# Discrete Analytical Hyperplanes

@article{Andres1997DiscreteAH, title={Discrete Analytical Hyperplanes}, author={Eric Andres and Raj S. Acharya and Claudio H. Sibata}, journal={CVGIP Graph. Model. Image Process.}, year={1997}, volume={59}, pages={302-309} }

This paper presents the properties of the discrete analytical hyperplanes. They are defined analytically in the discrete domain by Diophantine equations. We show that the discrete hyperplane is a generalization of the classical digital hyperplanes. We present original properties such as exact point localization and space tiling. The main result is the links made between the arithmetical thickness of a hyperplane and its topology.

## 123 Citations

On the Connectedness of Rational Arithmetic Discrete Hyperplanes

- MathematicsDGCI
- 2006

An algorithm is provided which computes the thickness of the thinnest 0-connected arithmetic plane with normal vector n, thanks to an arithmetic reduction on a given integer vector n.

Arithmetic Discrete Hyperspheres and Separating Capacity

- Mathematics
- 2019

In the framework of arithmetic discrete geometry, a discrete object is provided with its own analytical definition corresponding to a discretization scheme. It can thus be considered as the…

Arithmetic Discrete Hyperspheres and Separatingness

- MathematicsDGCI
- 2006

A general definition of discrete hyperspheres and the characterization of the k-minimal ones thanks to an arithmetic definition based on a non-constant thickness function and a link to adjacency and separatingness with norms is linked.

Characterization of the Best Discrete Approximation of a Line in the 3-Dimentional Space

- Computer Science, Mathematics
- 2020

This work focuses on the minimal 0-connected set of closest integer points to a Euclidean line, which leads to geometric, arithmetic and algorithmic characterizations of naive discrete lines in the 3-dimensional space.

Characterization of the Closest Discrete Approximation of a Line in the 3-Dimensional Space

- Mathematics, Computer ScienceISVC
- 2006

A definition of the minimal 0-connected set of closest integer points to a Euclidean line is proposed which leads to geometric, arithmetic and algorithmic characterizations of naive discrete lines in the 3-dimensional space.

Periodic graphs and connectivity of the rational digital hyperplanes

- Computer Science, MathematicsTheor. Comput. Sci.
- 2002

Minimal arithmetic thickness connecting discrete planes

- Mathematics, Computer ScienceDiscret. Appl. Math.
- 2009

Combinatorial Topologies for Discrete Planes

- MathematicsDGCI
- 2003

This paper defines a discrete combinatorsial plane DCP and shows relations between DAPs and DCPs such that a DCP is a combinatorial surface of a DAP.

Arithmetic Discrete Planes Are Quasicrystals

- MathematicsDGCI
- 2009

A substitution rule acting on discrete planes is introduced, which maps faces of unit cubes to unions of faces, and some applications to discrete geometry are discussed.

On the Connecting Thickness of Arithmetical Discrete Planes

- Mathematics, Computer ScienceDGCI
- 2009

The lower bound of the thicknesses 2-connecting the arithmetical discrete planes with normal vector v is computed, and it is shown how the translation parameter operates in the connectedness of theArithmeticals discrete planes.

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