Line drawing algorithm

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.

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.

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

The Cartesian slope-intercept equation for a straight line is ${\displaystyle Y=mx+b}$ With m representing the slope of the line and b as the y-intercept. Given that the two endpoints of the line segment are specified at positions ${\displaystyle (x1,y1)}$ and ${\displaystyle (x2,y2)}$, we can determine values for the slope m and y-intercept b with the following calculations: ${\displaystyle m=(y2-y1)/(x2-x1)}$ so, ${\displaystyle b=y1-m*x1}$.

List of line drawing algorithms

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

The following is a partial list of line drawing algorithms:

• naive algorithm
• Digital Differential Analyzer (graphics algorithm) — Similar to the naive line-drawing algorithm, with minor variations.
• Bresenham's line algorithm — optimized to use only additions (i.e. no divisions or multiplications); it also avoids floating-point computations.
• Xiaolin Wu's line algorithm — can perform spatial anti-aliasing, appears "ropey" from brightness varying along the length of the line
• Gupta-Sproull algorithm

A naive line-drawing algorithm

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 ${\displaystyle x_{2}>x_{1}}$. This algorithm works just fine when ${\displaystyle dx>=dy}$ (i.e., slope is less than or equal to 1), but if ${\displaystyle dx<dy}$ (i.e., slope greater than 1), the line becomes quite sparse with lots of gaps, and in the limiting case of ${\displaystyle dx=0}$, a division by zero exception will occur.

The naïve 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. Line drawing algorithms such as Bresenham's or Wu's are preferred instead.

References

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