Space-Filling Curves

Volume 9 of the series Texts in Computational Science and Engineering pp 31-45


Grammar-Based Description of Space-Filling Curves

  • Michael BaderAffiliated withDepartment of Informatics, Technische Universität München

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To construct the iterations of the Hilbert curve, we recursively subdivided squares into subsquares, and sequentialised the respective subsquares by the recursive patterns given by the iterations. The patterns were obtained by respective rotations and reflections of the original pattern. We will now figure out how many different patterns actually occur in these iterations. For that purpose we will identify the basic patterns within the iterations – each basic pattern being a section of the iteration that either corresponds to a scaled down 0-th iteration (sequence “up–right–down”) or results from a rotation or reflection of this pattern.