DDA Line generation Algorithm in Computer Graphics

Introduction :

DDA (Digital Differential Analyzer) is a line drawing algorithm used in computer graphics to generate a line segment between two specified endpoints. It is a simple and efficient algorithm that works by using the incremental difference between the x-coordinates and y-coordinates of the two endpoints to plot the line.

The steps involved in DDA line generation algorithm are:

1. Input the two endpoints of the line segment, (x1,y1) and (x2,y2).
2. Calculate the difference between the x-coordinates and y-coordinates of the endpoints as dx and dy respectively.
3. Calculate the slope of the line as m = dy/dx.
4. Set the initial point of the line as (x1,y1).
5. Loop through the x-coordinates of the line, incrementing by one each time, and calculate the corresponding y-coordinate using the equation y = y1 + m(x - x1).
6. Plot the pixel at the calculated (x,y) coordinate.
7. Repeat steps 5 and 6 until the endpoint (x2,y2) is reached.

DDA algorithm is relatively easy to implement and is computationally efficient, making it suitable for real-time applications. However, it has some limitations, such as the inability to handle vertical lines and the need for floating-point arithmetic, which can be slow on some systems. Nonetheless, it remains a popular choice for generating lines in computer graphics.

In any 2-Dimensional plane, if we connect two points (x0, y0) and (x1, y1), we get a line segment. But in the case of computer graphics, we can not directly join any two coordinate points, for that, we should calculate intermediate points' coordinates and put a pixel for each intermediate point, of the desired color with the help of functions like putpixel(x, y, K) in C, where (x,y) is our co-ordinate and K denotes some color.

Examples: 

Input: For line segment between (2, 2) and (6, 6) :
Output: we need (3, 3) (4, 4) and (5, 5) as our intermediate points.

Input: For line segment between (0, 2) and (0, 6) :
Output: we need (0, 3) (0, 4) and (0, 5) as our intermediate points.

For using graphics functions, our system output screen is treated as a coordinate system where the coordinate of the top-left corner is (0, 0) and as we move down our y-ordinate increases, and as we move right our x-ordinate increases for any point (x, y). Now, for generating any line segment we need intermediate points and for calculating them we can use a basic algorithm called DDA(Digital differential analyzer) line generating algorithm.
 

DDA Algorithm: 

Consider one point of the line as (X0, Y0) and the second point of the line as (X1, Y1). 

// calculate dx , dy

dx = X1 - X0;
dy = Y1 - Y0;

// Depending upon absolute value of dx & dy
// choose number of steps to put pixel as

// steps = abs(dx) > abs(dy) ? abs(dx) : abs(dy)
steps = abs(dx) > abs(dy) ? abs(dx) : abs(dy);

// calculate increment in x & y for each steps

Xinc = dx / (float) steps;
Yinc = dy / (float) steps;

// Put pixel for each step

X = X0;
Y = Y0;

for (int i = 0; i <= steps; i++)
{
    putpixel (round(X),round(Y),WHITE);
    X += Xinc;
    Y += Yinc;
}

Below is the implementation of the above approach:

// C program for DDA line generation

#include <graphics.h>
#include <math.h>
#include <stdio.h>
// Function for finding absolute value
int abs(int n) { return ((n > 0) ? n : (n * (-1))); }

// DDA Function for line generation
void DDA(int X0, int Y0, int X1, int Y1)
{
    // calculate dx & dy
    int dx = X1 - X0;
    int dy = Y1 - Y0;

    // calculate steps required for generating pixels
    int steps = abs(dx) > abs(dy) ? abs(dx) : abs(dy);

    // calculate increment in x & y for each steps
    float Xinc = dx / (float)steps;
    float Yinc = dy / (float)steps;

    // Put pixel for each step
    float X = X0;
    float Y = Y0;
    for (int i = 0; i <= steps; i++) {
        putpixel(round(X), round(Y),
                 RED); // put pixel at (X,Y)
        X += Xinc; // increment in x at each step
        Y += Yinc; // increment in y at each step
        delay(100); // for visualization of line-
                    // generation step by step
    }
}

// Driver program
int main()
{
    int gd = DETECT, gm;

    // Initialize graphics function
    initgraph(&gd, &gm, "");

    int X0 = 2, Y0 = 2, X1 = 14, Y1 = 16;

    // Function call
    DDA(2, 2, 14, 16);
    return 0;
}

// C++ program for DDA line generation

#include <bits/stdc++.h>
using namespace std;

// function for rounding off the pixels
int round(float n)
{
    if (n - (int)n < 0.5)
        return (int)n;
    return (int)(n + 1);
}

