camera extrinsics

Camera extrinsics describe where the camera is and which way it points in the world — its pose. While intrinsics are baked into the lens and sensor, extrinsics change every time you move or rotate the camera. They are the bridge between a shared world coordinate frame (say, a room or a map) and the camera's own frame, in which projection math is done.

Concretely, extrinsics are a rotation R (a 3x3 orthonormal matrix giving the camera's orientation) and a translation t (a 3-vector), assembled as the 3x4 matrix [R|t]. They transform a world point X_world into the camera frame by X_camera = R X_world + t. Two cautions on interpretation: t is not the camera's position in the world; rather, the camera center in world coordinates is C = -R^T t. And R maps world directions into the camera, so its transpose R^T maps camera directions back out into the world. Getting these conventions right is a perennial source of bugs.

Together with intrinsics, extrinsics complete the projection equation x ~ K[R|t] X_world (in homogeneous coordinates), the master formula of multi-view geometry. Estimating extrinsics is the core of camera localization, visual odometry, SLAM and structure-from-motion, where the relative poses between many images are solved jointly with the 3D scene. In neural and point-based reconstruction (NeRF, Gaussian splatting) the per-image extrinsics are the camera poses that tell the model from which viewpoint each training image was taken.