Bresenham’s Line Generation Algorithm

Given the coordinate of two points A(x1, y1) and B(x2, y2). The task is 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 the algorithm simple. 

1. We draw lines 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.

Naive Approach:

C++

// 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);
    }
}

Java

/*package whatever //do not write package name here */
import java.io.*;
class GFG {

    // A naive way of drawing line
    public static 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);
        }
    }
    public static void main(String[] args) {}
}

// This code is contributed by akashish__

Python3

# A naive way of drawing line
def naiveDrawLine(x1, x2, y1, y2):
    m = (y2 - y1) / (x2 - x1)

    # for (x = x1; x <= x2; x++) {
    for x in range(x1, x2 + 1):

        # Assuming that the round function finds
        # closest integer to a given float.
        y = round(mx + c)

        print(x, y)

# This code is contributed by akashish__

C#

using System;

public class GFG {

  // A naive way of drawing line
  public static 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);
    }
  }

  static public void Main()
  {

    // Code
  }
}

// This code is contributed by akashish__

Javascript

// A naive way of drawing line

function 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 = Math.round(mx + c);

        print(x, y);
    }
}

// This code is contributed by garg28harsh.

The 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 compute the 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 a decision parameter to decide whether to pick Y_k + 1 or Y_k as the next point. The idea is to keep track of slope error from the 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 the error.

C++

// 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;
    }
}

Java

/*package whatever //do not write package name here */
import java.io.*;

class GFG {

  // Modifying the naive way to use a parameter
  // to decide next y.
  public static 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;
    }
  }

  public static void main (String[] args) {
  }
}

// This code is contributed by akashish__

Python3

# Modifying the naive way to use a parameter
# to decide next y.
def withDecisionParameter(x1, x2, y1, y2):
    m = (y2 - y1) / (x2 - x1)
    slope_error = [Some Initial Value]
    for x in range(0.5,x1) and y in range(y1):
        y += 1
        slope_error -= 1.0


# This code is contributed by akashish__

C#

using System;

public class GFG {

    // Modifying the naive way to use a parameter
    // to decide next y.
    public static 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;
        }
    }

    static public void Main() {}
}

// This code is contributed by akashish__

Javascript

// Modifying the naive way to use a parameter
// to decide next y.
function 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;
    }
}

// This code is contributed by akashish__

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. 
  

C++

// 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);
    }
}

Java

public static void bresenham(int x1, int x2, int y1, int y2) {
    int m_new = 2 * (y2 - y1);
    int slope_error_new = 0;
    for (int x = x1, y = y1; x = 0😉 {
        y++;
        slope_error_new -= 2 * (x2 - x1);
    }
}

// This code is contributed by ishankhandelwals.

Python3

# Modifying the above algorithm to avoid floating
# point arithmetic and use comparison with 0.
def bresenham(x1, x2, y1, y2):
  m_new = 2 * (y2 - y1)
  slope_error_new = 0
  y = y1
  for x in range(x1, 0, -1) {
      y += 1
      slope_error_new -= 2 * (x2 - x1)

# This code is contributed by akashish__

C#

using System;

public class GFG{

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

    static public void Main (){

        // Code
    }
}

// This code is contributed by akashish__

Javascript

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

// This code is contributed by akashish__

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

Below is the implementation of the 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 code
int main()
{
    int x1 = 3, y1 = 2, x2 = 15, y2 = 5;

    // Function call
    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
                & quot;);

            // 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& gt; = 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;

        // Function call
        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 code
if __name__ == '__main__':
    x1 = 3
    y1 = 2
    x2 = 15
    y2 = 5

    # Function call
    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
                                 & quot;);

            // 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& gt; = 0) {
                y++;
                slope_error_new -= 2 * (x2 - x1);
            }
        }
    }

    // Driver code
    public static void Main()
    {
        int x1 = 3, y1 = 2, x2 = 15, y2 = 5;

        // Function call
        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;

// Function call
bresenham($x1, $y1, $x2, $y2);

// This code is contributed by nitin mittal.
?>

Javascript

// Javascript program for Bresenhams Line Generation

function plotPixel(x1, y1, x2,
                                y2, dx, dy,
                                decide)
    {
        // pk is initial decision making parameter
        // Note:x1&y1,x2&y2, dx&dy values are interchanged
        // and passed in plotPixel function so
        // it can handle both cases when m>1 & m<1
        let pk = 2 * dy - dx;
        for (let i = 0; i <= dx; i++) {
              if(decide == 0){
              console.log(x1 + "," + y1);
            }
              else{
              console.log(y1 + "," + x1);
            }

            // checking either to decrement or increment the
            // value if we have to plot from (0,100) to
            // (100,0)
            if (x1 < x2)
                x1++;
            else
                x1--;
            if (pk < 0) {
                // decision value will decide to plot
                // either x1 or y1 in x's position
                if (decide == 0) {
                    pk = pk + 2 * dy;
                }
                else
                    pk = pk + 2 * dy;
            }
            else {
                if (y1 < y2)
                    y1++;
                else
                    y1--;
                pk = pk + 2 * dy - 2 * dx;
            }
        }
    }

    // Driver code
        let x1 = 100, y1 = 110, x2 = 125, y2 = 120, dx, dy;
        dx = Math.abs(x2 - x1);
        dy = Math.abs(y2 - y1);
        // If slope is less than one
        if (dx > dy) {
            // passing argument as 0 to plot(x,y)
            plotPixel(x1, y1, x2, y2, dx, dy, 0);
        }
        // if slope is greater than or equal to 1
        else {
            // passing argument as 1 to plot (y,x)
            plotPixel(y1, x1, y2, x2, dy, dx, 1);
        }

// This code is contributed by akashish__ and kalil

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)

Time Complexity: O(x2 – x1)
Auxiliary Space: O(1) 
The above explanation is to provide a rough idea behind the algorithm. For detailed explanation and proof, readers can refer below references.

The above program only works if the slope of the line is less than 1. Here is a program implementation for any kind of slope.

Output

100,110
101,110
102,111
103,111
104,112
105,112
106,112
107,113
108,113
109,114
110,114
111,114
112,115
113,115
114,116
115,116
116,116
117,117
118,117
119,118
120,118
121,118
122,119
123,119
124,120
125,120

This article is contributed by Shivam Pradhan (anuj_charm). If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.