I am a novice in the 3D math.currently , I am tortured by a problem about the rotation of a FPS camera.hope someone can help me .
please allow me to assume that the discussion is based on OpenGL conversion.
suppose the FPS camera is placed at the initial state in which the three axis of camera are aligned with axis of world frame.
I firstly pitch the camera around the x axis of world frame, and then yaw the camera around the y axis of world frame.so I can get the following equation(column-major) :
camera_orientation_matrix_1 = yaw_yworld_matrix * pitch_xworld_matrix = rotateY(yaw_angle) * rotateX(pitch_angle);
some reference informed me that I can get the same orientation by using the local rotation matrix but applied in a reverse order.
so I firstly yaw the camera with local y axis of camera, then pitch the camera around the local x axis of the camrea.
ie: camera_orientation_matrix _2 = pitch_xlocal_matrix * yaw_ylocal_matrix = rotateX(pitch_angle) * rotateY(yaw_angle) ;
since the camera_orientation_matrix_1 equals camera_orientation_matrix _2, I derive the following:
rotateY(yaw_angle) * rotateX(pitch_angle) === rotateX(pitch_angle) * rotateY(yaw_angle) ;
but the above equation is a fallacy,Is there anyone can tell me where is wrong? Thanks!
From Wikipedia (general case): For n greater than 2, multiplication of n×n rotation matrices is not commutative.
Though this is not a proof, that a certain sub-set of nxn matrices is not commutative. E.g. the set of all pure homogenes 4x4 translation matrices are commutative.
Hehe, the above statement is at least a proof, that I fail to read carefully a post sometimes