# 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-

## 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 = (Xold, Yold)
• Shearing parameter towards X direction = Shx
• Shearing parameter towards Y direction = Shy
• New coordinates of the object O after shearing = (Xnew, Ynew)

### Shearing in X Axis-

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

• Xnew = Xold + Shx x Yold
• Ynew = Yold

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-

• Xnew = Xold
• Ynew = Yold + Shy x Xold

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-

## 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 (Shx) = 2
• Shearing parameter towards Y direction (Shy) = 2

## Shearing in X Axis-

### For Coordinates A(1, 1)

Let the new coordinates of corner A after shearing = (Xnew, Ynew).

Applying the shearing equations, we have-

• Xnew = Xold + Shx x Yold = 1 + 2 x 1 = 3
• Ynew = Yold = 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 = (Xnew, Ynew).

Applying the shearing equations, we have-

• Xnew = Xold + Shx x Yold = 0 + 2 x 0 = 0
• Ynew = Yold = 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 = (Xnew, Ynew).

Applying the shearing equations, we have-

• Xnew = Xold + Shx x Yold = 1 + 2 x 0 = 1
• Ynew = Yold = 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 = (Xnew, Ynew).

Applying the shearing equations, we have-

• Xnew = Xold = 1
• Ynew = Yold + Shy x Xold = 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 = (Xnew, Ynew).

Applying the shearing equations, we have-

• Xnew = Xold = 0
• Ynew = Yold + Shy x Xold = 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 = (Xnew, Ynew).

Applying the shearing equations, we have-

• Xnew = Xold = 1
• Ynew = Yold + Shy x Xold = 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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Summary
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2D Shearing in Computer Graphics | Definition | Examples
Description
2D Shearing in Computer Graphics is a process of modifying the shape of an object in 2D plane. Shearing Transformation in Computer Graphics Definition, Solved Examples and Problems.
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Gate Vidyalay
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