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{{Short description|Mathematical strategy}}
{{cleanup rewrite|date=January 2024}}
[[Rotation formalisms in three dimensions|Spatial rotations in three dimensions]] can be [[Coordinate system|parametrized]] using both [[Euler angles]] and [[Quaternions and spatial rotation|unit quaternions]]. This article explains how to convert between the two representations. Actually this simple use of "quaternions" was first presented by [[Leonhard Euler|Euler]] some seventy years earlier than [[William Rowan Hamilton|Hamilton]] to solve the problem of [[magic square]]s. For this reason the dynamics community commonly refers to quaternions in this application as "Euler parameters".
==Definition==
There are [[Quaternions_and_spatial_rotation#Alternative_conventions|two representations]] of quaternions. This article uses the more popular Hamilton.
A quaternion has 4 real values: {{mvar|q<sub>w</sub>}} (the real part or the scalar part) and {{mvar|q<sub>x</sub> q<sub>y</sub> q<sub>z</sub>}} (the imaginary part).
:<math>|\mathbf{q}|^2 = q_w^2 + q_x^2 + q_y^2 + q_z^2 = 1</math>▼
Defining the [[Quaternion#Conjugation,_the_norm,_and_reciprocal|norm of the quaternion]] as follows:
▲
A ''unit quaternion'' satisfies:
<math display=block>\lVert q \rVert = 1</math>
We can associate a [[quaternion]] with a rotation around an axis by the following expression
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:<math>\mathbf{q}_y = \sin(\alpha/2)\cos(\beta_y)</math>
:<math>\mathbf{q}_z = \sin(\alpha/2)\cos(\beta_z)</math>
where α is a simple rotation angle (the value in radians of the [[angle of rotation]]) and cos(β<sub>''x''</sub>), cos(β<sub>''y''</sub>) and cos(β<sub>''z''</sub>) are the "[[direction cosine]]s" of the angles between the three coordinate axes and the axis of rotation. ([[Euler's rotation theorem|Euler's Rotation Theorem]]).
==Intuition==
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== Rotation matrices ==
The [[orthogonal matrix]] (post-multiplying a [[Row and column vectors|column vector]]) corresponding to a clockwise/[[Right-hand rule|left-handed]] (looking along positive axis to origin) rotation by the unit [[quaternion]] <math>q=q_w+iq_x+jq_y+kq_z</math> is given by the [[Quaternions and spatial rotation#Quaternion-derived rotation matrix|inhomogeneous expression]]:
:<math>R = \begin{bmatrix}
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};
// This is not in game format, it is in mathematical format.
Quaternion ToQuaternion(double roll, double pitch, double yaw) // roll (x), pitch (
{
// Abbreviations for the various angular functions
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:<math>\vec{t} = 2\vec{q} \times \vec{v}</math>
:<math>\vec{v}^{\,\prime} = \vec{v} + q_w \vec{t} + \vec{q} \times \vec{t}</math>
where <math>\times</math> indicates a three-dimensional vector [[cross product]]. This involves fewer multiplications and is therefore computationally faster. Numerical tests indicate this latter approach may be up to 30% <ref>{{cite journal |pmc=4435132|year=2015|last1=Janota|first1=A|title=Improving the Precision and Speed of Euler Angles Computation from Low-Cost Rotation Sensor Data|journal=Sensors|volume=15|issue=3|pages=7016–7039|last2=Šimák|first2=V|last3=Nemec|first3=D|last4=Hrbček|first4=J|doi=10.3390/s150307016|pmid=25806874|bibcode=2015Senso..15.7016J |doi-access=free}}</ref> faster than the original for vector rotation.
=== Proof ===
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