Digital Two-Dimensional Bijective Reflection and Associated Rotation

In this paper, a new bijective reflection algorithm in two dimensions is proposed along with an associated rotation. The reflection line is defined by an arbitrary Euclidean point and a straight line passing through this point. The reflection line is digitized and the 2D space is paved by digital perpendicular (to the reflection line) straight lines. For each perpendicular line, integer points are reflected by central symmetry with respect to the reflection line. Two consecutive digital reflections are combined to define a digital bijective rotation about arbitrary center, i.e. bijective digital rigid motion.

[1]  Eric Rémila,et al.  Characterization of Bijective Discretized Rotations , 2004, IWCIA.

[2]  Gaëlle Largeteau-Skapin,et al.  Decomposition of nD-rotations: Classification, properties and algorithm , 2011, Graph. Model..

[3]  Eric Rémila,et al.  Incremental and Transitive Discrete Rotations , 2006, IWCIA.

[4]  Victor Ostromoukhov,et al.  Characterization of Bijective Digitized Rotations on the Hexagonal Grid , 2018, Journal of Mathematical Imaging and Vision.

[5]  Rémy Malgouyres,et al.  Parallelization of a Discrete Radiosity Method , 2006, Euro-Par.

[6]  Gonzalo Navarro,et al.  Rotation and lighting invariant template matching , 2007, Inf. Comput..

[7]  Pat Hanrahan,et al.  Beam tracing polygonal objects , 1984, SIGGRAPH.

[8]  Eric Andres,et al.  The Quasi-Shear Rotation , 1996, DGCI.

[9]  Nicolas Passat,et al.  Bijective Digitized Rigid Motions on Subsets of the Plane , 2017, Journal of Mathematical Imaging and Vision.

[10]  David Coeurjolly,et al.  Characterization of bijective discretized rotations by Gaussian integers , 2016 .

[11]  E. Andres,et al.  Shear based Bijective Digital Rotation in Triangular Grids , 2018 .

[12]  Nicolas Passat,et al.  Bijective Rigid Motions of the 2D Cartesian Grid , 2016, DGCI.

[13]  Eric Andres,et al.  Pedagogic discrete visualization of electromagnetic waves , 2003, Eurographics.

[14]  Jean-Pierre Reveillès Géométrie discrète, calcul en nombres entiers et algorithmique , 1991 .

[15]  Nicolas Passat,et al.  Bijectivity Certification of 3D Digitized Rotations , 2016, CTIC.

[16]  Walter A. Deuber,et al.  Geometrical Bijections in Discrete Lattices , 1999, Combinatorics, Probability and Computing.

[17]  Gaëlle Largeteau-Skapin,et al.  Shear Based Bijective Digital Rotation in Hexagonal Grids , 2018, DGMM.

[18]  Roe Goodman,et al.  Alice through Looking Glass after Looking Glass: The Mathematics of Mirrors and Kaleidoscopes , 2004, Am. Math. Mon..