Rotation of a coordinate system

X.B.3 Rotation of a coordinate system

A transformation of coordinate systems commonly done is a rotation of the coordinate system around a specified axis passing through the origin of the coordinate system. The resulting coordinate system has the same origin, but its base vectors are modified. A classical use of this operation corresponds to the transformation of vector or tensor results in material, ply or element axes of 2D elements.

For example, let $v$ be a unit vector defining the rotation axis and $θ$ the rotation angle. Then the three transformed base vectors are given by:

e_{i}^{'} = cos θ e_{i} + (1 - cos θ) (v \cdot e_{i}) v + sin θ (v \times e_{i}) .

(X.B.1)

In particular, if $v$ corresponds to $e_{3}$ , the three

\begin{align} e_{1}^{'} & = cos θ e_{1} + (1 - cos θ) (e_{3} \cdot e_{1}) e_{3} + sin θ (e_{3} \times e_{1}), \\ = cos θ e_{1} + sin θ (e_{2}), \\ = cos θ e_{1} + sin θ e_{2} . \end{align}

\begin{align} e_{2}^{'} & = cos θ e_{2} + (1 - cos θ) (e_{3} \cdot e_{2}) e_{3} + sin θ (e_{3} \times e_{2}), \\ = cos θ e_{2} + sin θ (- e_{1}), \\ = cos θ e_{2} - sin θ e_{1} . \end{align}

\begin{align} e_{3}^{'} & = cos θ e_{3} + (1 - cos θ) (e_{3} \cdot e_{3}) e_{3} + sin θ (e_{3} \times e_{3}), \\ = cos θ e_{3} + (1 - cos θ) (1) e_{3}, \\ = e_{3} . \end{align}

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