## 3D Transformations in Computer Graphics-

We have discussed-

• Transformation is a process of modifying and re-positioning the existing graphics.
• 3D Transformations take place in a three dimensional plane.

In computer graphics, various transformation techniques are-

In this article, we will discuss about 3D Shearing in Computer Graphics.

## 3D Shearing in Computer Graphics-

 In Computer graphics, 3D Shearing is an ideal technique to change the shape of an existing object in a three dimensional plane.

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

So, there are three versions of shearing-

1. Shearing in X direction
2. Shearing in Y direction
3. Shearing in Z direction

Consider a point object O has to be sheared in a 3D plane.

Let-

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

### Shearing in X Axis-

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

• Xnew = Xold
• Ynew = Yold + Shy x Xold
• Znew = Zold + Shz x Xold

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

### Shearing in Y Axis-

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

• Xnew = Xold + Shx x Yold
• Ynew = Yold
• Znew = Zold + Shz x Yold

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

### Shearing in Z Axis-

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

• Xnew = Xold + Shx x Zold
• Ynew = Yold + Shy x Zold
• Znew = Zold

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

## Problem-01:

Given a 3D triangle with points (0, 0, 0), (1, 1, 2) and (1, 1, 3). Apply shear parameter 2 on X axis, 2 on Y axis and 3 on Z axis and find out the new coordinates of the object.

## Solution-

Given-

• Old corner coordinates of the triangle = A (0, 0, 0), B(1, 1, 2), C(1, 1, 3)
• Shearing parameter towards X direction (Shx) = 2
• Shearing parameter towards Y direction (Shy) = 2
• Shearing parameter towards Y direction (Shz) = 3

## Shearing in X Axis-

### For Coordinates A(0, 0, 0)

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

Applying the shearing equations, we have-

• Xnew = Xold = 0
• Ynew = Yold + Shy x Xold = 0 + 2 x 0 = 0
• Znew = Zold + Shz x Xold = 0 + 3 x 0 = 0

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

### For Coordinates B(1, 1, 2)

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

Applying the shearing equations, we have-

• Xnew = Xold = 1
• Ynew = Yold + Shy x Xold = 1 + 2 x 1 = 3
• Znew = Zold + Shz x Xold = 2 + 3 x 1 = 5

Thus, New coordinates of corner B after shearing = (1, 3, 5).

### For Coordinates C(1, 1, 3)

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

Applying the shearing equations, we have-

• Xnew = Xold = 1
• Ynew = Yold + Shy x Xold = 1 + 2 x 1 = 3
• Znew = Zold + Shz x Xold = 3 + 3 x 1 = 6

Thus, New coordinates of corner C after shearing = (1, 3, 6).

Thus, New coordinates of the triangle after shearing in X axis = A (0, 0, 0), B(1, 3, 5), C(1, 3, 6).

## Shearing in Y Axis-

### For Coordinates A(0, 0, 0)

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

Applying the shearing equations, we have-

• Xnew = Xold + Shx x Yold = 0 + 2 x 0 = 0
• Ynew = Yold = 0
• Znew = Zold + Shz x Yold = 0 + 3 x 0 = 0

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

### For Coordinates B(1, 1, 2)

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

Applying the shearing equations, we have-

• Xnew = Xold + Shx x Yold = 1 + 2 x 1 = 3
• Ynew = Yold = 1
• Znew = Zold + Shz x Yold = 2 + 3 x 1 = 5

Thus, New coordinates of corner B after shearing = (3, 1, 5).

### For Coordinates C(1, 1, 3)

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

Applying the shearing equations, we have-

• Xnew = Xold + Shx x Yold = 1 + 2 x 1 = 3
• Ynew = Yold = 1
• Znew = Zold + Shz x Yold = 3 + 3 x 1 = 6

Thus, New coordinates of corner C after shearing = (3, 1, 6).

Thus, New coordinates of the triangle after shearing in Y axis = A (0, 0, 0), B(3, 1, 5), C(3, 1, 6).

## Shearing in Z Axis-

### For Coordinates A(0, 0, 0)

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

Applying the shearing equations, we have-

• Xnew = Xold + Shx x Zold = 0 + 2 x 0 = 0
• Ynew = Yold + Shy x Zold = 0 + 2 x 0 = 0
• Znew = Zold = 0

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

### For Coordinates B(1, 1, 2)

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

Applying the shearing equations, we have-

• Xnew = Xold + Shx x Zold = 1 + 2 x 2 = 5
• Ynew = Yold + Shy x Zold = 1 + 2 x 2 = 5
• Znew = Zold = 2

Thus, New coordinates of corner B after shearing = (5, 5, 2).

### For Coordinates C(1, 1, 3)

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

Applying the shearing equations, we have-

• Xnew = Xold + Shx x Zold = 1 + 2 x 3 = 7
• Ynew = Yold + Shy x Zold = 1 + 2 x 3 = 7
• Znew = Zold = 3

Thus, New coordinates of corner C after shearing = (7, 7, 3).

Thus, New coordinates of the triangle after shearing in Z axis = A (0, 0, 0), B(5, 5, 2), C(7, 7, 3).

To gain better understanding about 3D Shearing in Computer Graphics,

Watch this Video Lecture

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