Question

1. A tetrahedron originally has coordinates given in the table below. Assume the tetrahedron is to be rotated 90 deg. about the x-axis (positive rotation) such that point 1 remains fixed. Compute the combined transformation matrix that performs this operation. Compute the new coordinates of points 1-4

Point 1 x=1.5, y=.20, z=1.5

Point 2 x=2.0, y=0, z=0

Point 3 x=1.0, y=0, z=0

Point 4 x=1.6, y=2.5, z=.80

2. For the tetrahedron of problem 1, compute the transformation matrix that rotates the tetrahedron about the vector direction [1,1,1] positive 45 degrees. Point 4 of the tetrahedron is to remain in its original position after the rotation. Compute the combined transformation matrix that performs this operation. Compute the new coordinates for points 1 thru 4.

Answer #1

for problem 1

T = [Tr] [R] [Tr]^{-1}

this will shift point1 to origin

this will rotate the tereahedron about X axix

then apply

now get T and apply on [x y z 1] , where x,y,z are coordinates of points

we will get [x' y' z' 1] as output after applying T

x' , y' , z' are the required coordinates.

for problem 2

1)Translate (1.6, 2.5 , .80) so that the point is at origin

2) Make appropriate rotations (i.e.45 degree each about Z- axis and Y - axis Because (1,1,1) makes angle 45 degree with both axes) to make the vector (1,1,1) coincide with one of the axes, say x-axis

3)Rotate the object about x-axis by required angle

4)Apply the inverse of step 2

5)Apply the inverse of step 1

matrix will be [T] = [Tr] [R_{45}] [R_{45}]
[R_{45}] [R_{45}]^{-1}
[R_{45}]^{-1} [Tr]^{-1} using this you can
compute all the required coordinates.

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