// Function for line generation
void DDALine(int x0, int y0, int x1, int y1)
{

    // Calculate dx and dy
    int dx = x1 - x0;
    int dy = y1 - y0;

    int step;

    // If dx > dy we will take step as dx
    // else we will take step as dy to draw the complete
    // line
    if (abs(dx) > abs(dy))
        step = abs(dx);
    else
        step = abs(dy);

    // Calculate x-increment and y-increment for each step
    float x_incr = (float)dx / step;
    float y_incr = (float)dy / step;

    // Take the initial points as x and y
    float x = x0;
    float y = y0;

    for (int i = 0; i < step; i++) {

        // putpixel(round(x), round(y), WHITE);
        cout << round(x) << " " << round(y) << "\n";
        x += x_incr;
        y += y_incr;
        // delay(10);
    }
}

// Driver code
int main()
{

    int x0 = 200, y0 = 180, x1 = 180, y1 = 160;

    // Function call
    DDALine(x0, y0, x1, y1);

    return 0;
}

// all functions regarding to graphichs.h are commented out
// contributed by hopelessalexander

// Java Code for DDA line generation 
public class Solution {

  // function for rounding off the pixels
  public static int round(float n) {
    if (n - (int) n < 0.5)
      return (int) n;
    return (int) (n + 1);
  }

  // Function for line generation
  public static void DDALine(int x0, int y0, int x1, int y1) {

    // Calculate dx and dy
    int dx = x1 - x0;
    int dy = y1 - y0;

    int step;

    // If dx > dy we will take step as dx
    // else we will take step as dy to draw the complete
    // line
    if (Math.abs(dx) > Math.abs(dy))
      step = Math.abs(dx);
    else
      step = Math.abs(dy);

    // Calculate x-increment and y-increment for each step
    float x_incr = (float) dx / step;
    float y_incr = (float) dy / step;

    // Take the initial points as x and y
    float x = x0;
    float y = y0;

    for (int i = 0; i < step; i++) {

      // putpixel(round(x), round(y), WHITE);
      System.out.println(round(x) + " " + round(y));
      x += x_incr;
      y += y_incr;
      // delay(10);
    }
  }

  // Driver code
  public static void main(String[] args) {

    int x0 = 200, y0 = 180, x1 = 180, y1 = 160;

    // Function call
    DDALine(x0, y0, x1, y1);

  }
}

// This code is contributed by ishankhandelwals.

# Python program for DDA line generation

from matplotlib import pyplot as plt

# DDA Function for line generation


def DDA(x0, y0, x1, y1):

    # find absolute differences
    dx = abs(x0 - x1)
    dy = abs(y0 - y1)

    # find maximum difference
    steps = max(dx, dy)

    # calculate the increment in x and y
    xinc = dx/steps
    yinc = dy/steps

    # start with 1st point
    x = float(x0)
    y = float(y0)

    # make a list for coordinates
    x_coorinates = []
    y_coorinates = []

    for i in range(steps):
        # append the x,y coordinates in respective list
        x_coorinates.append(x)
        y_coorinates.append(y)

        # increment the values
        x = x + xinc
        y = y + yinc

    # plot the line with coordinates list
    plt.plot(x_coorinates, y_coorinates, marker="o",
             markersize=1, markerfacecolor="green")
    plt.show()


# Driver code
if __name__ == "__main__":

    # coordinates of 1st point
    x0, y0 = 20, 20

    # coordinates of 2nd point
    x1, y1 = 60, 50

    # Function call
    DDA(x0, y0, x1, y1)

    # This code is contributed by 111arpit1

// C# code for DDA line generation
using System;
public class Solution 
{
  
  // function for rounding off the pixels
  public static int Round(float n)
  {
    if (n - (int)n < 0.5)
      return (int)n;
    return (int)(n + 1);
  }

  // Function for line generation
  public static void DDALine(int x0, int y0, int x1,
                             int y1)
  {

    // Calculate dx and dy
    int dx = x1 - x0;
    int dy = y1 - y0;

    int step;

    // If dx > dy we will take step as dx
    // else we will take step as dy to draw the complete
    // line
    if (Math.Abs(dx) > Math.Abs(dy))
      step = Math.Abs(dx);
    else
      step = Math.Abs(dy);

    // Calculate x-increment and y-increment for each
    // step
    float x_incr = (float)dx / step;
    float y_incr = (float)dy / step;

    // Take the initial points as x and y
    float x = x0;
    float y = y0;

    for (int i = 0; i < step; i++) {

      // putpixel(round(x), round(y), WHITE);
      Console.WriteLine(Round(x) + " " + Round(y));
      x += x_incr;
      y += y_incr;
      // delay(10);
    }
  }

