Editing Bresenham's line algorithm (section)

==Method==
[[File:Bresenham.svg|right|300px|thumb|Illustration of the result of Bresenham's line algorithm. (0,0) is at the top left corner of the grid, (1,1) is at the top left end of the line and (11, 5) is at the bottom right end of the line.]]
The following conventions will be applied:
* the top-left is (0,0) such that pixel coordinates increase in the right and down directions (e.g. that the pixel at (7,4) is directly above the pixel at (7,5)), and
* the pixel centers have integer coordinates.
The endpoints of the line are the pixels at <math>(x_0,y_0)</math> and <math>(x_1,y_1)</math>, where the first coordinate of the pair is the column and the second is the row.

The algorithm will be initially presented only for the [[octant (plane geometry)|octant]] in which the segment goes down and to the right (<math>x_0 \leq x_1</math> and <math>y_0 \leq y_1</math>), and its horizontal projection <math>x_1-x_0</math> is longer than the vertical projection <math>y_1-y_0</math> (the line has a positive [[slope]] less than 1).
In this octant, for each column ''x'' between <math>x_0</math> and <math>x_1</math>, there is exactly one row ''y'' (computed by the algorithm) containing a pixel of the line, while each row between <math>y_0</math> and <math>y_1</math> may contain multiple rasterized pixels.

Bresenham's algorithm chooses the integer ''y'' corresponding to the [[pixel center]] that is closest to the ideal (fractional) ''y'' for the same ''x''; on successive columns ''y'' can remain the same or increase by 1.
The general equation of the line through the endpoints is given by:
:<math>\frac{y - y_0}{y_1-y_0} = \frac{x-x_0}{x_1-x_0}</math>.

Since we know the column, ''x'', the pixel's row, ''y'', is given by rounding this quantity to the nearest integer:

:<math>y = \frac{y_1-y_0}{x_1-x_0} (x-x_0) + y_0</math>.

The slope <math>(y_1-y_0)/(x_1-x_0)</math> depends on the endpoint coordinates only and can be precomputed, and the ideal ''y'' for successive integer values of ''x'' can be computed starting from <math>y_0</math> and repeatedly adding the slope.

In practice, the algorithm does not keep track of the y coordinate, which increases by ''m'' = ''∆y/∆x'' each time the ''x'' increases by one; it keeps an error bound at each
stage, which represents the negative of the distance from (a) the point where the line exits the pixel to (b) the top edge of the pixel. 
This value is first set to <math>y_0-0.5</math> (due to using the pixel's center coordinates), and is incremented by ''m'' each time the ''x'' coordinate is incremented by one. If the error becomes greater than ''0.5'', we know that the line has moved upwards
one pixel, and that we must increment our ''y'' coordinate and readjust the error to represent the distance from the top of the new pixel – which is done by subtracting one from error.<ref>{{Cite web|url=https://www.ercankoclar.com/wp-content/uploads/2016/12/Bresenhams-Algorithm.pdf|title=Bresenham's Algorithm|last=Joy|first=Kenneth|publisher=Visualization and Graphics Research Group, Department of Computer Science, University of California, Davis|access-date=20 December 2016}}</ref>