2D Transformation in Computer Graphics-

 

In Computer graphics, Transformation is a process of modifying and re-positioning the existing graphics.

 

• 2D Transformations take place in a two dimensional plane.
• Transformations are helpful in changing the position, size, orientation, shape etc of the object.

 

Transformation Techniques-

 

In computer graphics, various transformation techniques are-

 

 

1. Translation
2. Rotation
3. Scaling
4. Reflection
5. Shear

 

In this article, we will discuss about 2D Translation in Computer Graphics.

 

2D Translation in Computer Graphics-

 

In Computer graphics, 2D Translation is a process of moving an object from one position to another in a two dimensional plane.

 

Consider a point object O has to be moved from one position to another in a 2D plane.

 

Let-

• Initial coordinates of the object O = (X_old, Y_old)
• New coordinates of the object O after translation = (X_new, Y_new)
• Translation vector or Shift vector = (T_x, T_y)

 

Given a Translation vector (T_x, T_y)-

• T_x defines the distance the X_old coordinate has to be moved.
• T_y defines the distance the Y_old coordinate has to be moved.

 

 

This translation is achieved by adding the translation coordinates to the old coordinates of the object as-

• X_new = X_old + T_x     (This denotes translation towards X axis)
• Y_new = Y_old + T_y     (This denotes translation towards Y axis)

 

In Matrix form, the above translation equations may be represented as-

 

 

• The homogeneous coordinates representation of (X, Y) is (X, Y, 1).
• Through this representation, all the transformations can be performed using matrix / vector multiplications.

 

The above translation matrix may be represented as a 3 x 3 matrix as-

 

 

PRACTICE PROBLEMS BASED ON 2D TRANSLATION IN COMPUTER GRAPHICS-

 

Problem-01:

 

Given a circle C with radius 10 and center coordinates (1, 4). Apply the translation with distance 5 towards X axis and 1 towards Y axis. Obtain the new coordinates of C without changing its radius.

 

Solution-

 

Given-

• Old center coordinates of C = (X_old, Y_old) = (1, 4)
• Translation vector = (T_x, T_y) = (5, 1)

 

Let the new center coordinates of C = (X_new, Y_new).

 

Applying the translation equations, we have-

• X_new = X_old + T_x = 1 + 5 = 6
• Y_new = Y_old + T_y = 4 + 1 = 5

 

Thus, New center coordinates of C = (6, 5).

 

Alternatively,

 

In matrix form, the new center coordinates of C after translation may be obtained as-

 

 

Thus, New center coordinates of C = (6, 5).

 

 

Problem-02:

 

Given a square with coordinate points A(0, 3), B(3, 3), C(3, 0), D(0, 0). Apply the translation with distance 1 towards X axis and 1 towards Y axis. Obtain the new coordinates of the square.

 

Solution-

 

Given-

• Old coordinates of the square = A (0, 3), B(3, 3), C(3, 0), D(0, 0)
• Translation vector = (T_x, T_y) = (1, 1)

 

For Coordinates A(0, 3)

 

Let the new coordinates of corner A = (X_new, Y_new).

 

Applying the translation equations, we have-

• X_new = X_old + T_x = 0 + 1 = 1
• Y_new = Y_old + T_y = 3 + 1 = 4

 

Thus, New coordinates of corner A = (1, 4).

 

For Coordinates B(3, 3)

 

Let the new coordinates of corner B = (X_new, Y_new).

 

Applying the translation equations, we have-

• X_new = X_old + T_x = 3 + 1 = 4
• Y_new = Y_old + T_y = 3 + 1 = 4

 

Thus, New coordinates of corner B = (4, 4).

 

For Coordinates C(3, 0)

 

Let the new coordinates of corner C = (X_new, Y_new).

 

Applying the translation equations, we have-

• X_new = X_old + T_x = 3 + 1 = 4
• Y_new = Y_old + T_y = 0 + 1 = 1

 

Thus, New coordinates of corner C = (4, 1).

 

For Coordinates D(0, 0)

 

Let the new coordinates of corner D = (X_new, Y_new).

 

Applying the translation equations, we have-

• X_new = X_old + T_x = 0 + 1 = 1
• Y_new = Y_old + T_y = 0 + 1 = 1

 

Thus, New coordinates of corner D = (1, 1).

Thus, New coordinates of the square = A (1, 4), B(4, 4), C(4, 1), D(1, 1).

 

 

To gain better understanding about 2D Translation in Computer Graphics,

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Next Article- 2D Rotation in Computer Graphics

 

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