4.b) Explain shear and reflection transformation techniques.
Answer:
Reflection: –
2D reflection
This transformation retains x values, but “flips” the y values of coordinate positions. The resulting orientation of an object after it has been reflected about the x axis is shown in Figure.
A reflection about the line x = 0 (the y axis) flips x coordinates while keeping y coordinates the same. The matrix for this transformation is
Figure illustrates the change in position of an object that has been reflected about the line x = 0.
We flip both the x and y coordinates of a point by reflecting relative to an axis that is perpendicular to the xy plane and that passes through the coordinate origin. The matrix representation for this reflection is
An example of reflection about the origin is shown in Figure.
3D reflection
When the reflection plane is a coordinate plane (xy, xz, or yz), transformation can be thought as a 180◦ rotation in four dimensional space with a conversion between a left-handed frame and a right-handed frame.
The matrix representation for this reflection relative to the xy plane is
Shear:
2D shear
We can generate x-direction shears relative to other reference lines with
3D shear
For three-dimensional, shears can be generated relative to the z axis. A general z-axis shearing transformation relative to a selected reference position is produced with the following matrix:
