Quaternion normalization

Let's consider the following quaternion:

$$ Q=a+b.i+c.j+d.k = \left[ \begin{matrix} a && b && c && d \end{matrix} \right]$$

The normalization is given by the following formula:

$$ Q_{normalized}=\frac{Q}{| Q | } $$

where:

$$ | Q | = \sqrt{ Q.\overline{Q} } = \sqrt{ \overline{Q}.Q }=\sqrt {a^2 + b^2 + c^2 + d^2} $$

The general representation is given by:

$$ Q_{normalized}= \left[ \begin{matrix} \dfrac{a}{| Q |} && \dfrac{b}{| Q |} && \dfrac{c}{| Q |} && \dfrac{d}{| Q |} \end{matrix} \right] $$

$$ Q_{normalized}= \left[ \begin{matrix} A && B && C && D \end{matrix} \right] $$

with: $$ A = \dfrac{a}{\sqrt {a^2 + b^2 + c^2 + d^2}} $$ $$ B = \dfrac{b}{\sqrt {a^2 + b^2 + c^2 + d^2}} $$ $$ C = \dfrac{c}{\sqrt {a^2 + b^2 + c^2 + d^2}} $$ $$ D = \dfrac{d}{\sqrt {a^2 + b^2 + c^2 + d^2}} $$

\begin{multline} Q_{normalized}= \left[ \begin{matrix} \dfrac{a}{\sqrt {a^2 + b^2 + c^2 + d^2}} \end{matrix}\right. \\ \dfrac{b}{\sqrt {b^2 + b^2 + c^2 + d^2}} \quad \dfrac{c}{\sqrt {a^2 + b^2 + c^2 + d^2}} \\ \left. \begin{matrix} \dfrac{d}{\sqrt {a^2 + b^2 + c^2 + d^2}} \end{matrix} \right] \end{multline}

Quaternion normalization

See also