Camera & View Matrix

Camera Coordinate Axes

Given eye ($E$), target center ($C$), and up ($U$), the camera system axes are:

• Forward vector: $F = \frac{C - E}{\|C - E\|}$
• Right vector: $R = \frac{F \times U}{\|F \times U\|}$
• Orthogonal Up: $U' = R \times F$

The View Matrix

The resulting matrix combines translation and rotation into camera space. The composite matrix $V$ maps world coordinates into camera-relative locations: $$V = \begin{bmatrix} R_x & R_y & R_z & -R \cdot E \\ U'_x & U'_y & U'_z & -U' \cdot E \\ -F_x & -F_y & -F_z & F \cdot E \\ 0 & 0 & 0 & 1 \end{bmatrix}$$