LECTURE SEVEN TWO-DIMENSIONAL TRANSFORMATIONS 2
Assist. Lecturer: Alaa shawqi jaber
Outlines:
2D-Transformation: Reflection
Matrix Representations and Homogeneous Coordinates
A reflection is a transformation that produces a mirror image of an object. The mirror image is generated relative to an axis of reflection. Objects can be reflected about the x axis using the following transformation matrix:
This transformation keeps x values the same, but "flips" the y values. The resulting orientation of an object after it has been reflected about the x axis is shown in Fig.
Reflection of an object about the x axis. A reflection about the y axis flips the x coordinates while keeping y coordinates the same. We can perform this reflection with the transformation matrix: The following figure illustrates the change in position of an object that has been reflected about the y axis.
Reflection of an object about the y axis. Another type of reflection is one that flips both the x and y coordinates by reflecting relative to the coordinate origin. (The axis of reflection in this case is the line perpendicular to the xy plane and passing through the origin.) We write the transformation matrix for this type of reflection as:
An example of reflection about the origin is shown in the next Fig.
Reflection on of an object relative to the coordinate origin. The following matrix performs a reflection transformation about the line y = x .
Reflection of an object with respect to the line y = x. This transformation could be obtained from a sequence of rotations and coordinate axis reflections.
First we perform a clockwise rotation through a 45° angle, which rotates the line y = x onto the x axis. Next, we perform a reflection with respect to the x axis. The final step is to rotate the line y = x back to its original position with a counterclockwise rotation through 45°. Concatenating this sequence of three transformations yields the matrix : • Sequence of basic transformations to produce reflection about the line y = x: (a) Clockwise rotation of 45°; (b) reflection about the x axis; (c) Counterclockwise rotation by 45°. An equivalent sequence of transformations is first to reflect the object about the x axis, then rotate counterclockwise 90°.
To obtain a reflection about the line y = -x, we perform the transformation sequence: (1) clockwise rotation by 45° (2) reflection about the y axis, and (3) counterclockwise rotation by 45°. This sequence produces the transformation matrix:
The figure shows the original and final positions for an object reflected about the line y = - x.
Reflection with respect to the line y = -x.
To be continue in the word booklet.