Bresenham's Line Drawing Algorithm is a fundamental algorithm in computer graphics for rendering a straight line on a pixelated display. Introduced by Jack Bresenham in 1962, it has become a staple in the field due to its efficiency and simplicity. The algorithm's primary advantage lies in its ability to use only integer arithmetic, making it particularly suitable for implementation in hardware and low-level software environments where floating-point operations are costly.

Introduction

In the world of computer graphics, drawing a straight line between two points on a raster display (a grid of pixels) is a common but non-trivial task. The naive approach of calculating the slope and using the equation of a line can lead to inefficiencies and inaccuracies due to floating-point arithmetic. Bresenham's Line Drawing Algorithm addresses these issues by providing an efficient way to approximate a line using only integer operations.

The Algorithm

Bresenham's algorithm works by determining which pixel on the raster grid should be lit to best approximate the desired line. It incrementally plots the line from the start point to the end point, making decisions at each step about which pixel to choose next based on the accumulated error from the ideal line.

Algorithm Steps

1. Initialization:

   • Start at the initial point (𝑥0,𝑦0)(x0​,y0​).
   • Calculate the differences Δ𝑥=𝑥1−𝑥0Δx=x1​−x0​ and Δ𝑦=𝑦1−𝑦0Δy=y1​−y0​.
   • Determine the absolute values of these differences and the signs of the increments for x and y (𝑠𝑡𝑒𝑝𝑥stepx​ and 𝑠𝑡𝑒𝑝𝑦stepy​).

2. Decision Variable:

Initialize the decision variable 𝑑=2Δ𝑦−Δ𝑥d=2Δy−Δx.

1.  

2. Iterate and Plot:

   • For each x from 𝑥0x0​ to 𝑥1x1​:
     • Plot the current pixel (𝑥,𝑦)(x,y).
     • If 𝑑≥0d≥0, increment 𝑦y by 𝑠𝑡𝑒𝑝𝑦stepy​ and update 𝑑d by subtracting 2Δ𝑥2Δx.
     • Increment 𝑥x by 𝑠𝑡𝑒𝑝𝑥stepx​ and update 𝑑d by adding 2Δ𝑦2Δy.

Consider drawing a line from (2,3)(2,3) to (10,8)(10,8):

1. Initialization:

   • Δ𝑥=10−2=8Δx=10−2=8
   • Δ𝑦=8−3=5Δy=8−3=5
   • 𝑠𝑡𝑒𝑝𝑥=1stepx​=1, 𝑠𝑡𝑒𝑝𝑦=1stepy​=1
   • Initial decision variable 𝑑=2Δ𝑦−Δ𝑥=2⋅5−8=2d=2Δy−Δx=2⋅5−8=2

2. Plotting:

   • Start at (2,3)(2,3)
   • For each step in 𝑥x:
     • Plot (2,3)(2,3), 𝑑=2d=2
     • 𝑑≥0d≥0, so increment y: 𝑦=4y=4, update 𝑑=𝑑−2Δ𝑥=2−16=−14d=d−2Δx=2−16=−14
     • Increment x: 𝑥=3x=3, update 𝑑=𝑑+2Δ𝑦=−14+10=−4d=d+2Δy=−14+10=−4
     • Plot (3,4)(3,4), 𝑑=−4d=−4
     • 𝑑<0d<0, so increment x: 𝑥=4x=4, update 𝑑=𝑑+2Δ𝑦=−4+10=6d=d+2Δy=−4+10=6
     • Plot (4,4)(4,4), 𝑑=6d=6
     • Continue this process until reaching (10,8)(10,8).

Conclusion

Bresenham’s Line Drawing Algorithm remains a critical technique in computer graphics due to its balance of efficiency, simplicity, and precision. Its reliance on integer calculations makes it particularly suited for environments where performance is crucial, ensuring it remains relevant even with advances in graphical processing power. Whether in simple pixel art or complex graphical applications, Bresenham’s algorithm continues to be a foundational tool for rendering straight lines on raster displays.