Rotations: Difference between revisions

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The most natural representation of rotations are Quaternions.
The most natural representation of rotations are Quaternions.
However rotations can also be represented in various other forms.
However rotations can also be represented in various other forms.
See https://www.andre-gaschler.com/rotationconverter/ to convert between representations


===Angle Axis===
===Angle Axis===
Also known as axis-angle
===Rotation Vector===
===Rotation Vector===
===Euler Angles===
===Euler Angles===
===Quaternion===
===Quaternion===
See [[Quaternion]].
===Matrix===
===Matrix===
A \(3\times 3\) matrix is the most convenient form of a rotation since applying the rotation to a vector is simply a matrix multiplication.
A \(3\times 3\) matrix is the most convenient form of a rotation since applying the rotation to a vector is simply a matrix multiplication.
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This section is on converting between different forms of rotations.<br>
This section is on converting between different forms of rotations.<br>
How to construct a rotation.<br>
How to construct a rotation.<br>
See [http://www.euclideanspace.com/maths/geometry/rotations/conversions/index.htm http://www.euclideanspace.com/maths/geometry/rotations/conversions/index.htm] for a list of conversions.


===Angle Axis to Matrix===
===Angle Axis to Matrix===
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\mathbf{R} = \mathbf{I} + (\sin\theta)\mathbf{K} + (1-\cos\theta)\mathbf{K}^2
\mathbf{R} = \mathbf{I} + (\sin\theta)\mathbf{K} + (1-\cos\theta)\mathbf{K}^2
\end{equation}
\end{equation}
\]
===Angle Axis to Quaternion===
Based on [http://www.euclideanspace.com/maths/geometry/rotations/conversions/angleToQuaternion/index.htm http://www.euclideanspace.com/maths/geometry/rotations/conversions/angleToQuaternion/index.htm]
\[
\begin{align}
q_x &= k_x * \sin(\theta/2)\\
q_y &= k_y * \sin(\theta/2)\\
q_z &= k_z * \sin(\theta/2)\\
q_w &= \cos(\theta/2)
\end{align}
\]
\]


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* [https://docs.unity3d.com/ScriptReference/Quaternion.html Unity Quaternion Script Reference]
* [https://docs.unity3d.com/ScriptReference/Quaternion.html Unity Quaternion Script Reference]
* [https://threejs.org/docs/#api/en/math/Quaternion three.js Quaternion]
* [https://threejs.org/docs/#api/en/math/Quaternion three.js Quaternion]
** [https://github.com/mrdoob/three.js/blob/master/src/math/Quaternion.js three.js/Quaternion.js source code]
* [https://docs.scipy.org/doc/scipy/reference/generated/scipy.spatial.transform.Rotation.html scipy.spatial.transform.Rotation]
* [https://docs.scipy.org/doc/scipy/reference/generated/scipy.spatial.transform.Rotation.html scipy.spatial.transform.Rotation]


==References==
==References==

Latest revision as of 20:56, 20 September 2024

This article is about rotations in 3D space.

Representations

The most natural representation of rotations are Quaternions. However rotations can also be represented in various other forms.

See https://www.andre-gaschler.com/rotationconverter/ to convert between representations

Angle Axis

Also known as axis-angle

Rotation Vector

Euler Angles

Quaternion

See Quaternion.

Matrix

A \(3\times 3\) matrix is the most convenient form of a rotation since applying the rotation to a vector is simply a matrix multiplication.

Construction

This section is on converting between different forms of rotations.
How to construct a rotation.
See http://www.euclideanspace.com/maths/geometry/rotations/conversions/index.htm for a list of conversions.

Angle Axis to Matrix

Apply Wikipedia: Rodrigues' rotation formula

Suppose \(\mathbf{k}=(k_x, k_y, k_z)\) is the vector around which you want to rotate.
Let \[ \mathbf{K} = [\mathbf{k}]_\times = \begin{pmatrix} 0 & -k_z & k_y\\ k_z & 0 & -k_x\\ -k_y & k_x & 0 \end{pmatrix} \] Then our rotation matrix is: \[ \begin{equation} \mathbf{R} = \mathbf{I} + (\sin\theta)\mathbf{K} + (1-\cos\theta)\mathbf{K}^2 \end{equation} \]

Angle Axis to Quaternion

Based on http://www.euclideanspace.com/maths/geometry/rotations/conversions/angleToQuaternion/index.htm

\[ \begin{align} q_x &= k_x * \sin(\theta/2)\\ q_y &= k_y * \sin(\theta/2)\\ q_z &= k_z * \sin(\theta/2)\\ q_w &= \cos(\theta/2) \end{align} \]

Resources

References