Homogeneous Coordinates

Abstract

Suppose the position of the camera's center in world coordinates is a 3D point C w . If we wish to transform any other point X w into the camera's coordinate system, we first subtract off C w and then we perform a rotation: X c = R(X w − C w) . Use 3D vectors and 3 × 3 matrices, we can write this as follows, X c = RX w − RC w . In homogeneous coordinates, we would write so the transformation from world to camera coordinates is the product of a 4 × 4 translation matrix and a 4 × 4 rotation matrix. R −RC w 0 1 = R 0 0 1 I −C w 0 1 where I is the 3 × 3 identity matrix. The rotation matrix R and the translation vector C w define the camera's extrinsic coordinates, namely its orientation and position, respectively, in world coordinates. The matrix R transforms from world to camera coordinates, and so you can think of it as R c←w .