Characterization of Bijective Digitized Rotations on the Hexagonal Grid

Self Cite

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.

Section: Introductionmentioning

confidence: 64%

Visiting Bijective Digitized Reflections and Rotations Using Geometric Algebra

Breuils¹,

Kenmochi²,

Sugimoto³

2021

Self Cite

“…3.1. Rotation decomposition into shears in the hexagonal grid K. Pluta showed that there exists a subset of angles for which the digitized rotation of an hexagonal grid is bijective (Pluta et al (2017(Pluta et al ( , 2018). Our idea is to use shear transforms to define a digital rotation is the hexagonal grid that works for all angles by pushing grid points into specific directions.…”

Section: Bijective Digital Rotation In the Hexagonal Gridmentioning

confidence: 99%

“…Transformations in hexagonal grids are still a largely open problem with few references (see Her (1995) for a discussion on transforms on hexagonal grids). Recently K. Pluta et al showed that, as for the square grid (Jacob and Andres (1995), Nouvel and Remila (2005)), there are angles for which the digitized rotation is bijective (Pluta et al (2017(Pluta et al ( , 2018) in the hexagonal grid. The problem is that these angles represent only a subset of all angles (Pluta et al (2017(Pluta et al ( , 2018).…”

Section: Introductionmentioning

confidence: 99%

Shear Based Bijective Digital Rotation in Hexagonal Grids

Andrès

Largeteau-Skapin

Zrour

2021

In this paper, a new bijective digital rotation algorithm for the hexagonal grid is proposed. The method is based on an original decomposition of rotations into shear transforms. It works for any angle with an hexagonal centroid as rotation center and is easily invertible. The algorithm achieves an average distance between the digital rotated point and the continuous rotated point of about 0.42 (for 1.0 the distance between two neighboring hexagons).

“…Interesting links have been made using Gaussian integers between twin Pythagorean triplets and the angles of digitized bijective rotations [14]. More recently, Pluta et al [13] have brought a new light into this research subject by showing that similar results using Eisenstein integers exist for the hexagonal grid.…”

Section: Introductionmentioning

confidence: 99%

Conjecture on Characterisation of Bijective 3D Digitized Reflections and Rotations

Breuils

Kenmochi

Andrès

et al. 2023

Self Cite

Bijectivity of digitized linear transformations is crucial when transforming 2D/3D objects in computer graphics and computer vision. Although characterisation of bijective digitized rotations in 2D is well known, the extension to 3D is still an open problem. A certification algorithm exists that allows to verify that a digitized 3D rotation defined by a quaternion is bijective. In this paper, we use geometric algebra to represent a bijective digitized rotation as a pair of bijective digitized reflections. Visualization of bijective digitized reflections in 3D using geometric algebra leads to a conjectured characterization of 3D bijective digitized reflections and, thus, rotations. So far, any known quaternion that defines a bijective digitized rotation verifies the conjecture. An approximation method of any digitized reflection by a conjectured bijective one is also proposed.