2D Shearing in Computer Graphics | Definition | Examples

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-

 

 

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

 

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-

 

 

1. Shearing in X direction
2. 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).

 

 

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

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