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Unit 6: 2-D Transformation
The figure 6.8 shows the original rectangle and X-shear transformed rectangle for shear factors 1 and -1.
Figure 6.8: X-Shear Transformation
Plot a pentagon on the 2-D space and apply shear transformation in X and Y directions.
Let the X-shear value be -1 a Y shear value be 1. Compare the shapes of the pentagon
objects before and after shear transformation.
6.2 Homogeneous Coordinate System
Homogeneous coordinate system is developed based on the concept of infinity. With respect to point
coordinate system, infinity is a point that does not exist. However, many mathematicians have
mathematically proved that, with the concept of infinity many geometric concepts and computations
can be simplified and computed easily. Homogeneous coordinate system is one such mathematical
technique that makes use of the concept of infinity to work with 2-D objects in computer graphics and
computer-aided design.
Consider two real numbers ‘b’ and ‘z’.
p=b/z, where ‘p’ is a real number.
Let the value of ‘b’ be constant and the value of ‘z’ be varied, i.e., as the value of ‘z’ decreases the value
of ‘p’ increases. When the value of ‘z’ tends to zero the value of ‘p’ tends to infinity.
Therefore, if the value of ‘z’ is non-zero then the value of ‘p’ is equal to ‘b/z’ and is represented as p=
(b, z). If the value of ‘z’ is zero, the value of ‘p’ is infinity and is represented as p = (b, 0). Thus, the
infinity concept can be represented as (b, z) or b/z.
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