3D Transformations of translation, scaling and rotation

What are 3D Transformations?

• In computer graphics, 3D transformations are fundamental operations that change the position, size, and orientation of objects in a three-dimensional space.
• These transformations play a crucial role in creating realistic and dynamic visual scenes.

What is Translation?

• Translation involves moving an object from one position to another in 3D space without changing its orientation.
• This movement is defined by a vector (dx,dy,dz), where dx, dy, and dz represent the distances the object will move along the x, y, and z axes, respectively.

1| 1  0  0  t_x |
2| 0  1  0  t_y |
3| 0  0  1  t_z |
4| 0  0  0  1   |

What is Scaling?

• Scaling in computer graphics is like resizing or changing the size of an image or object on a computer screen.
• Example: Imagine you have a picture of a smiley face on your computer. If you make it twice as big, you're scaling it up. If you make it half as big, you're scaling it down.
• So, you can scale things not just uniformly, but differently in width, height, or depth. It's like stretching the image in different directions.

1| s_x  0   0   0 |
2| 0   s_y  0   0 |
3| 0    0  s_z  0 |
4| 0    0   0   1 |
5

What is Rotation?

• Rotation involves turning an object around a specified axis.
• The rotation is described by an angle θ and an axis of rotation (x, y, or z).
• Example: Take a book lying on a table.
• If we apply a rotation of 90 degrees around the z-axis, the book will appear as if it's standing upright on one of its covers.
• The pages will now be parallel to the x-y plane.

1| 1       0          0       0 |
2| 0   cos(θ)  -sin(θ)  0 |
3| 0   sin(θ)   cos(θ)  0 |
4| 0       0          0       1 |
5

1| cos(θ)  0  sin(θ)  0 |
2| 0       1  0       0 |
3| sin(θ)  0  cos(θ)  0 |
4| 0       0  0       1 |
5

What is Shearing?

• Shearing is like giving a little push to the object, and depending on the direction, it stretches it in that particular way.
• Unlike scaling, it changes the angles between the axes, causing stretching or compression in a particular direction.
• In computer graphics, shearing is often used to create cool visual effects by distorting shapes in specific directions.

What is Reflection?

• Reflection is a transformation that mirrors an object across a line.
• It changes the orientation of the object but does not alter its size or shape.
• There are different types of reflection, such as reflection across the x-axis, y-axis, or a diagonal line.
  

Examples:

• Reflection across the x-axis:
• Original point: (2, 3)
• Reflected point: (2, -3)
• The y-coordinate is negated while the x-coordinate remains the same.
  
• Reflection across the y-axis:
• Original point: (4, -1)
• Reflected point: (-4, -1)
• The x-coordinate is negated while the y-coordinate remains the same.
  
• Reflection across the origin (0,0):
• Original point: (-3, 5)
• Reflected point: (-3, -5)
• Explanation: Both the x and y coordinates are negated.

What is Transformation Concatenation?

• When applying multiple transformations to an object, the order in which they are performed matters and can lead to different results.
• This concept is known as transformation concatenation.

• Example Scenario:

• Suppose we want to move a chair represented by a 3D model:
• Translate (Move): First, we translate the chair to the desired room location.
• Rotate (Adjust Orientation): Next, we rotate the chair to face a specific direction.
• Scale (Resize): Finally, we scale the chair to fit the room's dimensions.
• If we change the order of these transformations, for instance, scaling before rotation, the chair might end up with a different orientation and size.

Conclusion