Consistent digital line
Euclid laid down the fundamental axioms of geometry. However, in today's digital world, everything became discrete, segments are represented by a finite number of pixels. How much can we save from his axioms?
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.
All the proofs will be very elementary, easily understandable by a high school student. We end by some interesting open questions.
This is joint work with Tobias Christ and Milos Stojakovic.