3D Shearing in Computer Graphics | Definition | Examples

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-

 

 

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

 

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-

 

 

PRACTICE PROBLEMS BASED ON 3D SHEARING IN COMPUTER GRAPHICS-

 

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

 

Get more notes and other study material of Computer Graphics.

Watch video lectures by visiting our YouTube channel LearnVidFun.

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3D Shearing in Computer Graphics | Definition | Examples
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3D Shearing in Computer Graphics | Definition | Examples
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3D Shearing in Computer Graphics is a process of modifying the shape of an object in 3D plane. Shearing Transformation in Computer Graphics Definition, Solved Examples and Problems.
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