Line drawing algorithm

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Two rasterized lines. The colored pixels are shown as circles. Above: monochrome screening; below: Gupta-Sproull anti-aliasing; the ideal line is considered here as a surface.

In computer graphics, a line drawing algorithm is an algorithm for approximating a line segment on discrete graphical media, such as pixel-based displays and printers. On such media, line drawing requires an approximation (in nontrivial cases). Basic algorithms rasterize lines in one color. A better representation with multiple color gradations requires an advanced process, spatial anti-aliasing.

On continuous media, by contrast, no algorithm is necessary to draw a line. For example, cathode-ray oscilloscopes use analog phenomena to draw lines and curves.

List of line drawing algorithms[edit]

Lines using Xiaolin Wu's algorithm, showing "ropey" appearance

The following is a partial list of line drawing algorithms:

A naive line-drawing algorithm[edit]

The simplest method of screening is the direct drawing of the equation defining the line.

dx = x2 − x1
dy = y2 − y1
for x from x1 to x2 do
    y = y1 + dy × (x − x1) / dx
    plot(x, y)

It is here that the points have already been ordered so that . This algorithm works just fine when (i.e., slope is less than or equal to 1), but if (i.e., slope greater than 1), the line becomes quite sparse with many gaps, and in the limiting case of , a division by zero exception will occur.

The naive line drawing algorithm is inefficient and thus, slow on a digital computer. Its inefficiency stems from the number of operations and the use of floating-point calculations. Algorithms such as Bresenham's line algorithm or Xiaolin Wu's line algorithm are preferred instead.

Gupta and Sproull algorithm[edit]

The Gupta-Sproull algorithm is based on Bresenham's line algorithm but adds antialiasing.

An optimized variant of the Gupta-Sproull algorithm can be written in pseudocode as follows:

DrawLine(x1, x2, y1, y2) {
    x = x1;
    y = y1;
    dx = x2 − x1;
    dy = y2 − y1;
    d = 2 * dy − dx; // discriminator
    // Euclidean distance of point (x,y) from line (signed)
    D = 0; 
    // Euclidean distance between points (x1, y1) and (x2, y2)
    length = sqrt(dx * dx + dy * dy); 
    sin = dx / length;     
    cos = dy / length;
    while (x <= x2) {
        IntensifyPixels(x, y − 1, D + cos);
        IntensifyPixels(x, y, D);
        IntensifyPixels(x, y + 1, D − cos);
        x = x + 1
        if (d <= 0) {
            D = D + sin;
            d = d + 2 * dy;
        } else {
            D = D + sin − cos;
            d = d + 2 * (dy − dx);
            y = y + 1;

The IntensifyPixels(x,y,r) function takes a radial line transformation and sets the intensity of the pixel (x,y) with the value of a cubic polynomial that depends on the pixel's distance r from the line.


  • Fundamentals of Computer Graphics, 2nd Edition, A.K. Peters by Peter Shirley