Linear Algebra Review

Computer graphics uses linear algebra as its working language. Points, directions, frames, projections, and camera motion can all be expressed with vectors and matrices.

Vectors

A vector stores magnitude and direction. The dot product measures alignment:

$$ a \cdot b = |a| |b| \cos \theta $$

The cross product produces a direction perpendicular to two input directions:

$$ a \times b = \begin{bmatrix} a_y b_z - a_z b_y \ a_z b_x - a_x b_z \ a_x b_y - a_y b_x \end{bmatrix}. $$

Matrices and transformations

Affine transformations are usually represented with homogeneous coordinates:

$$ \begin{bmatrix} x' \ y' \ z' \ 1 \end

\begin{bmatrix} R & t \ 0 & 1 \end{bmatrix} \begin{bmatrix} x \ y \ z \ 1 \end{bmatrix}. $$

• Translation moves a point.
• Rotation changes orientation while preserving length.
• Scaling changes distances along one or more axes.
• Projection maps 3D geometry to an image plane.

model space -> world space -> view space -> clip space -> screen space