Bresenham’s Line Generation Algorithm

Given coordinate of two points A(x1, y1) and B(x2, y2). The task to find all the intermediate points required for drawing line AB on the computer screen of pixels. Note that every pixel has integer coordinates.

Examples:

Input  : A(0,0), B(4,4)
Output : (0,0), (1,1), (2,2), (3,3), (4,4)

Input  : A(0,0), B(4,2)
Output : (0,0), (1,0), (2,1), (3,1), (4,2)

Below are some assumptions to keep algorithm simple.

We draw line from left to right.
x1 < x2 and y1< y2
Slope of the line is between 0 and 1. We draw a line from lower left to upper right.

Let us understand the process by considering the naive way first.

// A naive way of drawing line
void naiveDrawLine(x1, x2, y1, y2)
{
   m = (y2 - y1)/(x2 - x1)
   for (x  = x1; x <= x2; x++) 
   {    
      // Assuming that the round function finds
      // closest integer to a given float.
      y = round(mx + c);    
      print(x, y); 
   }
}

Above algorithm works, but it is slow. The idea of Bresenham’s algorithm is to avoid floating point multiplication and addition to compute mx + c, and then computing round value of (mx + c) in every step. In Bresenham’s algorithm, we move across the x-axis in unit intervals.

We always increase x by 1, and we choose about next y, whether we need to go to y+1 or remain on y. In other words, from any position (X_k, Y_k) we need to choose between (X_k + 1, Y_k) and (X_k + 1, Y_k + 1).
We would like to pick the y value (among Y_k + 1 and Y_k) corresponding to a point that is closer to the original line.

We need to a decision parameter to decide whether to pick Y_k + 1 or Y_k as next point. The idea is to keep track of slope error from previous increment to y. If the slope error becomes greater than 0.5, we know that the line has moved upwards one pixel, and that we must increment our y coordinate and readjust the error to represent the distance from the top of the new pixel – which is done by subtracting one from error.

// Modifying the naive way to use a parameter
// to decide next y.
void withDecisionParameter(x1, x2, y1, y2)
{
   m = (y2 - y1)/(x2 - x1)
   slope_error = [Some Initial Value]
   for (x = x1, y = y1; x = 0.5)  
   {       
      y++;       
      slope_error  -= 1.0;    
   }
}

How to avoid floating point arithmetic
The above algorithm still includes floating point arithmetic. To avoid floating point arithmetic, consider the value below value m.

m = (y2 – y1)/(x2 – x1)

We multiply both sides by (x2 – x1)

We also change slope_error to slope_error * (x2 – x1). To avoid comparison with 0.5, we further change it to slope_error * (x2 – x1) * 2.

Also, it is generally preferred to compare with 0 than 1.

// Modifying the above algorithm to avoid floating 
// point arithmetic and use comparison with 0.
void bresenham(x1, x2, y1, y2)
{
   m_new = 2 * (y2 - y1)
   slope_error_new = [Some Initial Value]
   for (x = x1, y = y1; x = 0)  
   {       
      y++;       
      slope_error_new  -= 2 * (x2 - x1);    
   }
}

The initial value of slope_error_new is 2*(y2 – y1) – (x2 – x1). Refer this for proof of this value

Below is the implementation of above algorithm.

C++

// C++ program for Bresenham’s Line Generation 
// Assumptions : 
// 1) Line is drawn from left to right. 
// 2) x1 < x2 and y1 < y2 
// 3) Slope of the line is between 0 and 1. 
//    We draw a line from lower left to upper 
//    right. 
#include<bits/stdc++.h> 
using namespace std; 
  
// function for line generation 
void bresenham(int x1, int y1, int x2, int y2) 
{ 
   int m_new = 2 * (y2 - y1); 
   int slope_error_new = m_new - (x2 - x1); 
   for (int x = x1, y = y1; x <= x2; x++) 
   { 
      cout << "(" << x << "," << y << ")\n"; 
  
      // Add slope to increment angle formed 
      slope_error_new += m_new; 
  
      // Slope error reached limit, time to 
      // increment y and update slope error. 
      if (slope_error_new >= 0) 
      { 
         y++; 
         slope_error_new  -= 2 * (x2 - x1); 
      } 
   } 
} 
  
// driver function 
int main() 
{ 
  int x1 = 3, y1 = 2, x2 = 15, y2 = 5; 
  bresenham(x1, y1, x2, y2); 
  return 0; 
} 

Java

// Java program for Bresenhams Line Generation 
// Assumptions : 
// 1) Line is drawn from left to right. 
// 2) x1 < x2 and y1 < y2 
// 3) Slope of the line is between 0 and 1. 
// We draw a line from lower left to upper 
// right. 
class GFG 
{ 
    // function for line generation 
    static void bresenham(int x1, int y1, int x2, 
                                         int y2) 
    { 
        int m_new = 2 * (y2 - y1); 
        int slope_error_new = m_new - (x2 - x1); 
      
        for (int x = x1, y = y1; x <= x2; x++) 
        { 
            System.out.print("(" +x + "," + y + ")\n"); 
  
            // Add slope to increment angle formed 
            slope_error_new += m_new; 
  
            // Slope error reached limit, time to 
            // increment y and update slope error. 
            if (slope_error_new >= 0) 
            { 
                y++; 
                slope_error_new -= 2 * (x2 - x1); 
            } 
        } 
    }          
  
    // Driver code  
    public static void main (String[] args) 
    { 
        int x1 = 3, y1 = 2, x2 = 15, y2 = 5;     
        bresenham(x1, y1, x2, y2); 
    } 
} 
  
// This code is contributed by Anant Agarwal. 

