3.B] Write a note on inverse transformations. Derive matrices for each.
Answer:
In computer graphics, transformations are used to manipulate objects and scenes in a 2D or 3D space. These transformations can include translation, rotation, scaling, and more complex operations. To efficiently manage these transformations and ensure objects appear correctly in different contexts (such as camera views or different coordinate systems), it is crucial to understand inverse transformations. Inverse transformations allow us to reverse the effects of a transformation, which is essential for operations like object positioning, animation, and camera manipulation.
For translation, inverse matrix is obtained by negating the translation distances. If the two dimensional translation distances are tx and ty , then inverse translation matrix is
An inverse rotation is accomplished by replacing the rotation angle by its negative. A two-dimensional rotation through an angle θ about the coordinate origin has the inverse transformation matrix
Negative values for rotation angles generate rotations in a clockwise direction.
The inverse matrix for any scaling transformation is obtained by replacing the scaling parameters with their reciprocals. For two-dimensional scaling with parameters sx and sy applied relative to the coordinate origin, the inverse transformation matrix is
The inverse matrix generates an opposite scaling transformation.