View representation as matrix

Camera parameterized by:

A center \(c\); right, up, front unit vectors
- expressed in global coordinates

Frame matrix:

Transform unit coordinates into camera vectors

\(Frame\times(1,0,0,0)=right\)
\(Frame\times(0,1,0,0)=up\)
\(Frame\times(0,0,1,0)=-front\)
\(Frame\times(0,0,0,1)=c\)

Matrix of column-oriented vectors of camera directions

Frame=\(\left(\begin{array}{cccc} | & | & | & | \\ right & up & -front & c \\ | & | & | & | \\ 0 & 0 & 0 & 1\end{array}\right)=\left(\begin{array}{cc} O & c \\ 0 & 1 \end{array} \right)\)

View matrix: inverse of the Frame matrix

Transforms the camera vectors into the unit coordinates

\(View\times right = (1,0,0,0)\)
\(View\times up = (0,1,0,0)\)
\(View\times -front = (0,0,1,0)\)
\(View\times c = (0,0,0,1)\)

Orientation \(O\) can be read on the rows

- View=\(\left(\begin{array}{cccc} ( \cdots & right & \cdots ) & | \\ ( \dots & up & \cdots ) & - O^{T}\,c \\ ( \cdots & -front & \cdots ) & | \\ 0 & 0 & 0 & 1 \end{array}\right)\)

assets/camera_space.png