3D Homogenous Transformations

3D Homogenous Transformations
1. Rotation about x-axis

2. Rotation about y-axis

3. General form of a composition of transformations

3D Homogenous Transformations

1. Translation

 

2. Scaling

3. Rotation

Composing Transformation

Applying several transforms in succession to form one overall transformation is known as Composing Transformation.

(M3 x (M2 x (M1 x P ))) = M3 x M2 x M1 x P

1. Transformation products may not be commutative.

    A x B != B x A

2. Some cases where A x B = B x A

 A                                    B

translation                        translation

scaling                             scaling

rotation                            rotation

uniform scaling                  rotation

(sx = sy)

3. Rotation and translation are not commutative.

Example :

Suppose we wish to reduce the square in the following image to half its size and rotate it by 45° about point P.

We need to remember that that scaling and rotation takes place with respect to the origin.

1– Translate square so that the point around which the rotation is to occur is at the origin

2– Scale

3– Rotate

4– Translate origin back to position P

Window to Viewport Transformation

To display the appropriate images on the screen (or other device) it is necessary to map from world coordinates to screen or device coordinates.



This transformation is known as the window to viewport transformation, the mapping from the world coordinate window to the viewport (which is given in screen coordinates)

In general the window to viewport transformation will involve a scaling and translation as in figure below here the scaling in non-uniform (the vertical axis has been stretched).

Computer Graphics and Geometric Transformations

2D Transformations
2D computer graphics is the computer-based generation of digital images—mostly from two-dimensional models.There are three types of transformations that are essential in computer graphics

1-Translation
Moving an object to a new location by adding to objects x and y co-ordinates.
2-Scaling
Changing the size of the object.This can be uniform where both dimensions are re-sized by same factor or non-uniform.
3-Rotation
Object is rotated around the origin by specified angle.

Homogeneous Transformations
In order to represent a translation as a matrix operation we use 3 x 3 matrices and pad our points to become 1 x 3 matrices.