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    If I have an infinitely large grid made of perfect squares aligned in the exact same way as normal squared paper, if I start at any arbitary point, and draw concentric circles, where I start on the corners of squares, and draw round 360 degrees, will the circle, no matter how large, only ever pass exactly (not just to good precision, but exactly) 4 corners of squares. I appreciate I probably haven't explained it well so i've attached a diagram. it takes ages to do though so only the first few are shown, you should get the idea I think.

    If your interested, i'm doing this to work out the RDF of a uniform lattice
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    If your circle has radius 5, won't it pass through (3,4)?
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    Imagine your radius as the hypotenuse of a right-angled triangle. Then at any point where r^2=a^2+b^2 where a and b are multiples of the side of a square's length, then it would also be true for all multiples of a, b and r so it would cross an infinite number of corners.
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    Taking the centre of the circles as the origin, doesn't the circle passing through (1,3) pass eight corners? In fact, if you imagine the radius as the hypotenuse of a right-angled triangle, as tgodkin suggested, then any time the circle passes through a corner where the shorter sides of the triangle are different lengths it will pass through eight points.

    E.g.: a circle drawn through (1,3) would also pass (3,1), (3,-1), (1,-3), (-1,-3), (-3,-1), (-3,1), (-1,3).

    However, a circle passing through a point where one of the coordinates is 0 [e.g. (2,0)] or where the absolute value of the x and y coordinates is the same [e.g. (2,-2)] will only pass through four corners.

    I might have misunderstood your question, but hopefully that helps.
 
 
 
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