Computer Graphics

Notation

• We will use exponent notation to abbreviate the dot product of a vector with itself , i.e.

$$\vec{v} \cdot \vec{v} = |\vec{v}|^2$$

• Where appropriate, for any vector $\vec{x}$ with $|\vec{x}| = 1$, we will alternatively use hat notation, $\hat{x}$, to explicitly indicate that we're dealing with a unit vector.

• We will use column-based notation for vectors. Row-vectors will be specified as transposed column vectors, e.g.

$$(1, 2)^T = \begin{pmatrix}1 \\2 \end{pmatrix}$$

• The identity matrix is represented by the symbol $\boldsymbol{I}$. If not immediately clear from the context, we will make use of the dimension as the subsript, e.g. $\boldsymbol{I}_2$ for the identity matrix with 2 rows and 2 columns.

• cw and ccw may be used to denote rotations in clockwise and counterclockwise directions, respectively

📄️ The Geometry of the Dot Product

The Geometry of the Dot Product and Applications to Visibility Modeling

📄️ Rotations as a Special Case of Vector Transformations

Rotations as a Special Case of Vector Transformations, with Applications to Game World Modeling