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 3. Construct a (2 ( n + 1) matrix, the G matrix, where the elements of the Frst row consist of the ordered ( n + 1)-tuplet, ( x 1 , x 2 , x 3 , …, x n , x 1 ), and those of the second row consists of the corresponding y coordinates ( n + 1)-tuplet. 4. Plot the second row of G as function of its Frst row. Example 9.1 Plot the trapezoid whose vertices are located at the points (2, 1), (6, 1), (5, 3), and (3, 3). Solution: Enter and execute the following commands: G=[2 6 5 3 2; 1 1 3 3 1]; plot(G(1,:),G(2,:)) To ensure that the exact geometrical shape is properly reproduced, remember to instruct your computer to choose the axes such that you have equal x -range and y -range and an aspect ratio of 1. If you would like to add any text anywhere in the Fgure, use the command gtext . 9.1.2 Inversion about the Origin and Refection about the Coordinate Axes We concern ourselves here with inversion with respect to the origin and with re±ection about the x- or y -axis. Inversion about other points or re±ection about other than the coordinate axes can be deduced from a composition of the present transformations and those discussed later. • The inversion about the origin changes the coordinates as follows: (9.1) In matrix form, this transformation can be represented by: (9.2) • ²or the re±ection about the x -axis, denoted by P x , and the re±ection about the y -axis, denoted by P y , the transformation matrices are given by: ′=− xx yy P = 10 01
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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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