 Open Access
 Total Downloads : 64
 Authors : Ankur Rana , Sandhya Samant, Abhishek Agarwal
 Paper ID : IJERTV8IS050398
 Volume & Issue : Volume 08, Issue 05 (May 2019)
 Published (First Online): 27052019
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Extensive Study of 2d Transformations in Computer Graphics
Ankur Rana
Assistant Professor, Department of Computer Science
Quantum University, Roorkee, India
Sandhya Samant
Assistant Professor, Department of Computer Science
Quantum University, Roorkee, India
Abhishek Agarwal Assistant Professor, Department of Mathematics
Quantum University, Roorkee , India
Abstract: This research paper gives a comprehensive overview of various 2D Transformations in Computer Graphics. 2D graphics uses a two dimensional representation of the real world objects, stored as images in the computer for being manipulated and rendered. Transformation, in graphics, is the process of manipulation of images. Majorly, it is used to reposition the image (Translation), to change its size (Scaling) or orientation (Rotation) and also to get the mirror image (Reflection).
Translation
Transformation
Scaling
Scaling
Shearing
Shearing
Roatation Reflection
Figure 1 Types of Transformation
Keywords: 2D Graphics, Objects, Images, Transformation, Translation, Scaling, Rotation, Reflection etc.
I. INTRODUCTION
Computer graphics is the field of computer science which deals with creation, storing, displaying and manipulation of images. It is concerned with digitally synthesizing and manipulating visual contents. An image is the 2D representation of a 3D object. Images, on screen, are made up of pixels, which is the smallest addressable unit of display.
Computer graphics is generally categorized into Interactive and Noninteractive graphics. Interactive Computer Graphics is a two way communication between the computer and user, where the image on the screen is controlled by giving signals or instructions to the computer through an input device, Whereas NonInteractive Computer Graphics is a passive system in which the user does not have any command or control over the image. Here the image is a product of already stored program and is operated by the instructions according to the program and not the user.
Example of Interactive system is Flight Simulator and the example of NonInteractive system is screen savers.
 . TRANSFORMATION
Transformation is the process of changing the original coordinates of the image with the new coordinates by applying different mathematical functions.[1]
 Translation: Translation is the transformation that moves every point by certain distance in horizontal and vertical directions[2]. This movement is based on translation factors tx and ty where, tx is the horizontal movement and ty is the vertical movement.
Let the point P be defined by (x,y) coordinates, then the new coordinates (x,y) after translation[3] will be:
x = x+ tx y = y+ ty
Figure 2 Translation
Figure 2 shows the translation applied on alphabet F.
Furthermore, the equation of translation can be explained using matrix, where the representation of the operation will be:
x 1 0 tx x
[y] = [0 1 ty].[y]1 0 0 1 1
Figure 3 Translation Matrix
 Scaling: Scaling is the transformation in which we resize the object. It refers to enlargement or shrinking of the object. Scaling depends on the scaling factors Sx and Sy where Sx and Sy are the scaling in horizontal and vertical directions respectively.
Let the point P be defined by (x,y) coordinates, then the new coordinates (x,y) after scaling will be:
x = x . Sx y = y . Sy
Figure 4 Scaling
Figure 3 shows the scaling applied on alphabet F.
Scaling can be categorized into uniform and nonuniform scaling. If Sx = Sy then it is Uniform Scaling and if Sx Sy then it is Nonuniform Scaling.
Furthermore, the matrix representation of the above operation can be given as under:
Figure 6 shows the operation of Rotation.
Furthermore, we can also represent the rotation on angle with the given matrix:
x cos sin 0 x [y]=[sin cos 0].[y] 1 0 0 1 1
Figure 7 Rotation Matrix
4. Reflection: Reflection is the transformation that produces the mirror image of an object. The mirror image of the object is produced with respect to the reflection axis by rotating the object at 180 degrees along the reflection axis. The axis of reflection can be anywhere in the xy plane.
If xaxis is the axis of reflection then the new coordinates after reflection will be:
X = X and Y = Y,
If yaxis is the axis of reflection then the new coordinates after reflection will be:
X = X and Y = Y,
If both xaxis and yaxis are the axis of reflection then the new coordinates will be:
X = X and Y = Y.
x sx 0 0 x
[y]=[ 0 sy 0].[y]1 0 0 1 1
Figure 5 Scaling Matrix
 Rotation: Rotation is the transformation in which we rotate each point through the origin (0,0). Every point is rotated through the same angle, which is known as the angle of rotation[4]. The unit of this angle can either be in degrees or in radians. If the angle of rotation is then it is taken positive for counter clockwise direction and negative for clockwise direction. The equation for the new coordinates of the point after rotation is:
X = x cos() y sin () Y = y cos () + x sin ()
Figure 8 Reflection
Figure 8 shows the reflection when both xaxis and yaxis are the axis of reflection.
x 1 0 0 x
[y]=[0 1 0].[y]1 0 0 1 1
Figure 9 Reflection Matrix (about x axis)
x 1 0 0 x
[y]=[ 0 1 0].[y]1 0 0 0 1
Figure 10 Reflection Matrix (about y axis)
x 1 0 0 x
[y]=[ 0 1 0].[y]1 0 0 1 1
Figure 11 Reflection Matrix (about xy axis)
Figure 6 Rotation
5. Shearing: Shearing is the transformation that produces the distortion of the object. In other words we can also say that shearing tilts the object. If the shearing is horizontal then it will tilt the object towards the left (negative shear) or right (positive shear). Whereas the vertical shear tilts the object up or down.
Figure 11 Shearing
Figure 9 shows an example of horizontal shear.
The equation for horizontal shear applied to a point (x,y), to obtain new coordinates (X,Y) is given by:
X = x + b.y and Y = y, Where b is shearing factor.
Whereas the equation for vertical shear with shearing factor c is given by:
X = x and Y = c . x + y
The matrix for X shear can be represented as:
1 0 0 Xsh=[shx 1 0] 0 0 1 Figure 12 Horizontal Shearing
Similarly, the matrix for vertical shearing can be represented as:
1 shy 0
Ysh=[0 1 0]
0 0 1
Figure 13 Vertical Shearing
 Translation: Translation is the transformation that moves every point by certain distance in horizontal and vertical directions[2]. This movement is based on translation factors tx and ty where, tx is the horizontal movement and ty is the vertical movement.
 . CONCLUSION
Here in this paper we have discussed various transformations along with their matrix representations. Among the transformation we have discussed, Translation, Rotation, Reflection and their combinations are the rigid transformations because in these transformations the preimage and image are of same size and shape (congruent). Scaling and Shearing are not rigid transformations because they produce images with different size or shape.
IV. REFERENCES

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No.2, March 2013
 David J.EckHobart and William Smith, Introdution to Computer Graphics Version 1.2, January 2018
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