  // Driver code
  public static void Main(string[] args)
  {

    int x0 = 200, y0 = 180, x1 = 180, y1 = 160;

    // Function call
    DDALine(x0, y0, x1, y1);
  }
}

// This code is contributed by ishankhandelwals

// JS program for DDA Line generation

function round(n) {
    if (n - Math.floor(n) < 0.5)
        return Math.floor(n);
    return Math.floor(n + 1);
};

function DDALine(x0, y0, x1, y1) {
    let dx = x1 - x0;
    let dy = y1 - y0;
    let step;

    if (Math.abs(dx) > Math.abs(dy))
        step = Math.abs(dx);
    else
        step = Math.abs(dy);

    let x_incr = (dx / step);
    let y_incr = (dy / step);

    let x = x0;
    let y = y0;

    for (let i = 0; i < step; i++) {
        console.log(round(x) + " " + round(y));
        x += x_incr;
        y += y_incr;
    }
};

let x0 = 200, y0 = 180, x1 = 180, y1 = 160;
DDALine(x0, y0, x1, y1);

// This code is contributed by ishankhandelwals.

Output: 

200 180
199 179
198 178
197 177
196 176
195 175
194 174
193 173
192 172
191 171
190 170
189 169
188 168
187 167
186 166
185 165
184 164
183 163
182 162
181 161

Advantages of DDA Algorithm: 

• It is a simple and easy-to-implement algorithm.
• It avoids using multiple operations which have high time complexities.
• It is faster than the direct use of the line equation because it does not use any floating point multiplication and it calculates points on the line.

Disadvantages of DDA Algorithm: 

• It deals with the rounding off operation and floating point arithmetic so it has high time complexity.
• As it is orientation-dependent, so it has poor endpoint accuracy.
• Due to the limited precision in the floating point representation, it produces a cumulative error.

Bresenham’s Line Generation Algorithm

Uses of DDA Algorithm: 

DDA (Digital Differential Analyzer) algorithm is commonly used in computer graphics for line drawing. It has a wide range of applications, including:

1. Creating basic graphics primitives: DDA algorithm can be used to draw simple shapes such as lines, polygons, and rectangles. By using a series of line segments generated using DDA, more complex shapes can also be created.
2. Computer-aided design (CAD): In CAD software, DDA algorithm is used to draw lines between two points, which are used to create 2D and 3D models.
3. Image processing: DDA algorithm can be used in image processing for tasks such as edge detection and image segmentation.
4. Video game development: DDA algorithm is used for rendering lines and polygons in real-time graphics rendering for video games.
5. Simulation and modeling: DDA algorithm is used to simulate physical phenomena such as ray tracing, which is used in computer graphics to create realistic images of 3D objects.
    

Issues in DDA Algorithm: 

some limitations and issues, which are:

1. Floating point arithmetic: The DDA algorithm requires floating-point arithmetic, which can be slow on some systems. This can be a problem when dealing with large datasets.
2. Limited precision: The use of floating-point arithmetic can lead to limited precision in some cases, especially when the slope of the line is very steep or shallow.
3. Round-off errors: Round-off errors can occur during calculations, which can lead to inaccuracies in the generated line. This is particularly true when the slope of the line is close to 1.
4. Inability to handle vertical lines: The DDA algorithm is unable to handle vertical lines, as the slope becomes undefined.
5. Slow for complex curves: The DDA algorithm is not suitable for generating complex curves such as circles and ellipses, as it requires a large number of line segments to approximate these curves accurately.
6. Aliasing: Aliasing occurs when the line segments generated using the DDA algorithm do not accurately represent the line being drawn, resulting in a jagged appearance.
7. Not suitable for thick lines: The DDA algorithm generates thin lines, which can be problematic when drawing thick lines, as the line segments may overlap or leave gaps.

Reference :

Here are some references for the DDA (Digital Differential Analyzer) algorithm in computer graphics:

1. Computer Graphics: Principles and Practice (3rd Edition) by James D. Foley, Andries van Dam, Steven K. Feiner, and John F. Hughes.
2. Computer Graphics: C Version (2nd Edition) by Donald Hearn and M. Pauline Baker.
3. Fundamentals of Computer Graphics (4th Edition) by Steve Marschner and Peter Shirley.
4. Computer Graphics with OpenGL (4th Edition) by Donald Hearn, M. Pauline Baker, and Warren Carithers.
5. Introduction to Computer Graphics: Using Java 2D and 3D by Frank Klawonn.