Finding the axis and angle that define a rotation matrix involves extracting specific properties from the matrix itself.
Understanding Axis-Angle Representation
A rotation in 3D space can be uniquely described by a single rotation around a specific axis. This representation consists of:
- The Axis: A unit vector $\mathbf{k}$ pointing along the axis of rotation.
- The Angle: An angle $\theta$ representing the amount of rotation around the axis $\mathbf{k}$.
Given a 3x3 rotation matrix $Q$, you can determine these two components.
Calculating the Rotation Angle (θ)
The easiest way to find the rotation angle $\theta$ from a rotation matrix $Q$ is by using its trace. The trace of a matrix is the sum of its diagonal elements. For a rotation matrix $Q$, the trace is related to the angle $\theta$ by the formula:
1 + 2cosθ = trace(Q)
From this formula, you can isolate $\cos\theta$:
cosθ = (trace(Q) - 1) / 2
Once you have cosθ, you can find the angle $\theta$ using the arccosine function:
θ = arccos( (trace(Q) - 1) / 2 )
Important Considerations:
- The trace of a 3x3 rotation matrix must be between -1 (for a 180-degree rotation) and 3 (for no rotation). If the trace is outside this range (due to floating-point inaccuracies or if the matrix isn't truly a rotation matrix), the argument to
arccosmight be invalid. - The
arccosfunction typically returns an angle between 0 and $\pi$ (or 0 and 180 degrees). This gives you the magnitude of the angle.
Example Trace Calculation:
As mentioned in the reference, if you had a specific matrix Q where trace(Q) = 0.36 + 0.60 + 0.60, then trace(Q) = 1.56.
Using the formula:
cosθ = (1.56 - 1) / 2 = 0.56 / 2 = 0.28
θ = arccos(0.28)
You would then calculate the numerical value of arccos(0.28) to find the angle $\theta$.
Finding the Rotation Axis (k)
The rotation axis $\mathbf{k}$ is a vector that does not change its direction when the rotation is applied. Mathematically, this means that rotating the vector $\mathbf{k}$ by the matrix $Q$ results in the same vector $\mathbf{k}$:
$Q\mathbf{k} = \mathbf{k}$
Rearranging this equation gives:
$Q\mathbf{k} - \mathbf{k} = \mathbf{0}$
$(Q - I)\mathbf{k} = \mathbf{0}$
where $I$ is the identity matrix. This is the definition of an eigenvector equation where $\mathbf{k}$ is the eigenvector and the eigenvalue is 1.
Therefore, the axis of rotation is an eigenvector of the rotation matrix Q corresponding to the eigenvalue 1.
Steps to Find the Axis:
- Form the matrix $(Q - I)$.
- Solve the linear system $(Q - I)\mathbf{k} = \mathbf{0}$ for the vector $\mathbf{k}$.
- The solution space of this system is the set of all vectors parallel to the rotation axis.
- Any non-zero vector found this way can be used as the axis vector.
- Typically, the axis is represented as a unit vector, so you should normalize the resulting vector $\mathbf{k}$ by dividing it by its magnitude: $\mathbf{k}_{\text{unit}} = \mathbf{k} / ||\mathbf{k}||$.
Practical Note:
For a proper rotation matrix (not a reflection), there will always be an eigenvalue of 1, and its corresponding eigenvector space will be one-dimensional, representing the line along the rotation axis. If the rotation is 0 degrees (identity matrix), all vectors are eigenvectors with eigenvalue 1, and the axis is undefined (or often taken as an arbitrary axis like [1, 0, 0]). If the rotation is 180 degrees, the eigenvalue 1 still exists, but finding the corresponding eigenvector might require careful handling of floating-point arithmetic as $(Q-I)$ will have rank 1.
Summary: Axis-Angle Extraction
To find the axis-angle representation (axis $\mathbf{k}$ and angle $\theta$) from a rotation matrix $Q$:
- Calculate the Angle: Use the formula
θ = arccos( (trace(Q) - 1) / 2 ). - Find the Axis: Find a non-zero eigenvector of $Q$ corresponding to the eigenvalue 1. Normalize this eigenvector to get the unit axis vector $\mathbf{k}$.
This process allows you to convert a rotation matrix into its equivalent axis-angle representation, which is often more intuitive for visualizing or describing a rotation.
