**2D Transformations in Computer Graphics-**

We have discussed-

- Transformation is a process of modifying and re-positioning the existing graphics.
- 2D Transformations take place in a two dimensional plane.

In computer graphics, various transformation techniques are-

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

**2D Shearing in Computer Graphics-**

In Computer graphics, 2D Shearing is an ideal technique to change the shape of an existing object in a two dimensional plane. |

In a two dimensional plane, the object size can be changed along X direction as well as Y direction.

So, there are two versions of shearing-

- Shearing in X direction
- Shearing in Y direction

Consider a point object O has to be sheared in a 2D plane.

Let-

- Initial coordinates of the object O = (X
_{old}, Y_{old}) - Shearing parameter towards X direction = Sh
_{x} - Shearing parameter towards Y direction = Sh
_{y} - New coordinates of the object O after shearing = (X
_{new}, Y_{new})

**Shearing in X Axis-**

Shearing in X axis is achieved by using the following shearing equations-

- X
_{new}= X_{old}+ Sh_{x}x Y_{old} - Y
_{new}= Y_{old}

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

For homogeneous coordinates, the above shearing matrix may be represented as a 3 x 3 matrix as-

**Shearing in Y Axis-**

Shearing in Y axis is achieved by using the following shearing equations-

- X
_{new}= X_{old} - Y
_{new}= Y_{old}+ Sh_{y}x X_{old}

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

For homogeneous coordinates, the above shearing matrix may be represented as a 3 x 3 matrix as-

**PRACTICE PROBLEMS BASED ON 2D SHEARING IN COMPUTER GRAPHICS-**

**Problem-01:**

Given a triangle with points (1, 1), (0, 0) and (1, 0). Apply shear parameter 2 on X axis and 2 on Y axis and find out the new coordinates of the object.

**Solution-**

Given-

- Old corner coordinates of the triangle = A (1, 1), B(0, 0), C(1, 0)
- Shearing parameter towards X direction (Sh
_{x}) = 2 - Shearing parameter towards Y direction (Sh
_{y}) = 2

**Shearing in X Axis-**

**For Coordinates A(1, 1)**

Let the new coordinates of corner A after shearing = (X_{new}, Y_{new}).

Applying the shearing equations, we have-

- X
_{new}= X_{old}+ Sh_{x}x Y_{old}= 1 + 2 x 1 = 3 - Y
_{new}= Y_{old}= 1

Thus, New coordinates of corner A after shearing = (3, 1).

**For Coordinates B(0, 0)**

Let the new coordinates of corner B after shearing = (X_{new}, Y_{new}).

Applying the shearing equations, we have-

- X
_{new}= X_{old}+ Sh_{x}x Y_{old}= 0 + 2 x 0 = 0 - Y
_{new}= Y_{old}= 0

Thus, New coordinates of corner B after shearing = (0, 0).

**For Coordinates C(1, 0)**

Let the new coordinates of corner C after shearing = (X_{new}, Y_{new}).

Applying the shearing equations, we have-

- X
_{new}= X_{old}+ Sh_{x}x Y_{old}= 1 + 2 x 0 = 1 - Y
_{new}= Y_{old}= 0

Thus, New coordinates of corner C after shearing = (1, 0).

Thus, New coordinates of the triangle after shearing in X axis = A (3, 1), B(0, 0), C(1, 0).

**Shearing in Y Axis-**

**For Coordinates A(1, 1)**

Let the new coordinates of corner A after shearing = (X_{new}, Y_{new}).

Applying the shearing equations, we have-

- X
_{new}= X_{old}= 1 - Y
_{new}= Y_{old}+ Sh_{y}x X_{old}= 1 + 2 x 1 = 3

Thus, New coordinates of corner A after shearing = (1, 3).

**For Coordinates B(0, 0)**

Let the new coordinates of corner B after shearing = (X_{new}, Y_{new}).

Applying the shearing equations, we have-

- X
_{new}= X_{old}= 0 - Y
_{new}= Y_{old}+ Sh_{y}x X_{old}= 0 + 2 x 0 = 0

Thus, New coordinates of corner B after shearing = (0, 0).

**For Coordinates C(1, 0)**

Let the new coordinates of corner C after shearing = (X_{new}, Y_{new}).

Applying the shearing equations, we have-

- X
_{new}= X_{old}= 1 - Y
_{new}= Y_{old}+ Sh_{y}x X_{old}= 0 + 2 x 1 = 2

Thus, New coordinates of corner C after shearing = (1, 2).

Thus, New coordinates of the triangle after shearing in Y axis = A (1, 3), B(0, 0), C(1, 2).

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