Which information does a homogeneous transformation matrix encode in robotics?

• A. Only rotation around a single axis
• B. Scaling and shearing of coordinates
• C. Only translation along a single axis
• D. Rotation and translation to represent the pose of a coordinate frame

Answer: (D)

A homogeneous transformation matrix encodes how a coordinate frame sits in space—its orientation and position relative to another frame. In robotics, this is captured with a 4x4 matrix where the top-left 3×3 block is a rotation matrix describing how the axes are oriented, the top-right 3×1 column is the translation describing the frame’s origin, and the bottom row is [0 0 0 1] to support homogeneous coordinates. This structure lets you represent the pose of a frame (rotation plus translation) and to compose multiple motions by matrix multiplication. Scaling or shearing would distort distances and angles and are not part of a rigid-frame pose, so they’re not encoded in this transformation. Thus rotation and translation together precisely represent the pose of a coordinate frame.