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The pyramid defined by the coordinates A(0,0,0), B(1,0,0), C(0,1,0) and D(0,0,1) is rotated 45 deg about the line L that has the direction V = J + K and passing through point C(0,1,0).

Find the coordinates of the rotated figure.

Mumbai university > mechanical engineering > sem 7 > cad/cam/cae

Marks: 10M

Difficulty: Easy

1 Answer
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enter image description here

Step 1: Translating line (i.e Point C) to the origin O(0,0,0)

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{tx = 0, ty = -1, tz =0}

Step 2: Aligning enter image description here with Positive z-axis

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On comparing with V = ai + bj + ck ; a = 0, b =1, c =1

also enter image description here

and enter image description here

Now,

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Step 3: Rotation of the object about z-axis by an angle of θ= 45 deg in CCW direction

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Step 4: Aligning Z with V

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Step 5: Inverse Translation

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Now,

The combined transformation matrix,

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[T] = [Tr] [R] [M] [R]-1 [Tr]-1

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Now, new resultant coordinates are:

[X'] = [X] [T]

enter image description here

Hence,

A' = (0.5, 0.15, -0.146)

B' = (1.207, 0.6, -0.646)

C' = (0, 1, 0)

D' = (1, 0.296, 0.704)

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