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Chapter 18: Look-at Matrix

glm::lookat(vec3 eye, vec3 center, vec3 up)

• with only eye and center, infinite possibilities
• up tells us which one

How do we construct the matrix?

Camera Basis Vectors

U: “side”, “right”, $camera_x$

V: $camera_y$

W: “direction”, “forward”, $camera_z$

Actually, the W vector is the opposite of the “forward” vector of the camera. This is because we are working in a right-handed coordinate system.

$$forward = \frac{center - eye}{|| center - eye ||}$$ $$U = forward \times up$$ $$V = U \times forward$$ $$W = -forward$$

Camera Matrix

$$C = \begin{bmatrix}1 & 0 & 0 & P_x\\0 & 1 & 0 & p_y\\0 & 0 & 1 & p_z\\0 & 0 & 0 & 1\end{bmatrix} * \begin{bmatrix}U_x & V_x & W_x & 0\\U_y & V_y & W_y & 0\\U_z & V_z & W_z & 0\\0 & 0 & 0 & 1\end{bmatrix} = \begin{bmatrix}U_x & V_x & W_x & P_x\\U_y & V_y & W_y & P_y\\U_z & V_z & W_z & P_z\\0 & 0 & 0 & 1\end{bmatrix}$$

View Matrix

What we want is the view matrix, inverse of the camera matrix.

$$V = C^{-1}$$

Because $U$, $V$, and $W$ are orthonormal basis vectors (and other matrix theorems etc.) the inverse of the above matrix is:

$$V = C^{-1} = \begin{bmatrix}U_x & U_y & U_z & -\vec P \cdot \vec U\\V_x & V_y & V_z & -\vec P \cdot \vec V\\W_x & W_y & W_z & -\vec P \cdot \vec W\\0 & 0 & 0 & 1\end{bmatrix}$$

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