Transformation in Computer Graphics

Transformation in Computer Graphics is utilized to modify the position, orientation, or size of objects within a graphical scene. By employing mathematical operations, vertices or points in a 3D or 2D space are manipulated to achieve the desired changes.

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In this article, we cover Transformation in Computer Graphics explaining 2d Transformation, rotation, translation, scaling, reflection, shearing and the difference between 2d and 3d Transformation.

What is 2d Transformation in Computer Graphics?

2d Transformation in Computer Graphics is utilized to modify the position, orientation, or size of objects within a two-dimensional space. These transformations involve applying mathematical operations to the coordinates of points or vertices in order to achieve the desired changes.

These transformations can be applied in a sequence to achieve more complex effects. For example, a combination of translation, rotation, and scaling operations can be used to animate an object’s movement, rotation, and resizing in a 2D animation.

The various 2d Transformation in Computer Graphics examples include object manipulation, computer-aided design (CAD), image processing, and graphical user interfaces (GUIs). By manipulating the position, orientation, and size of objects within a 2D space, these transformations enable a wide range of visual effects and graphical applications.

The fundamental geometrical 2d Transformation in Computer Graphics include:

• Rotation
• Translation
• Scaling

The derived geometrical 2d Transformation in Computer Graphics are:

• Reflection
• Shearing

2D Rotation in Computer Graphics

2D rotation is a fundamental concept that involves changing the orientation of an object or a coordinate system in a 2D plane. It enables us to rotate graphical elements around a specified point or axis by a certain angle.

To perform a 2D rotation, we need to consider two key components: the rotation angle and the rotation center. The rotation angle, denoted as θ (in radians), represents the amount of rotation to be applied. The rotation center, represented as (cx, cy), defines the point around which the rotation will occur.

The original coordinates of points P, A, and B can be represented using standard trigonometric functions.

Similarly, the points P’, A’, and B’ can be represented as follows:

Upon substituting equations (1) and (2) into equations (3) and (4) correspondingly, we obtain the following expressions:

In matrix form, the above equation can be represented as follows:

P′ = P . R

Where R is the rotation matrix, the equation can be represented in matrix form as follows:

The above matrix can be used in the case of a rotation angle which is positive, however for a rotation angle which is negative, the matrix is shown below:

2D Translation

2D Translation is a transformation technique that changes the position of each point in an object or a coordinate system by a specified distance in the x and y axes. Applying 2D translation, we can say:

X’ = X + tx

Y’ = Y + ty

(tx, ty) represents the shift or the translation vector. The equations can be expressed using column vectors for efficient representation and computation.

P=[X]/[Y] p’ = [X′]/[Y′] T = [tx]/[ty]

This can also be written as:

P’ = P + T

2D Scaling in Computer Graphics

2D Scaling in Computer Graphics involves resizing objects or coordinate systems in a 2D plane. It allows us to change the size of each point in the object or coordinate system by applying scaling factors in the x and y directions.

To perform 2D scaling, we utilize scaling factors: sx for the x-axis and sy for the y-axis. These factors determine how much each coordinate should be scaled along its respective axis.

If the scaling factor (SX and SY) is greater than 1, the object is enlarged and moves away from the origin. A scaling factor of 1 leaves the object unchanged, while a scaling factor less than 1 shrinks the object and moves it closer to the origin.

The equations for scaling are X’ = X * SX and Y’ = Y * SY, where X and Y are the original coordinates of a point, and X’ and Y’ are the scaled coordinates after the transformation.

These equations can also be represented in matrix form as:

OR

P’ = P . S

The scaling process is depicted using the scaling matrix S in the given figure:

2D Reflection in Computer Graphics

2D reflection is a transformation technique that involves flipping or mirroring an object or coordinate system across a specific axis in a 2D plane. It allows us to change the orientation of each point in the object or coordinate system in relation to the reflection axis.

The figures depict the X and Y axes and the origin:

2D Shearing in Computer Graphics

2D Shearing transformation slants or distorts an object or coordinate system along either the x-axis or y-axis in a 2D plane. It involves shifting the position of points in a specific direction based on their original coordinates.

To shear the given image along the x-axis, we use the shearing parameter sh_x.

The equation will be:

X’= X + Y.shx

Y’ = X

The shearing matrix along with the x-axis is:

The shearing matrix along with the y-axis is:

To shear the provided image along the Y-axis, we utilize the shearing parameter shy.

The equation is now:

Y’= X.shy + Y

X’ = X

The shearing matrix along with the x-axis is:

Difference between 2d and 3d Transformation in Computer Graphics

The difference between 2D and 3D transformations in computer graphics lies in the dimensionality of the space in which the transformations are applied. The prime difference between 2d and 3d Transformation in Computer Graphics are listed in the table below:

The Transformation in Computer Graphics can be combined and applied in a specific order to achieve more complex effects. They serve as fundamental operations in computer graphics, enabling the manipulation of objects, creation of animations, simulation of virtual environments, and various other graphical tasks.