In robotics, a sensor is rarely at the origin. It is mounted on a robot base or mechanical arm. Hand-Eye Calibration is the process of finding the exact transform between the robot's base and the sensor's optical center.
In a Static setup, the sensor is fixed relative to the base (e.g., a camera mounted on a tripod next to a conveyor). In a Dynamic setup (In-hand), the sensor moves with the robot arm, requiring the system to solve for the constant offset X given varying base poses A and target observations B.
Hand-eye alignment requires solving for 6 degrees of freedom: 3 for translation (x, y, z) and 3 for rotation (roll, pitch, yaw). Small errors in these parameters lead to massive "reprojection" errors in the actual vision task.
Transform (X) = Base-to-Sensor Offset