Bresenham's algorithm for drawing digital straight lines is an implementation of the Euclidean algorithm

Rasterizing a straight line onto a pixel grid requires deciding which pixels to activate along the run. For a line spanning n columns and k rows, the sequence of steps follows the same even-distribution logic as Bjorklund’s algorithm: Harris and Reingold showed the well-known Bresenham line-drawing algorithm is an implementation of the Euclidean algorithm. So the exact structure behind Euclidean rhythms also governs computer-graphics line drawing — one more domain (alongside calendars, nuclear-accelerator timing, and number theory) where maximal evenness is the operative principle. Concretely, the upper boundary of a digital line from (0,0) to (16,5) traces the interval sequence (33334) = E(5,16), the Bossa-Nova necklace.

Examples

A digital line with horizontal span 16 and vertical span 5 yields interval sequence (33334) = E(5,16). The same pattern is the Bossa-Nova rhythm necklace.

Assessment

State in one sentence the common problem Bresenham’s and Bjorklund’s algorithms both solve. Explain how this cross-domain appearance supports calling Euclidean rhythms ‘natural’.