Visiting Bijective Digitized Reflections and Rotations Using Geometric Algebra

Abstract

Geometric algebra has become popularly used in applications dealing with geometry. This framework allows us to reformulate and redefine problems involving geometric transformations in a more intuitive and general way. In this paper, we focus on 2D bijective digitized reflections and rotations. After defining the digitization through geometric algebra, we characterize the set of bijective digitized reflections in the plane. We derive new bijective digitized rotations as compositions of bijective digitized reflections since any rotation is represented as the composition of two reflections. We also compare them with those obtained through geometric transformations by computing their distributions.

Conjecture on Characterisation of Bijective 3D Digitized Reflections and Rotations

“…Let us briefly review reflections and rotations with geometric algebra. For more details, see [3,6,11]. We here focus on reflections and rotations expressed as two reflections.…”

“…However, there exist subsets of digitized transformations that are bijective. The characterization of these subsets were shown for digitized reflections in [3] and for rotations in [14,8].…”

“…In [3], the characterization of digitized reflections using the bijectivity condition is presented. The idea there consists of expressing the bijectivity condition using a geometric algebra rotation Q in 2D and expressing the set of remainders of digitized reflections by the set of remainders of digitized rotations.…”

“…Meanwhile Andres et al [2] proposed an algorithm for bijective digitized reflections in 2D which easily deduces a method that generates bijective digitized rotations. Breuils et al [3] used geometric algebra to reformulate the problem and characterized 2D bijective digitized reflections. This is the starting point of the present paper.…”

Homotopic Affine Transformations in the 2D Cartesian Grid

“…In Cartesian grids, various subfamilies of affine transformations have been investigated, namely translations [7,33], scalings [1,3], reflections [5,12], rotations [2,4,5,8,18,25,44,45,49,54], rigid motions [38, 39, 41-43, 46, 50], combined scalings and rotations [22], and affine transformations [11,21,23,24,27,29,37]. The purposes were manifold: describing the combinatorial structure of these transformations with respect to R n versus Z n [1-3, 7, 8, 18, 21-23, 33, 38, 48, 56], guaranteeing their bijectivity [4,5,12,25,44,49,50,54] or their transitivity [45] in Z n , preserving geometrical properties [41,42] and, less frequently, ensuring their topological invariance [39,43] in Z n . These are non-trivial questions, and their difficulty increases with the dimension of the Cartesian grid [46].…”