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There are several definitions of 3D discrete lines, e.g. http://diwww.epfl.ch/w3lsp/publications/discretegeo/nratddl.html , http://dx.doi.org/10.1007/978-3-642-19867-0_4 . However, I know of none that has all the following properties:

• Minimal 6-connectedness: The discrete line is 6-connected. Also, assuming WLOG that the direction vector is in the first octant, for any integer N there is exactly one voxel $(x,y,z)$ of the line $(x,y,z)$ such that $x+y+z=N$ (the "minimal" part).
• Well-behaved projections: The projections onto the $xy$, $xz$, and $yz$ planes are all 4-connected 2D discrete lines, such as would be produced by a modified Bresenham's algorithm.
• Convexity: The voxels of the discrete line form a convex lattice set in $\mathbb Z^3$; that is, it is equal to the intersection of $\mathbb Z^3$ with its own convex hull.

Is there a kind of discrete line that has all these properties?

It's surprising and intersting to me that discrete lines are so easy to define in 2D, but so hard to pin down in 3D. I guess being codimension-1 makes it easy.

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# Minimally 6-connected 3D discrete lines that are convex lattice sets

There are several definitions of 3D discrete lines, e.g. http://diwww.epfl.ch/w3lsp/publications/discretegeo/nratddl.html , http://dx.doi.org/10.1007/978-3-642-19867-0_4 . However, I know of none that has all the following properties:

• Minimal 6-connectedness: The discrete line is 6-connected. Also, assuming WLOG that the direction vector is in the first octant, for any integer N there is exactly one voxel of the line $(x,y,z)$ such that $x+y+z=N$ (the "minimal" part).
• Well-behaved projections: The projections onto the $xy$, $xz$, and $yz$ planes are all 4-connected 2D discrete lines, such as would be produced by a modified Bresenham's algorithm.
• Convexity: The voxels of the discrete line form a convex lattice set in $\mathbb Z^3$; that is, it is equal to the intersection of $\mathbb Z^3$ with its own convex hull.

Is there a kind of discrete line that has all these properties?

It's surprising and intersting to me that discrete lines are so easy to define in 2D, but so hard to pin down in 3D. I guess being codimension-1 makes it easy.
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