Direction Cosines

Direction cosines contain information that allows transformation between local and global axis sets. Given a set of orthogonal unit axis vectors the direction cosine array is defined as

$$D = \left\lbrack x|y|z \right\rbrack$$

Any vector or tensor can then be transformed from local to global through

$$\begin{aligned}v_{g} &= Dv_{l} \\ t_{g} &= Dt_{l}D^{T}\end{aligned}$$

The inverse transformation uses the transpose of the direction cosine array

$$\begin{aligned}v_{l} &= D^{T}v_{g}\\ t_{l} &= D^{T}t_{g}D\end{aligned}$$