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 = (X_old, Y_old, Z_old)
• Shearing parameter towards X direction = Sh_x
• Shearing parameter towards Y direction = Sh_y
• Shearing parameter towards Z direction = Sh_z
• New coordinates of the object O after shearing = (X_new, Y_new, Z_new)

 

Shearing in X Axis-

 

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

• X_new = X_old
• Y_new = Y_old + Sh_y x X_old
• Z_new = Z_old + Sh_z x X_old

 

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-

• X_new = X_old + Sh_x x Y_old
• Y_new = Y_old
• Z_new = Z_old + Sh_z x Y_old

 

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-

• X_new = X_old + Sh_x x Z_old
• Y_new = Y_old + Sh_y x Z_old
• Z_new = Z_old

 

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 (Sh_x) = 2
• Shearing parameter towards Y direction (Sh_y) = 2
• Shearing parameter towards Y direction (Sh_z) = 3

 

Shearing in X Axis-

 

For Coordinates A(0, 0, 0)

 

Let the new coordinates of corner A after shearing = (X_new, Y_new, Z_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
• Z_new = Z_old + Sh_z x X_old = 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 = (X_new, Y_new, Z_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
• Z_new = Z_old + Sh_z x X_old = 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 = (X_new, Y_new, Z_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
• Z_new = Z_old + Sh_z x X_old = 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 = (X_new, Y_new, Z_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
• Z_new = Z_old + Sh_z x Y_old = 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 = (X_new, Y_new, Z_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
• Z_new = Z_old + Sh_z x Y_old = 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 = (X_new, Y_new, Z_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
• Z_new = Z_old + Sh_z x Y_old = 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 = (X_new, Y_new, Z_new).

 

Applying the shearing equations, we have-

• X_new = X_old + Sh_x x Z_old = 0 + 2 x 0 = 0
• Y_new = Y_old + Sh_y x Z_old = 0 + 2 x 0 = 0
• Z_new = Z_old = 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 = (X_new, Y_new, Z_new).

 

Applying the shearing equations, we have-

• X_new = X_old + Sh_x x Z_old = 1 + 2 x 2 = 5
• Y_new = Y_old + Sh_y x Z_old = 1 + 2 x 2 = 5
• Z_new = Z_old = 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 = (X_new, Y_new, Z_new).

 

Applying the shearing equations, we have-

• X_new = X_old + Sh_x x Z_old = 1 + 2 x 3 = 7
• Y_new = Y_old + Sh_y x Z_old = 1 + 2 x 3 = 7
• Z_new = Z_old = 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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