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Computer graphics – bresenham line drawing algorithm DERIVATION • Starting from the left endpoint (x0, y0) of a given line, we step to each. Assumption: Y=mX+b where b is the intercept cut by line at Y axis and m is the slope of line (0 Derivation: Initially we have plotted a. To derive Bresenham’s algorithm, two steps must be taken. and then using this new equation for a line to draw a line based on the.

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Unsourced material may be challenged and removed. This is a function of only x and it would be useful to make this equation written as a function of both x and y.

With slopes greater than 1 or less than -1, we must take the previous implementation and swap all x and y values to “move” the calculations back into the “First Octant”. The algorithm is used in hardware such as plotters and in the graphics chips of modern graphics cards. The label “Bresenham” is used today for a family of algorithms extending or modifying Bresenham’s original algorithm.


In addition to this optimization, Bresenham Algorithm suggests to optimize more. It is an incremental error algorithm. This process is called rasterization. This decision can be generalized by accumulating the error.

derivation of bresenham line algorithm

The basic idea of the Bresenham Algorithm is shown is the previous sectionbut the algorithm can be easily extended to all other algoruthm, not just the lines with slope between 0 and 1. This article needs additional citations for verification.


To find the best “next pixel”, first we must find the distances to the two available choices from the ideal location of the real line. However, as mentioned above this is only for octant zero, that is lines starting at the origin with a gradient between 0 and 1 where x increases by exactly 1 per iteration and y increases by 0 or 1.

It shows a screen-like pixel grid and a few examples of lines drawn using the Bresenham Algorithm. Consider a line with initial point x1y1 and terminal point x2y2 in device space.

MATH Project: The Bresenham Line-Drawing Algorithm

In order to develop a fast way of doing this, we will not be comparing these values in such a manner, instead we will create a decision variable that can be used to quickly determine which point to use.

The algorithm basically approximates real valued line by calculating what pixels to illuminate and to provide “illusion” of line. If the error becomes greater than 0. If we now evaluate a new decision variable P kwe get: By using this site, you agree to the Terms of Use and Privacy Policy.

Bresenham’s line algorithm

The plotting can be viewed by plotting at the intersection of lines blue circles or filling in pixel boxes yellow squares. In Algotithm wrote: Distance between pixel-to-right and ideal pixel is: To find out more, including how to control cookies, see here: Simplifying this expression yields:. It is one of the earliest algorithms developed in the field of computer graphics. Therefore the final recursive definition for pi will be based on choice, as follows remember that the sign of pi is the same as the sign of d1 — d2: However, we can do better than this, by defining pi recursively.


This site uses cookies. The summary of the basic steps of the algorithm for “First Octant” is following: This observation is crucial to understand! The general solution of the Bresenham Algorithm must check for the slope of the line, is it within our previous bounds where x is independent variable or is it where y is independent variable.

Note that the underlined part is constant it does not change during iterationwe call it c, i. Retrieved from ” https: In general, the Bresenham Algorithm, have no floating point numbers, no divisions and it can be implemented using bit shifting as multiplying operation.