Question

correct answers

135. Computer Graphics. One of the most important applications of linear transformations is computer graphics where we wish to view 3-dimensional objects (for example a crystal) on a 2-dimensional screen. The screen is the ry-plane. The aim is to rotate the crystal and orthogonally project it onto the ry-plane to obtain different views of it. We consider 3 possible rotations: 0 . A rotation of θ round the x-axis using the matrix R2-10 cos θ -sin θ cos θ 0 sin θ cos θ 0 sin θ . A rotation of θ round the y-axis using the matrix Ry-1 0 1 0 sin θ 0 cos θ cos θ -sin θ 0 . A rotation of θ round the 2-axis using the matrix R,-| sin θ cos θ 0 0 The matrix used to orthogonally project the object onto the computer screen is A erystal has vertices A: (0,0,1), B (1,0,0), c: (0.1,0), D:(-1.0,0), E (0 -1,0), F : (0,0,-1) with edges the line segments AB, AC, AD, AE, FB, FC, FD, FE, BC, CD, DE and EB Projecting onto the ry-plane, we find P(A)A' is given by In the same way B (1,0), C (0,1), D (1,0), E (0,-1) and F the line segments appropriately we find the view on the computer screen is (0,0). Connecting up A & F Draw the picture that would appear on the computer screen if (a) the crystal is rotated 45° around the r-axis before projection, (b) the crystal is rotated 45° around the r-axis and then 30° around the z-axis, (c) the crystal is rotated 45 around the r-axis, 30° around the z-axis and then -60 around the y-axis, You might like to think about whether or not orthogonal projection is the best way to project from 3-D to 2-D 135 (c) A':( -436,612) B':(.433, .500) C:(-.789, .612) D':(-433,-.500) E': (.789,-.612) F': (.436,.612) (b) A: ( .354,-.612) B"( .866, .500) C':(-.354, .612) D':(-.866,-.500) E': (.354,-612) F:(-.354, .612) (a) A':(0,-.707) C:(0, .707) E':(0,-.707) F:(0,.707) C'&F' C'&F В' A'&E

Answer 1

Mathematics Sabyasachi May 29, 2019 Solution (a). The required transformation matrix is: 0 0 PRx(45°)=P10 cos 45° -sin 45°1-1 -で 0 sin 45S 45 - (0,-.707), B' 1,0, | = (0, .707), 17- = | | = (0,707). = | (b). The required transformation matrix is: 01「1 cos 30° -sin 30° 0 PR:(30°) Ra (45°) P sin 30 cos30 00 cos 45si45 1 0 sin 45 cos 45 (.354,-612), B (866,.500), D' (-35.612), = = (-866,-.500), ng | = (.354,-.612), F' = = (-354,-612)

Please wait...