Discrete Analytical Hyperplanes

  title={Discrete Analytical Hyperplanes},
  author={Eric Andres and Raj S. Acharya and Claudio H. Sibata},
  journal={CVGIP Graph. Model. Image Process.},
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. 

Figures from this paper

On the Connectedness of Rational Arithmetic Discrete Hyperplanes
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
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
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
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
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
  • Y. Gérard
  • Computer Science, Mathematics
    Theor. Comput. Sci.
  • 2002
Minimal arithmetic thickness connecting discrete planes
Combinatorial Topologies for Discrete Planes
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
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
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.


Discrete Combinatorial Surfaces
  • J. Françon
  • Mathematics
    CVGIP Graph. Model. Image Process.
  • 1995
There are as many concepts of triangulated surfaces as there are neighborhood relations; thus, the same concepts, algorithms, and methods can be used in computer imagery and in the field of topology-based geometric modeling.
Digital plane and grid point segments
  • S. Forchhammer
  • Computer Science, Physics
    Comput. Vis. Graph. Image Process.
  • 1989
Discrete circles, rings and spheres
  • E. Andres
  • Mathematics, Computer Science
    Comput. Graph.
  • 1994
Three-Dimensional Digital Planes
  • C. Kim
  • Mathematics
    IEEE Transactions on Pattern Analysis and Machine Intelligence
  • 1984
Definitions of 3-D digital surface and plane are introduced and it is shown that digital convexity is neither a necessary nor a sufficient condition for a digital surface element to be a convex digital plane element.
An Introduction to the Theory of Numbers
Divisibility congruence quadratic reciprocity and quadratic forms some functions of number theory some diophantine equations Farey fractions and irrational numbers simple continued fractions primes
An Introduction to the Theory of Numbers
This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford,
Three-Dimensional Digital Line Segments
  • C. Kim
  • Art, Mathematics
    IEEE Transactions on Pattern Analysis and Machine Intelligence
  • 1983
It is shown that a digital line segment may be characterized by the chord property holding for its projections onto the coordinate planes, and that a Digital arcs in 3-D digital pictures may not be characterize by its own chord property.
An introduction to the theory of lists
In these lectures a notation and a calculus for specifying and manipulating computable functions over lists are introduced, used to derive efficient solutions for a number of problems, including problems in text processing.
New methods in oblique slice generation
This paper considers the result of the voxel space as if it would be cut by a scalpel, simulating the action of a surgeon, and shows that the supercover of the continuous oblique plane is a 6-connected discrete plane that has many very interesting properties that can be usefully exploited.
Discrete ray tracing
It is shown that RRT operates in two phases: preprocessing voxel and discrete ray tracing, which employs a discrete variation of the conventional recursive ray tracer in which 3-D discrete rays are traversed through the3-D raster to find the first surface voxels.