Publication Date:
2013-12-12
Description:
A convex plane set S is discretized by first mapping the centre of S to a point ( u , v ), preserving orientation, enlarging by a factor t to obtain the image S ( t , u , v ) and then taking the discrete set J ( t , u , v ) of integer points in S ( t , u , v ). Let N ( t , u , v ) be the size of the ‘configuration’ J ( t , u , v ). Let L ( N ) be the number of different configurations (up to equivalence by translation) of size N ( t , u , v ) = N and let M ( N ) be the number of different configurations with 1 ≤ N ( t , u , v ) ≤ N . Then L ( N ) ≤ 2 N –1, M ( N ) ≤ N 2 , with equality if S satisfies the Quadrangle Condition, that no image S ( t , u , v ) has four or more integer points on the boundary. For the circle, which does not satisfy the Quadrangle Condition, we expect that L ( N ) should be asymptotic to 2 N , despite the numerical evidence.
Print ISSN:
0024-6115
Electronic ISSN:
1460-244X
Topics:
Mathematics
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