To map XYZ to PQR using rigid motions (rotations and translations), you can use the following transformations: (a) ∠Y → Rotation about the Y-axis, (b) Translation along the X-axis → ∠P, (c) YZ → Rotation about the X-axis, (d) Translation along the Z-axis → PR, and (e) Rotation and translation → ΔRPQ.
To map the XYZ coordinate system to the PQR coordinate system using rigid motions, a sequence of transformations is applied. Each transformation involves either a rotation about one of the axes or a translation along one of the axes. Let's analyze each transformation:
(a) The transformation "∠Y → Rotation about the Y-axis" implies rotating the entire system around the Y-axis. This reorients the coordinate system while maintaining the same spatial relationships between points.
(b) The transformation "Translation along the X-axis → ∠P" suggests moving the entire system along the X-axis, introducing a displacement. This aligns with the ∠P orientation.
(c) The transformation "YZ → Rotation about the X-axis" involves rotating the system about the X-axis, altering the orientation of the Y and Z axes.
(d) The transformation "Translation along the Z-axis → PR" shifts the system along the Z-axis, aligning the origin with point P in the PQR coordinate system.
(e) The transformation "Rotation and translation → ΔRPQ" combines both rotation and translation to position the XYZ coordinate system into the PQR coordinate system, forming the ΔRPQ.