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Non-urgent, non-exam-related geometry question watch

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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)?

    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.

    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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Updated: January 15, 2010

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