 307</span></div>
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A rectilinear grid is an orthogonal grid where the spacing between the gridlines does not have to be equidistant.<BR />
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An example of such grid is logarithmic graph paper.
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</p>
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<P>
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Consider rectilinear grids in the Cartesian coordinate system with the following properties:<br /><ul><li>The gridlines are parallel to the axes of the Cartesian coordinate system.</li><li>There are N+2 vertical and N+2 horizontal gridlines. Hence there are (N+1) x (N+1) rectangular cells.</li><li>The equations of the two outer vertical gridlines are x = -1 and x = 1.</li><li>The equations of the two outer horizontal gridlines are y = -1 and y = 1.</li><li>The grid cells are colored red if they overlap with the <dfn title="The unit circle is the circle that has radius 1 and is centered at the origin">unit circle</dfn>, black otherwise.</li></ul>For this problem we would like you to find the postions of the remaining N inner horizontal and N inner vertical gridlines so that the area occupied by the red cells is minimized.
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E.g. here is a picture of the solution for N = 10:
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<img src=project/images/p392_gridlines.png></P>
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</p>
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The area occupied by the red cells for N = 10 rounded to 10 digits behind the decimal point is 3.3469640797.
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<p>
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Find the positions for N = 400.<BR />
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Give as your answer the area occupied by the red cells rounded to 10 digits behind the decimal point.
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