Abstract
Digital circles not only play an important role in various technological settings, but also provide a lively playground for more fundamental number-theoretical questions. In this paper, we present a new algorithm for the construction of digital circles on the integer lattice \(\mathbb {Z}^2\), which makes sole use of the signum function. By briefly elaborating on the nature of discretization of circular paths, we then find that this algorithm recovers, in a space endowed with \(\ell ^1\)-norm, the defining constant \(\pi \) of a circle in \(\mathbb {R}^2\).
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Acknowledgements
Research supported by CNRS. This project has received funding from the European Union’s Horizon 2020 Framework Programme for Research and Innovation under Grant Agreement No. 720270 (Human Brain Project SGA1). The author wishes to thank L.S. Dee, J. Antolik, D. Holstein, J.A.G. Willow, S. Hower and C.O. Cain for valuable discussions and comments.
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Rudolph-Lilith, M. On a Recursive Construction of Circular Paths and the Search for \(\pi \) on the Integer Lattice \(\mathbb {Z}^2\) . Discrete Comput Geom 59, 643–662 (2018). https://doi.org/10.1007/s00454-017-9930-7
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DOI: https://doi.org/10.1007/s00454-017-9930-7