Quaternion Functions

inline Quaternion QDynamics::operator*(const Quaternion &lhs, const Quaternion &rhs)

The crucial quaternion multiplication operation, defined such that:

\(\mathsf{a} \otimes \mathsf{b} = \begin{pmatrix} a_0 b_0 - \vec{a} \cdot \vec{b} \\ a_0 \vec{b} + b_0 \vec{a} + \vec{a} \times \vec{b}\end{pmatrix}\) . Note that this product is highly non-commutative in general.

Parameters

• lhs – The first argument of the operation ( \(\mathsf{a}\) )

• rhs – The second argument ( \(\mathsf{b}\) )

Returns

The quaternion product ( \(\mathsf{a}\otimes\mathsf{b}\) )

inline Quaternion QDynamics::operator/(const Quaternion &lhs, const Quaternion &rhs)

Quaternion division operation, such that:

\(\mathsf{a} \oslash \mathsf{b} = \frac{1}{|\mathsf{b}|^2} \mathsf{a} \otimes \overline{\mathsf{b}}\) .

Parameters

• lhs – The first argument of the operation ( \(\mathsf{a}\) )

• rhs – The second argument ( \(\mathsf{b}\) )

Returns

The quaternion division ( \(\mathsf{a}\oslash\mathsf{b}\) )

inline Quaternion QDynamics::operator*(const JSL::Matrix &lhs, const Quaternion &rhs)

The (slightly dodgy) matrix-quaternion product. In reality, casts the quaternion to R^4, multiplies, then casts back.

Parameters

• lhs – A JSL::Matrix object to be multiplied

• rhs – A Quaternion object to be multiplied

Returns

A Quaternion object of the resulting product

inline Quaternion QDynamics::exp(const Quaternion &a)

The quaternion exponential, defined such that:

\(\exp(\mathsf{a}) = \sum_{n = 0}^\infty \frac{\mathsf{a}^n}{n!}\)

Parameters

a – The argument of the operation ( \(\mathsf{a}\) )

Returns

The (analytically computed) quaternion exponential