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