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 Rotation in Computer Graphics.

 

3D Rotation in Computer Graphics-

 

In Computer graphics, 3D Rotation is a process of rotating an object with respect to an angle in a three dimensional plane.

 

Consider a point object O has to be rotated from one angle to another in a 3D plane.

 

Let-

• Initial coordinates of the object O = (X_old, Y_old, Z_old)
• Initial angle of the object O with respect to origin = Φ
• Rotation angle = θ
• New coordinates of the object O after rotation = (X_new, Y_new, Z_new)

 

In 3 dimensions, there are 3 possible types of rotation-

• X-axis Rotation
• Y-axis Rotation
• Z-axis Rotation

 

For X-Axis Rotation-

 

This rotation is achieved by using the following rotation equations-

• X_new = X_old
• Y_new = Y_old x cosθ – Z_old x sinθ
• Z_new = Y_old x sinθ + Z_old x cosθ

 

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

 

 

For Y-Axis Rotation-

 

This rotation is achieved by using the following rotation equations-

• X_new = Z_old x sinθ + X_old x cosθ
• Y_new = Y_old
• Z_new = Y_old x cosθ – X_old x sinθ

 

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

 

 

For Z-Axis Rotation-

 

This rotation is achieved by using the following rotation equations-

• X_new = X_old x cosθ – Y_old x sinθ
• Y_new = X_old x sinθ + Y_old x cosθ
• Z_new = Z_old

 

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

 

 

PRACTICE PROBLEMS BASED ON 3D ROTATION IN COMPUTER GRAPHICS-

 

Problem-01:

 

Given a homogeneous point (1, 2, 3). Apply rotation 90 degree towards X, Y and Z axis and find out the new coordinate points.

 

Solution-

 

Given-

• Old coordinates = (X_old, Y_old, Z_old) = (1, 2, 3)
• Rotation angle = θ = 90º

 

For X-Axis Rotation-

 

Let the new coordinates after rotation = (X_new, Y_new, Z_new).

 

Applying the rotation equations, we have-

• X_new = X_old = 1
• Y_new = Y_old x cosθ – Z_old x sinθ = 2 x cos90° – 3 x sin90° = 2 x 0 – 3 x 1 = -3
• Z_new = Y_old x sinθ + Z_old x cosθ = 2 x sin90° + 3 x cos90° = 2 x 1 + 3 x 0 = 2

 

Thus, New coordinates after rotation = (1, -3, 2).

 

For Y-Axis Rotation-

 

Let the new coordinates after rotation = (X_new, Y_new, Z_new).

 

Applying the rotation equations, we have-

• X_new = Z_old x sinθ + X_old x cosθ = 3 x sin90° + 1 x cos90° = 3 x 1 + 1 x 0 = 3
• Y_new = Y_old = 2
• Z_new = Y_old x cosθ – X_old x sinθ = 2 x cos90° – 1 x sin90° = 2 x 0 – 1 x 1 = -1

 

Thus, New coordinates after rotation = (3, 2, -1).

 

For Z-Axis Rotation-

 

Let the new coordinates after rotation = (X_new, Y_new, Z_new).

 

Applying the rotation equations, we have-

• X_new = X_old x cosθ – Y_old x sinθ = 1 x cos90° – 2 x sin90° = 1 x 0 – 2 x 1 = -2
• Y_new = X_old x sinθ + Y_old x cosθ = 1 x sin90° + 2 x cos90° = 1 x 1 + 2 x 0 = 1
• Z_new = Z_old = 3

 

Thus, New coordinates after rotation = (-2, 1, 3).

 

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

Watch this Video Lecture

 

Next Article- 3D Scaling in Computer Graphics

 

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