From prima quadraginta octant to lattice sphere through primitive integer operations

We present here the first integer-based algorithm for constructing a well-defined lattice sphere specified by integer radius and integer center. The algorithm evolves from a unique correspondence between the lattice points comprising the sphere and the distribution of sum of three square numbers in...

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Bibliographic Details
Published inTheoretical computer science Vol. 624; pp. 56 - 72
Main Authors Biswas, Ranita, Bhowmick, Partha
Format Journal Article
LanguageEnglish
Published Elsevier B.V 18.04.2016
Subjects
Aids
Algorithms
Computational efficiency
Confining
Construction specifications
Digital geometry
Digital sphere
Geometry of numbers
Integer algorithm
Integers
Intervals
Lattice sphere
Lattices
Geometry of numbers
Integer algorithm
Digital geometry
Lattice sphere
Digital sphere
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ISSN0304-3975
1879-2294
DOI10.1016/j.tcs.2015.11.018

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Summary:We present here the first integer-based algorithm for constructing a well-defined lattice sphere specified by integer radius and integer center. The algorithm evolves from a unique correspondence between the lattice points comprising the sphere and the distribution of sum of three square numbers in integer intervals. We characterize these intervals to derive a useful set of recurrences, which, in turn, aids in efficient computation. Each point of the lattice sphere is determined by resorting to only a few primitive operations in the integer domain. The symmetry of its quadraginta octants provides an added advantage by confining the computation to its prima quadraginta octant. Detailed theoretical analysis and experimental results have been furnished to demonstrate its simplicity and elegance.
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ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2015.11.018