# 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.

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- 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.

// Anaive wayof 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.` `?>` |

## Javascript

`<script>` ` ` `// javascript 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.` ` ` ` ` `// function for line generation` ` ` `function` `bresenham(x1 , y1 , x2,y2)` ` ` `{` ` ` `var` `m_new = 2 * (y2 - y1);` ` ` `var` `slope_error_new = m_new - (x2 - x1);` ` ` ` ` `for` `(x = x1, y = y1; x <= x2; x++)` ` ` `{` ` ` `document.write(` `"("` `+x + ` `","` `+ y + ` `")<br>"` `);` ` ` ` ` `// 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 ` ` ` `var` `x1 = 3, y1 = 2, x2 = 15, y2 = 5; ` `bresenham(x1, y1, x2, y2);` ` ` `// This code is contributed by Amit Katiyar ` ` ` `</script>` |

**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 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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