Digital planarity—A review

Abstract

Digital planarity is defined by digitizing Euclidean planes in the three-dimensional digital space of voxels; voxels are given either in the grid-point or the grid-cube model. The paper summarizes results (also including most of the proofs) about different aspects of digital planarity, such as supporting or separating Euclidean planes, characterizations in arithmetic geometry, periodicity, connectivity, and algorithmic solutions. The paper provides a uniform presentation, which further extends and details a recent book chapter in (Klette and Rosenfeld 2004).

Optimization Schemes for the Reversible Discrete Volume Polyhedrization Using Marching Cubes Simplification

“…In other words, there exists (α, [6,3,10]. Thus we can define the preimage of V as the set of (α, β, γ) parameters fulfilling this condition [17,15,3]. In the following, we call digital plane segments (DPS for short) coplanar sets of voxels.…”

“…Consider a set of voxels V, this set is a piece of digital plane with x ≥ z, y ≥ z and z > 0 if and only if there exists a Euclidean plane containing V in its digitization. In other words, there exists (α, [6,3,10]. Thus we can define the preimage of V as the set of (α, β, γ) parameters fulfilling this condition [17,15,3].…”

Digital Analytical Geometry: How Do I Define a Digital Analytical Object?

“…See [41,47] for a survey of digital linearity and planarity with interesting historical perspectives and useful comments and references on digital analytical lines and hyperplanes. An important step in bringing dierent theoretical approaches together, was to establish a link between the thickness of digital hyperplanes and topology [25]: let us assume, w.l.o.g.…”

Thoughts on 3D Digital Subplane Recognition and Minimum-Maximum of a Bilinear Congruence Sequence

“…In this paper we are interested in the extension of this problem to dimension three: the Digital Subplane (DSP) Recognition Problem. Check the following papers for a general approach on Digital Plane Recognition [6,9,12,3,7]. In Section 2, we propose a recall on 2D results and the unsolved problems in 3D.…”