Consistent digital line segments

We introduce a novel and general approach for digitalization of line segments in the plane that satisfies a set of axioms naturally arising from Euclidean axioms. In particular, we show how to derive such a system of digital segments from any total order on the integers. As a consequence, using a well-chosen total order, we manage to define a system of digital segments such that all digital segments are, in Hausdorff metric, optimally close to their corresponding Euclidean segments, thus giving an explicit construction that resolves the main question of Chun et al.

💡 Research Summary

The paper addresses a fundamental problem in digital geometry: how to represent straight line segments on the integer lattice ℤ² so that the digital representation respects the natural geometric properties of Euclidean segments while staying as close as possible to the original line. Chun et al. previously formulated five axioms (S1–S5) that capture essential Euclidean characteristics—grid‑path connectivity, symmetry, subsegment inclusion, extendability, and monotonicity for horizontal/vertical lines. Existing digital segment constructions either violate the subsegment property (S3) or fail to guarantee a grid‑path (S1), making them unsuitable as “consistent” digital line segment systems (CDS).

The authors introduce a novel, highly general method: start from any strict total order ≺ on the integers and define a digital segment S≺(p,q) between two lattice points p=(px,py) and q=(qx,qy) (with px≤qx). If py≤qy, the algorithm walks from p to q by repeatedly moving either one step right or one step up. The total number of steps is (qx−px)+(qy−py); exactly (qy−py) of them must be upward moves. The decision at each intermediate point (x,y) is based on the sum s=x+y. Consider the interval I=