Chapter 9. Transformations

# Chapter 9. Transformations - 9 Transformations The theory...

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0-8493-????-?/00/\$0.00+\$.50 © 2000 by CRC Press LLC © 2001 by CRC Press LLC 9 Transformations The theory of transformations concerns itself with changes in the coordinates and shapes of objects upon the action of geometrical operations, dynamical boosts, or other operators. In this chapter, we deal only with linear transfor- mations, using examples from both plane geometry and relativistic dynamics (space-time geometry). We also show how transformation techniques play an important role in image processing. We formulate both the problems and their solutions in the language of matrices. Matrices are still denoted by bold- face type and matrix multiplication by an asterisk. 9.1 Two-Dimensional (2-D) Geometric Transformations We Frst concern ourselves with the operations of inversion about the origin of axes, re±ection about the coordinate axes, rotation around the origin, scal- ing, and translation. But prior to going into the details of these transforma- tions, we need to learn how to draw closed polygonal Fgures in MATLAB so that we can implement and graph the different cases. 9.1.1 Polygonal Figures Construction Consider a polygonal Fgure whose vertices are located at the points: The polygonal Fgure can then be thought off as line segments (edges) con- necting the vertices in a given order, including the edge connecting the last point to the initial point to ensure that we obtain a closed Fgure. The imple- mentation of the steps leading to the drawing of the Fgure follows: 1. Label all vertex points. 2. Label the path you follow. ( , ), ( , ), , ( , ) xy nn 11 22

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© 2001 by CRC Press LLC (9.3) (9.4) In-Class Exercise Pb. 9.1 Using the trapezoid of Example 9.1, obtain all the transformed

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Chapter 9. Transformations - 9 Transformations The theory...

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