6.5 Applications of Linear Transformations

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1 33 Chapter 6 Linear Transformations 6.5 Applications of Linear Transformations Identif linear transformations defined b reflections, epansions, contractions, or shears in. Use a linear transformation to rotate a figure in R 3. a. (, ) (, ) THE GEOMETRY OF LINEAR TRANSFORMATIONS IN This section gives geometric interpretations of linear transformations represented b 2 2 elementar matrices. Following a summar of the various tpes of 2 2 elementar matrices are eamples that eamine each tpe of matri in more detail. b. c. (, ) (, ) (, ) (, ) Elementar Matrices for Linear Transformations in Reflection in -Ais Reflection in Line Horiontal Epansion > or Contraction < < Horiontal Shear Reflections in Reflection in -Ais Vertical Epansion > or Contraction < < Vertical Shear Reflections in Figure 6. The transformations defined b the following matrices are called reflections. These have the effect of mapping a point in the -plane to its mirror image with respect to one of the coordinate aes or the line, as shown in Figure 6.. a. Reflection in the -ais: b. Reflection in the -ais: c. Reflection in the line : T,, T,, T,,

2 6.5 Applications of Linear Transformations 33 Epansions and Contractions in The transformations defined b the following matrices are called epansions or contractions, depending on the value of the positive scalar. a. Horiontal contractions and epansions: b. Vertical contractions and epansions: Note in Figures 6.2 and 6.3 that the distance the point, moves b a contraction or an epansion is proportional to its - or -coordinate. For instance, under the transformation represented b T, 2, the point, 3 would move one unit to the right, but the point 4, 3 would move four units to the right. Under the transformation represented b T,, 2 the point, 4 would move two units down, but the point, 2 would move one unit down. T,, T,, (, ) (, ) (, ) (, ) Contraction ( < < ) Figure 6.2 Epansion ( > ) (, ) (, ) (, ) (, ) Contraction ( < < ) Figure 6.3 Epansion ( > ) Another tpe of linear transformation in called a shear, as described in Eample 3. corresponding to an elementar matri is

3 332 Chapter 6 Linear Transformations Shears in The transformations defined b the following matrices are shears. T,, a. A horiontal shear represented b T, 2, is shown in Figure 6.4. Under this transformation, points in the upper half-plane shear to the right b amounts proportional to their -coordinates. Points in the lower half-plane shear to the left b amounts proportional to the absolute values of their -coordinates. Points on the -ais do not move b this transformation. b. A vertical shear represented b T,, 2 T,, Figure 6.4 is shown in Figure 6.5. Here, points in the right half-plane shear upward b amounts proportional to their -coordinates. Points in the left half-plane shear downward b amounts proportional to the absolute values of their -coordinates. Points on the -ais do not move (, ) ( + 2, ) (, + 2) (, ) Figure LINEAR ALGEBRA APPLIED The use of computer graphics is common in man fields. B using graphics software, a designer can see an object before it is phsicall created. Linear transformations can be useful in computer graphics. To illustrate with a simplified eample, onl 23 points in R 3 mae up the images of the to boat shown in the figure at the left. Most graphics software can use such minimal information to generate views of an image from an perspective, as well as color, shade, and render as appropriate. Linear transformations, specificall those that produce rotations in R 3, can represent the different views. The remainder of this section discusses rotation in R 3.

4 6.5 Applications of Linear Transformations 333 ROTATION IN R 3 (,, ) Figure 6.6 θ (,, ) In Eample 7 in Section 6., ou saw how a linear transformation can be used to rotate figures in. Here ou will see how linear transformations can be used to rotate figures in R 3. Suppose ou want to rotate the point,, counterclocwise about the -ais through an angle, as shown in Figure 6.6. Letting the coordinates of the rotated point be,,, ou have sin sin cos. cos cos sin sin cos Eample 4 shows how to use this matri to rotate a figure in three-dimensional space. Rotation About the -Ais The eight vertices of the rectangular prism shown in Figure 6.7 are as follows. V,, V 2,, V 3, 2, V 4, 2, V 5,, 3 V 6,, 3 V 7, 2, 3 V 8, 2, 3 a. Find the coordinates of the vertices after the prism is rotated counterclocwise about the -ais through (a) (b) and (c) 2. 6, 9, Figure 6.7 b. 6 SOLUTION a. The matri that ields a rotation of 6 is cos 6 sin 6 2 A sin 6 cos Multipling this matri b the column vectors corresponding to each verte produces the following rotated vertices. V,, V 2.5,.87, V 3.23,.87, V 4.73,, V 5,, 3 V 6.5,.87, 3 V 7.23,.87, 3 V 8.73,, 3 c. Figure Figure 6.8(a) shows a graph of the rotated prism. b. The matri that ields a rotation of 9 is cos 9 sin 9 A sin 9 cos 9 and Figure 6.8(b) shows a graph of the rotated prism. c. The matri that ields a rotation of 2 is cos 2 sin 2 A sin 2 cos and Figure 6.8(c) shows a graph of the rotated prism. 32 2

5 334 Chapter 6 Linear Transformations REMARK To illustrate the right-hand rule, imagine the thumb of our right hand pointing in the positive direction of an ais. The cupped fingers will point in the direction of counterclocwise rotation. The figure below shows counterclocwise rotation about the -ais. Eample 4 uses matrices to perform rotations about the -ais. Similarl, ou can use matrices to rotate figures about the - or -ais. The following summaries all three tpes of rotations. Rotation About the -Ais Rotation About the -Ais Rotation About the -Ais cos sin sin cos cos sin cos sin sin cos sin cos In each case, the rotation is oriented counterclocwise (using the right-hand rule ) relative to the indicated ais, as shown in Figure 6.9. Rotation about -ais Rotation about -ais Rotation about -ais Figure 6.9 Rotation About the -Ais and -Ais Simulation Eplore this concept further with an electronic simulation available at a. The matri that ields a rotation of 9 about the -ais is cos 9 sin 9 sin 9 cos 9 Figure 6.2(a) shows the prism from Eample 4 rotated 9 about the -ais. b. The matri that ields a rotation of 9 about the -ais is cos 9 sin A sin 9 cos 9 Figure 6.2(b) shows the prism from Eample 4 rotated 9 about the -ais... a. b Figure 6.2 Figure 6.2 Rotations about the coordinate aes can be combined to produce an desired view of a figure. For instance, Figure 6.2 shows the prism from Eample 4 rotated 9 about the -ais and then 2 about the -ais.