Translation of objects in computer graphics

In computer graphics, we have seen how to draw some basic figures like line and circles. In this post we will discuss on basics of an important operation in computer graphics as well as 2-D geometry, which is transformation.
In computer graphics, transformation of the coordinates consists of three major processes:

  • Translation
  • Rotation
  • Scaling

In this post we will discuss about translation only.

What is translation?

A translation process moves every point a constant distance in a specified direction. It can be described as a rigid motion. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system.
Suppose, If point (X, Y) is to be translated by amount Dx and Dy to a new location (X’, Y’) then new coordinates can be obtained by adding Dx to X and Dy to Y as:

X' = Dx + X
Y' = Dy + Y

or P' = T + P where

P' = (X', Y'),
T = (Dx, Dy ),
P = (X, Y)

Here, P(X, Y) is the original point. T(Dx, Dy) is the translation factor, i.e. the amount by which the point will be translated. P'(X’, Y’) is the coordinates of point P after translation.
Examples:

Input : P[] = {5, 6}, T = {1, 1}
Output : P'[] = {6, 7}

Input : P[] = {8, 6}, T = {-1, -1}
Output : P'[] = {7, 5}

Whenever we perform translation of any object we simply translate its each and every point. Some of basic objects along with their translation can be drawn as:

  1. Point Translation P(X, Y) : Here we only translate the x and y coordinates of given point as per given translation factor dx and dy respectively.
    Below is the C++ program to translate a point:

    // C++ program for translation
    // of a single coordinate
    #include<bits/stdc++.h>
    #include<graphics.h>
    
    using namespace std;
    
    // function to translate point
    void translatePoint ( int P[], int T[])
    {
        /* init graph and putpixel are used for 
           representing coordinates through graphical 
           functions 
        */
        int gd = DETECT, gm, errorcode;
        initgraph (&gd, &gm, "c:\\tc\\bgi"); 
        
        cout<<"Original Coordinates :"<<P[0]<<","<<P[1];
    
        putpixel (P[0], P[1], 1);
    
        // calculating translated coordinates
        P[0] = P[0] + T[0];
        P[1] = P[1] + T[1];
    
        cout<<"\nTranslated Coordinates :"<< P[0]<<","<< P[1];
       
        // Draw new coordinatses
        putpixel (P[0], P[1], 3);
        closegraph();
    }
    
    // driver program
    int main()
    {
        int P[2] = {5, 8}; // coordinates of point
        int T[] = {2, 1}; // translation factor
        translatePoint (P, T);
        return 0;
    } 
    

    Output:

    Original Coordinates : 5, 8
    Translated Coordinates : 7, 9
    
  2. Line Translation: The idea to translate a line is to translate both of the end points of the line by the given translation factor(dx, dy) and then draw a new line with inbuilt graphics function.
    Below is the C++ implementation of above idea:

    // cpp program for translation
    // of a single line
    #include<bits/stdc++.h>
    #include<graphics.h>
    
    using namespace std;
    
    // function to translate line
    void translateLine ( int P[][2], int T[])
    {
        /* init graph and line() are used for 
           representing line through graphical
           functions 
        */
        int gd = DETECT, gm, errorcode;
        initgraph (&gd, &gm, "c:\\tc\\bgi"); 
        
        // drawing original line using graphics functions
        setcolor (2);
        line(P[0][0], P[0][1], P[1][0], P[1][1]);
    
        // calculating translated coordinates
        P[0][0] = P[0][0] + T[0];
        P[0][1] = P[0][1] + T[1];
        P[1][0] = P[1][0] + T[0];
        P[1][1] = P[1][1] + T[1];
    
        // drawing translated line using graphics functions
        setcolor(3);
        line(P[0][0], P[0][1], P[1][0], P[1][1]);
        closegraph();
    }
    
    // driver program
    int main()
    {
        int P[2][2] = {5, 8, 12, 18}; // coordinates of point
        int T[] = {2, 1}; // translation factor
        translateLine (P, T);
        return 0;
    } 
    

    Output:
    translation 1

  3. Rectangle Translation : Here we translate the x and y coordinates of both given points A(top left ) and B(bottom right) as per given translation factor dx and dy respectively and then draw a rectangle with inbuilt graphics function

    // C++ program for translation
    // of a rectangle
    #include<bits/stdc++.h>
    #include<graphics.h>
    using namespace std;
    
    // function to translate rectangle
    void translateRectangle ( int P[][2], int T[])
    {
        /* init graph and rectangle() are used for 
        representing rectangle through graphical functions */
        int gd = DETECT, gm, errorcode;
        initgraph (&gd, &gm, "c:\\tc\\bgi"); 
        setcolor (2);
        // rectangle (Xmin, Ymin, Xmax, Ymax)
        // original rectangle
        rectangle (P[0][0], P[0][1], P[1][0], P[1][1]);
    
        // calculating translated coordinates
        P[0][0] = P[0][0] + T[0];
        P[0][1] = P[0][1] + T[1];
        P[1][0] = P[1][0] + T[0];
        P[1][1] = P[1][1] + T[1];
    
        // translated rectangle (Xmin, Ymin, Xmax, Ymax)
        // setcolor(3);
        rectangle (P[0][0], P[0][1], P[1][0], P[1][1]);
        // closegraph();
    }
    
    // driver program
    int main()
    {
        // Xmin, Ymin, Xmax, Ymax as rectangle
        // coordinates of top left and bottom right points
        int P[2][2] = {5, 8, 12, 18};
        int T[] = {2, 1}; // translation factor
        translateRectangle (P, T);
        return 0;
    } 
    

    Output:
    translation 2

References : http://math.hws.edu/graphicsbook/.

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 contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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