Tag: 3D Rotation Axis

3D Rotation 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 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 = (Xold, Yold, Zold)
  • Initial angle of the object O with respect to origin = Φ
  • Rotation angle = θ
  • New coordinates of the object O after rotation = (Xnew, Ynew, Znew)

 

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-

  • Xnew = Xold
  • Ynew = Yold x cosθ – Zold x sinθ
  • Znew = Yold x sinθ + Zold 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-

  • Xnew = Zold x sinθ + Xold x cosθ
  • Ynew = Yold
  • Znew = Yold x cosθ – Xold 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-

  • Xnew = Xold x cosθ – Yold x sinθ
  • Ynew = Xold x sinθ + Yold x cosθ
  • Znew = Zold

 

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 = (Xold, Yold, Zold) = (1, 2, 3)
  • Rotation angle = θ = 90º

 

For X-Axis Rotation-

 

Let the new coordinates after rotation = (Xnew, Ynew, Znew).

 

Applying the rotation equations, we have-

  • Xnew = Xold = 1
  • Ynew = Yold x cosθ – Zold x sinθ = 2 x cos90° – 3 x sin90° = 2 x 0 – 3 x 1 = -3
  • Znew = Yold x sinθ + Zold 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 = (Xnew, Ynew, Znew).

 

Applying the rotation equations, we have-

  • Xnew = Zold x sinθ + Xold x cosθ = 3 x sin90° + 1 x cos90° = 3 x 1 + 1 x 0 = 3
  • Ynew = Yold = 2
  • Znew = Yold x cosθ – Xold 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 = (Xnew, Ynew, Znew).

 

Applying the rotation equations, we have-

  • Xnew = Xold x cosθ – Yold x sinθ = 1 x cos90° – 2 x sin90° = 1 x 0 – 2 x 1 = -2
  • Ynew = Xold x sinθ + Yold x cosθ = 1 x sin90° + 2 x cos90° = 1 x 1 + 2 x 0 = 1
  • Znew = Zold = 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

 

Get more notes and other study material of Computer Graphics.

Watch video lectures by visiting our YouTube channel LearnVidFun.