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.
- We draw lines 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.
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 (Xk, Yk) we need to choose between (Xk + 1, Yk) and (Xk + 1, Yk + 1).
We would like to pick the y value (among Yk + 1 and Yk) corresponding to a point that is closer to the original line.
We need a decision parameter to decide whether to pick Yk + 1 or Yk 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
Related Articles:
This article is contributed byShivam Pradhan. If you like neveropen 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 neveropen 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.
Feeling lost in the world of random DSA topics, wasting time without progress? It’s time for a change! Join our DSA course, where we’ll guide you on an exciting journey to master DSA efficiently and on schedule.
Ready to dive in? Explore our Free Demo Content and join our DSA course, trusted by over 100,000 neveropen!
Ready to dive in? Explore our Free Demo Content and join our DSA course, trusted by over 100,000 neveropen!
