This is a practice project that was given a while
Question This is a practice project that was given a while ago. I am very interested in learning the answers because it will greatly help me with my final exam. Please do “Exercise and Exploration” thanks in advance. Hello attached is a set of past problems please answer all questions. Description is provided. Attachment 1 Attachment 2 Attachment 3 Attachment 4 Attachment 5 ATTACHMENT PREVIEW Download attachment MiniProjectTwo2.jpg Math 340 Mini Project 2: Homogeneous Coordinates Overview As we are seeing, linear algebra comes up in a lot of applications. One of the prime applications for vectors and matrix transformations is the area of computergraphics and the related fields of computer vision, robotics, and image processing. If fact, one of the best reviews of linear algebra may be found in the appendix of the Computer Graphics textbook [4]. Basic operations in computer graphics include translations, rotations, and scaling. In this project, we’ll investigate how these may be combined into a single matrix (multiplication) operation through the use of homogeneous coordinates. Background In computer graphics [4] (as well as in robotics [5]) object positions can be defined using vectors. Thus, changes in position (translations and rotations) can then be defined as matrix operations on these vectors. Likewise, scaling (resizing) the object can be also defined via matrix operations. These operations are represented as, pI = p T (Translation) (1) p’ = RP (Rotation) (2) P’ = SP (Scaling) (3) We see that the latter two of these operations are multiplications, while the first (translation) is a sum. Thus it is difficult to combine the three operations into a single matrix transformation. However, this combination may be achieved if the vectors are rewritten in homogeneous coordinates. In homogeneous coordinates, we add a third coordinate to a point (vector). Instead of being represented by a pair of numbers (x,y)1, each point (vector) is represented as a triple (x,y, w). Thus a point in 2D will be represented by the set of points {(x, y, w)|w at 0} and we say point points are equal (equivalent) if and only if one is a multiple of the other. For example, (2,3,6) and (4,6,12) represent the point in 2D. That is, each point has many different homogeneous coordinate representations. Also, at least one coordinate must be nonzero (that is, (0,0,0) is not allowed.) If the w coordinate is nonzero, we can divide through by it giving (i, a 1). That is, (i, i, 1) 1 In this project, we’ll deal with two dimensional vectors for ease of visualization, but similar techniques apply to three dimensional vectors (for example, see [2].) ATTACHMENT PREVIEW Download attachment MiniProjectTwo3.jpg represents the same point as (x,y, w) and the numbers % and fiare called the Cartesian coordinates of the homogeneous point. Hence we generally represent the point (x, y) in Cartesian coordinates as the triple (x, y, 1) in homogeneous coordinates. The points with w = 0 are called points at infinity and will not be discussed here. The advantage of homogeneous coordinates is that they allow us to combine the three operations above into a single matrix transformation. For example, the translation of (x,y) by the vector (h, k) (that is, (x, y) is translated to (x, y) (h, k) = (x h,y k)) may be represented as, 10hx OHM- 0011 A counterclockwise rotation through an angle 6 is given by multiplication by the matrix x h y k 1 (1′) cos I9 — sin 6 0 sin 6 cos 6 0 (2’) 0 0 1 And a scaling bys in the x direction and tin the y direction, is given by, s 0 0 0 t 0 (3’) 0 0 1 Hence, combinations of these three operations are simply products of matrices ofthese forms. Note that if the matrix is ofthe form, 71,1 71,2 tx 72,1 7’22 ty 0 0 1 where the upper 2X2 submatrix is orthogonal and thus preserves angles and lengths (scaling is one in all directions). Such transformations are called rigid-body transformations and are useful in robotics and other applications with no deformations (see [5] for more details). Example For example, the 3×3 matrix that corresponds to a scaling of one-half (0.5), a rotation of 90 degrees, and a translation of —1 in the x direction and by 2 in the y direction, is given by, 0 —0.5 —1 0.5 0 2 0 0 1 Here is the transformation applied to a (blue) rectangle in Matlab and the resulting (red) rectangle. ATTACHMENT PREVIEW Download attachment MiniProjectTwo4.jpg 1T1 Exercises and Explorations 1. {15] Verify that the operations (1), (2), and (3) are given by the matrix operations as described by (1’), (2’), and (3’)- 2. {5] Prove the two successive 2D rotations are additive (and thus commutative}, that is, show R(61) ‘ R092) = R091 62) [Hint: Recall the trig identities for sum of angles.] 3. {5] Matrix multiplication is generally not commutative, thus rotation in 3D are generally not commutative. Give an example of two rotations in opposite order that shows they are not commutative. [You do n_ot need to give the matrices —you may simply may describe the operations. For example, rotate the unit vector in the x direction, 45 degrees clockwise around the y axis, etc.] Give a drawing or plot of your results. 4. [10] Determine the 3X3 matrix that corresponds to a scaling of doubling in each direction, a rotation of 45 degrees, and a translation of 2 in the x direction and by —2 in the y direction. Plot your results using a rectangle or similarfigure as shown in the example. You may use software or an online application to do this. 5. [15] (Explore) Investigate and demonstrate the use of software or an online application to demonstrate the properties of homogeneous transformations. Make your results as general as possible. For example, allow user input of rotation angle, scaling factor, and translation vector ATTACHMENT PREVIEW Download attachment MiniProjectTwo6.jpg Appendix Matlab code used in the example: % Define and draw a square x=[0 2 2 0 0]; y=[0 0 1 1 0]; P10t(er: I’OI); axis([—6 6 —6 6]); % Change to homogeneous coordinates z=one5(1,5); H=[X:Y:Z]; % Define the matrix for the transformation T=[U —0.5 —1;0.5 0 2:0 0 1] % Multiply to do the scaling, rotation, and translation R=T*H; % Recover the Cartesian coordinates for plotting x1 = R(1,:); y1 = R(2r=)} hold on; P10t(X1:YL’-0′): ATTACHMENT PREVIEW Download attachment MiniProjectTwoFine.jpg and demonstrate the result on a figure of your choices. Demonstrate what happens if you change the order of these operations. Conclusion Sometime complicating a problem makes it simpler. In this case, adding a coordinate to our standard Cartesian coordinates to make them into homogeneous coordinates allows us to combine several operations (scaling, rotation, and translation) into a single matrix transformation. References 1. Wikipedia:
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