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Rotate object about axis y=4

Short answer

Translate axis to x-axis, rotate 90° about x, translate back. For θ=90°, points become (0,4,-4), (2,4,-1), (5,0,-4).

Rotate the object (0, 0, 0), (2, 3, 0), (5, 0, 4) about the rotation axis y = 4.

Assumptions: the rotation axis is the line y = 4 (z = 0), which is parallel to the x-axis. The angle is not given, so θ = 90° anticlockwise is used (the general matrix is given as well).

Method: translate the axis to the x-axis, rotate about x, translate back.

M = T(0, 4, 0) · Rx(θ) · T(0, −4, 0)

         | 1 0 0  0 |
Rx(θ) =  | 0 c -s 0 |      c = cosθ, s = sinθ
         | 0 s  c 0 |
         | 0 0 0  1 |

General result: y' = 4 + (y − 4)cosθ − z sinθ, z' = (y − 4)sinθ + z cosθ, x' = x.

For θ = 90° (c = 0, s = 1): y' = 4 − z, z' = y − 4.

M = | 1 0 0 0 |
    | 0 0 -1 4 |
    | 0 1 0 -4 |
    | 0 0 0 1 |
Pointy − 4y' = 4 − zz' = y − 4New point
A(0,0,0)−44−4A'(0, 4, −4)
B(2,3,0)−14−1B'(2, 4, −1)
C(5,0,4)−40−4C'(5, 0, −4)

Answer (θ = 90°): A'(0,4,−4), B'(2,4,−1), C'(5,0,−4). The x-coordinates do not change.

✓ Answer checked by Claude Opus 5.5 (high effort)

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