Digital Sphere Functionality

Ranita Biswas and Partha Bhowmick,
On the Functionality and Usefulness of Quadraginta Octants of Naive Sphere,
Journal of Mathematical Imaging and Vision, Vol. 59, pp. 69–83, 2017.


This paper presents a novel study on the functional gradation of coordinate planes in connection with the thinnest and tunnel-free (i.e., naive) discretization of sphere in the integer space.
For each of the 48-symmetric quadraginta octants of naive sphere with integer radius and integer center, we show that the corresponding voxel set forms a bijection with its projected pixel set on a unique coordinate plane, which thereby serves as its functional plane. We use this fundamental property to prove several other theoretical results for naive sphere.
First, the quadraginta octants form symmetry groups and subgroups with certain equivalent topological properties.
Second, a naive sphere is always unique and consists of fewest voxels.
Third, it is efficiently constructible from its functional-plane projection.
And finally, a special class of 4-symmetric discrete 3D circles can be constructed on a naive sphere based on back projection from the functional plane.

Figure aside shows a set of ortho-coordinate circles generated by our algorithm for r = 3, 6, 9, ..., 24 and plane normal ⟨0,1,3⟩. Shown in green are Steiner voxels. Parts drawn by symmetry are shown in white.

Figure on the right:
A demonstration of Algorithm NSH with r = 12.
(The step-by-step snapshots show how the naive sphere grows with changing values of i and j. Note that only the voxels of the 1st q-octant are computed by the algorithm and their other 47 symmetric voxels are generated by permuting the coordinate values and signs.
The voxels belonging to more than one q-octant are shown in white and the voxels belonging to a single q-octant are shown in a specific color. Different colors are to signify change in the coordinate space.)