**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-

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,

**Next Article-** **2D Rotation in Computer Graphics**

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