World space to camera space

I am confused on how to convert world space coordinates to camera coordinates.

My current understanding is that I would need to calculate the camera space vector where

n = eyepoint - lookat

u = up(0,1,0) X n(normalized)

v = n X u

Then once I have < U, V, N > would I simply multiply each point by ?

Lets assume:

• Eye position is E=(e_x, e_y, e_z),
• Viewing direction is D=(d_x, d_y, d_z)
• Up-Vector is UP=(up_x, up_y, up_z)

Now first construct an orthonormal frame:

• R = D X UP
• U = R X D
• Now normalize D,R,U and you have an orthonormal frame for the camera (D,R,U)

In order to transform the global coord frame into the cam-coord frame you can apply the following matrix M_R:

• | R_x, R_y, R_z, 0 |
• | U_x, U_y, U_z, 0 |
• | -D_x, -D_y, -D_z, 0|
• | 0.0, 0.0, 0.0, 1.0|

If your cam is not positioned at global origin you also have to apply a translation M_T:

• | 1, 0, 0, -e_x |
• | 0, 1, 0, -e_y |
• | 0, 0, 1, -e_z|
• | 0, 0, 0, 1|

In the end your complete transformation matrix from global to cam-coords is:

• M = M_R * M_T

• | R_x, R_y, R_z, (R dot -E) |
• | U_x, U_y, U_z, (U dot -E) |
• | -D_x, -D_y, -D_z, (D dot E)|
• | 0.0, 0.0, 0.0, 1.0|