Editing Bresenham's line algorithm (section)

==Similar algorithms==
The Bresenham algorithm can be interpreted as slightly modified [[Digital differential analyzer (graphics algorithm)|digital differential analyzer]] (using 0.5 as error threshold instead of 0, which is required for non-overlapping polygon rasterizing).

The principle of using an [[incremental error algorithm|incremental error]] in place of division operations has other applications in graphics. It is possible to use this technique to calculate the [[uv mapping|U,V co-ordinates]] during raster scan of texture mapped polygons.<ref>{{Cite patent|title=Apparatus and method for performing perspectively correct interpolation in computer graphics|country=US|number=5739818|inventor1-first=John Neil|inventor1-last=Spackman|assign=[[Canon_Inc.|Canon KK]]|pubdate=1998-04-14}}</ref> The [[voxel]] heightmap software-rendering engines seen in some PC games also used this principle.

Bresenham also published a Run-Slice computational algorithm: while the above described Run-Length algorithm runs the loop on the major axis, the Run-Slice variation loops the other way.<ref>{{cite web |title=Michael Abrash's Graphics Programming Black Book Special Edition: The Good, the Bad, and the Run-Sliced |url=http://www.phatcode.net/res/224/files/html/ch36/36-01.html |website=www.phatcode.net |access-date=13 February 2024}}; </ref>  This method has been represented in a number of US patents: 
* {{US patent reference
|number={{formatnum:5,815,163|R}}
|title=Method and apparatus to draw line slices during calculation
}}
* {{US patent reference
|number={{formatnum:5,740,345|R}}
|title=Method and apparatus for displaying computer graphics data stored in a compressed format with an efficient color indexing system
}}
* {{US patent reference
|number={{formatnum:5,657,435|R}}
|title=Run slice line draw engine with non-linear scaling capabilities
}}
* {{US patent reference
|number={{formatnum:5,627,957|R}}
|title=Run slice line draw engine with enhanced processing capabilities
}}
* {{US patent reference
|number={{formatnum:5,627,956|R}}
|title=Run slice line draw engine with stretching capabilities
}}
* {{US patent reference
|number={{formatnum:5,617,524|R}}
|title=Run slice line draw engine with shading capabilities
}}
* {{US patent reference
|number={{formatnum:5,611,029|R}}
|title=Run slice line draw engine with non-linear shading capabilities
}}
* {{US patent reference
|number={{formatnum:5,604,852|R}}
|title=Method and apparatus for displaying a parametric curve on a video display
}}
* {{US patent reference
|number={{formatnum:5,600,769|R}}
|title=Run slice line draw engine with enhanced clipping techniques
}}

The algorithm has been extended to:
* Draw lines of arbitrary thickness, an algorithm created by Alan Murphy at IBM.<ref>{{Cite web|url=http://homepages.enterprise.net/murphy/thickline/index.html|title=Murphy's Modified Bresenham Line Algorithm|website=homepages.enterprise.net|access-date=2018-06-09}} ('Line Thickening by Modification to Bresenham's Algorithm' in the IBM Technical Disclosure Bulletin Vol. 20 No. 12 May 1978 pages 5358-5366.)</ref>
* Draw multiple kinds curves (circles, ellipses, cubic, quadratic, and rational [[Bézier curve]]s) and antialiased lines and curves; a set of algorithms by Alois Zingl.<ref name=Zingl>{{cite report|last=Zingl|first=Alois|title=A Rasterizing Algorithm for Drawing Curves|date=2016|orig-date=Previously published in 2012|url=https://zingl.github.io/Bresenham.pdf}}<br>HTML abstract and demo: {{cite web|date=2020|orig-date=Previously published in 2012|last=Zingl|first=Alois|title=The Beauty of Bresenham's Algorithm|url=https://zingl.github.io/bresenham.html|website=zingl.github.io}}</ref>