Building rotation tensor from rotation vector

X.B.7 Building rotation tensor from rotation vector

A rotation can be defined with a vector. Then, the axis of rotation is parallel to the vector and the amplitude of rotation in radians corresponds to the magnitude of rotation vector:

r = θ k,

in which $k$ is the unit vector defining rotation axis. The rotation vector $r$ corresponds typically to the rotational displacements calculated by finite element solver.

This section explains how the rotation matrix $Q$ introduced in section X.B.6 can be derived from rotation vector $r$ . To do this, we first justify how a vector $u$ is transformed to rotated vector $v$ by rotation $r$ .

Note first that vector $v$ can be decomposed in a sum of two vectors respectively parallel and orthogonal to axis of rotation:

v_{∥} = (k \cdot v) k = (k k) \cdot v,

v_{⊥} = v - v_{∥} = (δ - k k) \cdot v .

A third vector perpendicular to the first two vectors is also built:

v_{⊤} = k \times v_{⊥} .

Note that as $k$ and $v_{⊥}$ are orthogonal, and $k$ has unit length, the magnitude of $v_{⊤}$ is the same as the magnitude of $v_{⊥}$ . Also, as $k \times v_{∥} = 0$ , one can also write:

v_{⊤} = k \times v .

(X.B.4)

We also have

v_{⊥} = - k \times v_{⊤} = - k \times k \times v .

In equation (X.B.4), we have a linear relation between vectors $v$ and $v_{⊤}$ . This means that a matrix $K$ can be introduced such that:

v_{⊤} = K \cdot v .

Using indicial notation and Einstein summation convention on repeated indices we write:

{v_{⊤}}_{i} = K_{i l} v_{l} = ϵ_{i j l} k_{j} v_{l},

and

K_{i l} = ϵ_{i j l} k_{j} = (\begin{matrix} 0 & - k_{z} & k_{y} \\ k_{z} & 0 & - k_{x} \\ - k_{y} & k_{x} & 0 \end{matrix}) .

Using matrix $K$ , we write:

v_{⊥} = - K^{2} \cdot v,

v_{∥} = (I + K^{2}) \cdot v,

v_{⊤} = K \cdot v .

The rotated vector $v^{'}$ is obtained as follows:

\begin{align} v^{'} & = v_{∥} + cos θ v_{⊥} + sin θ v_{⊤}, \\ = (I + K^{2}) \cdot v - cos θ K^{2} \cdot v + sin θ K \cdot v, \\ = (I + K^{2} - cos θ K^{2} + sin θ K) \cdot v, \\ = R \cdot v, \end{align}

where $R$ is defined as:

\begin{align} R & = I + K^{2} - cos θ K^{2} + sin θ K, \\ = I + sin θ K + (1 - cos θ) K^{2} . & (X.B.5) \end{align}

The rotation matrix $R$ corresponds to matrix $Q T$ in section X.B.6. Tensor $Q$ can be obtained by using $- r$ instead of $r$ in the derivation of $R$ . It can be used to transform the components of vectors and tensors as explained in section X.B.6.

The derivation of $R$ matrix via expression (X.B.5) corresponds to Rodrigues’ rotation formula.

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