Characterization of bijective digitized rotations on the hexagonal grid

Kacper Pluta, Tristan Roussillon, David Coeurjolly, Pascal Romon, Yukiko Kenmochi, Victor Ostromoukhov
June 2017
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Abstract

Digitized rotations on discrete spaces are usually defined as the composition of a Euclidean rotation and a rounding operator; they are in gen- eral not bijective. Nevertheless, it is well known that digitized rotations defined on the square grid are bijective for some specific angles. This infinite family of angles has been characterized by Nou- vel and Rémila and more recently by Roussillon and Cœurjolly. In this article, we characterize bijec- tive digitized rotations on the hexagonal grid using arithmetical properties of the Eisenstein integers.

Type
Report

Caption: Distribution of angles whose digitized rotations by them are bijective i the hexagonal grids. Angles obtain from generators of the form s > 0 and t = s + 1 are colored in blue, while angles generated by s = 1 and t > 0, are colored in green

@techreport{pluta17hexRot,
      author = {Pluta, Kacper and Roussillon, Tristan and Coeurjolly, David and Romon, Pascal and Kenmochi, Yukiko and Ostromoukhov, Victor},
      hal_id = {hal-01540772},
      hal_version = {v1},
      institution = {HAL},
      keywords = {digital topology ; honeycomb geometry ; digital
geometry ; bijective transformations ; digitized
rotations ; hexagonal grid},
      month = {June},
      note = {Submitted to Journal of Mathematical Imaging and
Vision.},
      pdf = {https://hal.archives-ouvertes.fr/hal-01540772/file/article.pdf},
      title = {Characterization of bijective digitized rotations on the hexagonal grid},
      url = {https://hal.archives-ouvertes.fr/hal-01540772},
      year = {2017}
}
digital topology ; honeycomb geometry ; digital geometry ; bijective transformations ; digitized rotations ; hexagonal grid