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. | |||
==Construction | ==Construction== | ||
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=== | ||
| Line 22: | Line 32: | ||
Let | Let | ||
\[ | \[ | ||
\mathbf{K} = [\mathbf{k}]_\times | \mathbf{K} = [\mathbf{k}]_\times = | ||
\begin{pmatrix} | \begin{pmatrix} | ||
0 & -k_z & k_y\\ | 0 & -k_z & k_y\\ | ||
| Line 35: | Line 45: | ||
\end{equation} | \end{equation} | ||
\] | \] | ||
===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] | |||
\[ | |||
\begin{align} | |||
q_x &= k_x * \sin(\theta/2)\\ | |||
q_y &= k_y * \sin(\theta/2)\\ | |||
q_z &= k_z * \sin(\theta/2)\\ | |||
q_w &= \cos(\theta/2) | |||
\end{align} | |||
\] | |||
==Resources== | ==Resources== | ||
* [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
\[ \begin{align} q_x &= k_x * \sin(\theta/2)\\ q_y &= k_y * \sin(\theta/2)\\ q_z &= k_z * \sin(\theta/2)\\ q_w &= \cos(\theta/2) \end{align} \]