Jack E. Bresenham

A linear algorithm for incremental digital display of circular arcs

Circular arcs can be drawn on an incremental display device such as a cathode ray tube, digital plotter, or matrix printer using only sign testing and elementary addition and subtraction. This paper describes methodology for producing dot or step patterns closest to the true circle.

Run Length Slice Algorithm for Incremental Lines

Lines displayed on devices such as incremental plotters, raster CRT or plasma panel displays, and matrix printers must be approximated by sequences of discrete axial and diagonal unit steps in which successive incremental movements are constrained to the movement pattern of the king piece in a game of chess. Described is a Freeman/Reggiori-like algorithm for generating directly the run lengths of constant direction movement within the step sequence in contrast to generating the sequence in its basic unit step elements. The repetitive loop for generating lengths of alternating runs of solely axial and solely diagonal steps requires only integer addition/subtraction together with a sign test and will be executed at most only half the number of times as the comparable loop used to generate the single unit move sequence one step at a time. The algorithm also can be used to examine repetitive patterns and cycles which occur in rastered lines.

Algorithms for Circular Arc Generation

By much the same methodology as is employed in algorithms which generate a quantized representation of a straight line, incremental algorithms using only addition/subtraction and sign testing can generate a close approximation of a circular arc. Several mathematical measures of ‘closeness’ to the true circle can be defined. Discussed will be closeness measures of function residue or squared error, normal deviation or radial error, and rounded or axial intercept error as they apply to both a tri-directional and a bi-directional algorithm for pel by pel generation of a quantized circular arc.