• Graphics
  • Line

The Bresenham line algorithm is an algorithm which determines which order to form a close approximation to a straight line between two given points. It is commonly used to draw lines on a computer screen, as it uses only integer addition, subtraction and bit shifting, all of which are very cheap operations in standard computer architectures. It is one of the earliest algorithms developed in the field of computer graphics. A minor extension to the original algorithm also deals with drawing circles.

While algorithms such as Wu's algorithm are also frequently used in modern computer graphics because they can support antialiasing, the speed and simplicity of Bresenham's line algorithm means that it is still important. The algorithm is used in hardware such as plotters and in the graphics chips of modern graphics cards. It can also be found in many software graphics libraries. Because the algorithm is very simple, it is often implemented in either the firmware or the graphics hardware of modern graphics cards.

The label "Bresenham" is used today for a whole family of algorithms extending or modifying Bresenham's original algorithm. See further references below.

The algorithm

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 common conventions will be used:

  • 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
  • that the pixel centers have integer coordinates.

The endpoints of the line are the pixels at (x0y0) and (x1y1), 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 in which the segment goes down and to the right (x0x1 andy0y1), and its horizontal projection x_1-x_0 is longer than the vertical projection y_1-y_0 (the line has a negativeslope whose absolute value is less than 1.) In this octant, for each column x between x_0 and x_1, there is exactly one rowy (computed by the algorithm) containing a pixel of the line, while each row between y_0 and y_1 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:

\frac{y - y_0}{y_1-y_0} = \frac{x-x_0}{x_1-x_0}.

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

y = \frac{y_1-y_0}{x_1-x_0} (x-x_0) + y_0.

The slope (y_1-y_0)/(x_1-x_0) 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 y_0 and repeatedly adding the slope.

In practice, the algorithm can track, instead of possibly large y values, a small error value between −0.5 and 0.5: the vertical distance between the rounded and the exact y values for the current x. Each time x is increased, the error is increased by the slope; if it exceeds 0.5, the rasterization y is increased by 1 (the line continues on the next lower row of the raster) and the error is decremented by 1.0.

C++ Other implementations: Java