Similarity transformation of projection matrix

I have a similarity transformation S from world coordinate system 1 to 2. I also have a set of 3D points x_i and projection matrices P_j (3x4 or 4x4) of cameras in either world coordinate system 1 or 2. Your question is unclear. Similarity transform is not conventional OpenGL terminology. The pipeline starts with object coordinates. A model transform M takes these to world coordinates (there is only one world coordinate system). A view transform V takes these to eye coordinates. Sometimes the view matrix is called the camera matrix because V takes a hypothetical camera line of sight vector and eye point to the world negative z axis and origin respectively. A projection transform P takes the eye coordinates to homogenous (4d) clip coordinates. In parallel projections, these are the same as normalized device coordinates. For perspective, the division needs still to be performed, but this can't be represented as 4x4 matrix ops and isn't important for what you want. So the whole pipe in 4d is where d is a clip/normalized device coordinate and x is an object coordinate. You can change any or all of the pipe as a scene is rendered. But it's unusual to change P or V. So your question doesn't make much sense. A camera matrix V is described with a point and vector in eye coordinates. If you want to see where those lie in object space, just multiply by inv(M). Perhaps (I am guessing) what you have is two object spaces and corresponding model matrices to take these to a common world: Now if you have an eye point and vector in object system 1 and need to get them to system 2, do the obvious thing. Solve for x_2 in terms of x_1: To say a camera matrix is 'in' any particular coordinate system doesn't have meaning. The matrix is between coordinate systems. Projection matrix P1 projects 3D points to image plane and thus to 2D points in the source coordinate system: To obtain the projection matrix P2 that transforms the 3D point in the destination system to the 2D points, which do not change: Source.


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