Abstract
A discrete spherical circle is a topologically well-connected 3D circle in the integer space, which belongs to a discrete sphere as well as a discrete plane. It is one of the most important 3D geometric primitives, but has not possibly yet been studied up to its merit. This paper is a maiden exposition of some of its elementary properties, which indicates a sense of its profound theoretical prospects in the framework of digital geometry. We have shown how different types of discretization can lead to forbidden and admissible classes, when one attempts to define the discretization of a spherical circle in terms of intersection between a discrete sphere and a discrete plane. Several fundamental theoretical results have been presented, the algorithm for construction of discrete spherical circles has been discussed, and some test results have been furnished to demonstrate its practicality and usefulness.
V.E. Brimkov—On leave from the Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria.
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Notes
- 1.
The meaning of thickness of some discretization is discussed in Sect. 2.
- 2.
For formal definitions and details on tunnels and gaps in discrete objects, we refer to [14]. In what follows, gap-free means 0-gap-free, since a 0-gap-free surface is also 1- and 2-gap-free.
- 3.
The term offset is equivalent to thickness in the context of our work.
- 4.
While the meaning of smoothness of discrete curve is perhaps intuitively clear, this will be formally defined and discussed in Sect. 3.2.
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Biswas, R., Bhowmick, P., Brimkov, V.E. (2015). On the Connectivity and Smoothness of Discrete Spherical Circles. In: Barneva, R., Bhattacharya, B., Brimkov, V. (eds) Combinatorial Image Analysis. IWCIA 2015. Lecture Notes in Computer Science(), vol 9448. Springer, Cham. https://doi.org/10.1007/978-3-319-26145-4_7
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