On the Functionality and Usefulness of Quadraginta Octants of Naive Sphere

Abstract

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.

Appendix: Notations for C-octants and Q-octants

C-oct	Q-octants	Notation
${\mathbb {C}}_{1}$	${\mathbb {Q}}_{1},\ldots ,{\mathbb {Q}}_{6}$	$+++$
${\mathbb {C}}_{2}$	${\mathbb {Q}}_{7},\ldots ,{\mathbb {Q}}_{12}$	$-++$
${\mathbb {C}}_{3}$	${\mathbb {Q}}_{13},\ldots ,{\mathbb {Q}}_{18}$	$+-+$
${\mathbb {C}}_{4}$	${\mathbb {Q}}_{19},\ldots ,{\mathbb {Q}}_{24}$	${-}{-}+$
${\mathbb {C}}_{5}$	${\mathbb {Q}}_{25},\ldots ,{\mathbb {Q}}_{30}$	$++-$
${\mathbb {C}}_{6}$	${\mathbb {Q}}_{31},\ldots ,{\mathbb {Q}}_{36}$	$-+-$
${\mathbb {C}}_{7}$	${\mathbb {Q}}_{37},\ldots ,{\mathbb {Q}}_{42}$	$+{-}{-}$
${\mathbb {C}}_{8}$	${\mathbb {Q}}_{43},\ldots ,{\mathbb {Q}}_{48}$	${-}{-}{-}$

Q-oct	Notation	Q-oct	Notation	Q-oct	Notation
${\mathbb {Q}}_{1}$	$(+1, +2, +3)$	${\mathbb {Q}}_{2}$	$(+2, +1, +3)$	${\mathbb {Q}}_{3}$	$(+2, +3, +1)$
${\mathbb {Q}}_{7}$	$(-1, +2, +3)$	${\mathbb {Q}}_{8}$	$(+2, -1, +3)$	${\mathbb {Q}}_{9}$	$(+2, +3, -1)$
${\mathbb {Q}}_{13}$	$(+1, -2, +3)$	${\mathbb {Q}}_{14}$	$(-2, +1, +3)$	${\mathbb {Q}}_{15}$	$(-2, +3, +1)$
${\mathbb {Q}}_{19}$	$(-1, -2, +3)$	${\mathbb {Q}}_{20}$	$(-2, -1, +3)$	${\mathbb {Q}}_{21}$	$(-2, +3, -1)$
${\mathbb {Q}}_{25}$	$(+1, +2, -3)$	${\mathbb {Q}}_{26}$	$(+2, +1, -3)$	${\mathbb {Q}}_{27}$	$(+2, -3, +1)$
${\mathbb {Q}}_{31}$	$(-1, +2, -3)$	${\mathbb {Q}}_{32}$	$(+2, -1, -3)$	${\mathbb {Q}}_{33}$	$(+2, -3, -1)$
${\mathbb {Q}}_{37}$	$(+1, -2, -3)$	${\mathbb {Q}}_{38}$	$(-2, +1, -3)$	${\mathbb {Q}}_{39}$	$(-2, -3, +1)$
${\mathbb {Q}}_{43}$	$(-1, -2, -3)$	${\mathbb {Q}}_{44}$	$(-2, -1, -3)$	${\mathbb {Q}}_{45}$	$(-2, -3, -1)$

Q-oct	Notation	Q-oct	Notation	Q-oct	Notation
${\mathbb {Q}}_{4}$	$(+3, +2, +1)$	${\mathbb {Q}}_{5}$	$(+3, +1, +2)$	${\mathbb {Q}}_{6}$	$(+1, +3, +2)$
${\mathbb {Q}}_{10}$	$(+3, +2, -1)$	${\mathbb {Q}}_{11}$	$(+3, -1, +2)$	${\mathbb {Q}}_{12}$	$(-1, +3, +2)$
${\mathbb {Q}}_{16}$	$(+3, -2, +1)$	${\mathbb {Q}}_{17}$	$(+3, +1, -2)$	${\mathbb {Q}}_{18}$	$(+1, +3, -2)$
${\mathbb {Q}}_{22}$	$(+3, -2, -1)$	${\mathbb {Q}}_{23}$	$(+3, -1, -2)$	${\mathbb {Q}}_{24}$	$(-1, +3, -2)$
${\mathbb {Q}}_{28}$	$(-3, +2, +1)$	${\mathbb {Q}}_{29}$	$(-3, +1, +2)$	${\mathbb {Q}}_{30}$	$(+1, -3, +2)$
${\mathbb {Q}}_{34}$	$(-3, +2, -1)$	${\mathbb {Q}}_{35}$	$(-3, -1, +2)$	${\mathbb {Q}}_{36}$	$(-1, -3, +2)$
${\mathbb {Q}}_{40}$	$(-3, -2, +1)$	${\mathbb {Q}}_{41}$	$(-3, +1, -2)$	${\mathbb {Q}}_{42}$	$(+1, -3, -2)$
${\mathbb {Q}}_{46}$	$(-3, -2, -1)$	${\mathbb {Q}}_{47}$	$(-3, -1, -2)$	${\mathbb {Q}}_{48}$	$(-1, -3, -2)$