Python3

# Python 3 program for Bresenham’s Line Generation  
# Assumptions :  
# 1) Line is drawn from left to right.  
# 2) x1 < x2 and y1 < y2  
# 3) Slope of the line is between 0 and 1.  
# We draw a line from lower left to upper  
# right.  
  
  
# function for line generation  
def bresenham(x1,y1,x2, y2):  
  
    m_new = 2 * (y2 - y1)  
    slope_error_new = m_new - (x2 - x1) 
  
    y=y1 
    for x in range(x1,x2+1):  
      
        print("(",x ,",",y ,")\n")  
  
        # Add slope to increment angle formed  
        slope_error_new =slope_error_new + m_new  
  
        # Slope error reached limit, time to  
        # increment y and update slope error.  
        if (slope_error_new >= 0):  
            y=y+1
            slope_error_new =slope_error_new - 2 * (x2 - x1)  
          
      
  
  
# driver function  
if __name__=='__main__': 
    x1 = 3
    y1 = 2
    x2 = 15
    y2 = 5
    bresenham(x1, y1, x2, y2)  
  
#This code is contributed by ash264 

C#

// C# program for Bresenhams Line Generation 
// Assumptions : 
// 1) Line is drawn from left to right. 
// 2) x1 < x2 and y1< y2 
// 3) Slope of the line is between 0 and 1. 
// We draw a line from lower left to upper 
// right. 
using System; 
  
class GFG { 
      
    // function for line generation 
    static void bresenham(int x1, int y1, int x2, 
                                        int y2) 
    { 
          
        int m_new = 2 * (y2 - y1); 
        int slope_error_new = m_new - (x2 - x1); 
      
        for (int x = x1, y = y1; x <= x2; x++) 
        { 
            Console.Write("(" + x + "," + y + ")\n"); 
  
            // Add slope to increment angle formed 
            slope_error_new += m_new; 
  
            // Slope error reached limit, time to 
            // increment y and update slope error. 
            if (slope_error_new >= 0) 
            { 
                y++; 
                slope_error_new -= 2 * (x2 - x1); 
            } 
        } 
    }          
  
    // Driver code  
    public static void Main () 
    { 
        int x1 = 3, y1 = 2, x2 = 15, y2 = 5;  
          
        bresenham(x1, y1, x2, y2); 
    } 
} 
  
// This code is contributed by nitin mittal. 

PHP

<?php 
// PHP program for Bresenham’s  
// Line Generation Assumptions : 
  
// 1) Line is drawn from 
// left to right. 
// 2) x1 < x2 and y1 < y2 
// 3) Slope of the line is  
// between 0 and 1. 
// We draw a line from lower  
// left to upper right. 
  
// function for line generation 
function bresenham($x1, $y1, $x2, $y2) 
{ 
$m_new = 2 * ($y2 - $y1); 
$slope_error_new = $m_new - ($x2 - $x1); 
for ($x = $x1, $y = $y1; $x <= $x2; $x++) 
{ 
    echo "(" ,$x , "," , $y, ")\n"; 
  
    // Add slope to increment 
    // angle formed 
    $slope_error_new += $m_new; 
  
    // Slope error reached limit,  
    // time to increment y and  
    // update slope error. 
    if ($slope_error_new >= 0) 
    { 
        $y++; 
        $slope_error_new -= 2 * ($x2 - $x1); 
    } 
} 
} 
  
// Driver Code 
$x1 = 3; $y1 = 2; $x2 = 15; $y2 = 5; 
bresenham($x1, $y1, $x2, $y2); 
  
// This code is contributed by nitin mittal. 
?> 

Output :

(3,2)
(4,3)
(5,3)
(6,3)
(7,3)
(8,4)
(9,4)
(10,4)
(11,4)
(12,5)
(13,5)
(14,5)
(15,5)

The above explanation is to provides a rough idea behind the algorithm. For detailed explanation and proof, readers can refer below references.

Related Articles:

Reference :
https://csustan.csustan.edu/~tom/Lecture-Notes/Graphics/Bresenham-Line/Bresenham-Line.pdf
https://en.wikipedia.org/wiki/Bresenham’s_line_algorithm
http://graphics.idav.ucdavis.edu/education/GraphicsNotes/Bresenhams-Algorithm.pdf

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