Rotors are geometric algebra's secret weapon for rotations. They're compact, efficient, and work in any dimension. Unlike Euler angles or matrices, rotors avoid gimbal lock and simplify complex rotations. They're the Swiss Army knife of rotation representations.
Applying rotors is a breeze: just sandwich your vector between a rotor and its reverse. This works for multivectors too, preserving their grade. Rotors can be composed, interpolated, and constructed from vectors or bivectors, making them incredibly versatile for all your rotation needs.
Rotors for Rotations
Rotor Definition and Properties
- Rotors are elements of the even subalgebra of a geometric algebra that can represent rotations in any dimension
- Rotors provide a compact and efficient way to represent rotations without the limitations of other representations (Euler angles, quaternions)
- Rotor composition is more efficient than matrix multiplication, as it involves fewer operations and maintains the unit norm property of rotors
- Rotors can be interpolated smoothly using techniques such as spherical linear interpolation (slerp), enabling the generation of continuous rotation paths
Applying Rotors to Vectors and Multivectors
- To rotate a vector $v$ by a rotor $R$, the operation is $v' = RvR̃$, where $R̃$ is the reverse of $R$
- The reverse of a rotor is obtained by reversing the order of its factors and negating the bivector components
- Rotors can also be applied to multivectors (bivectors, trivectors) using the same operation $M' = RMR̃$, where $M$ is the multivector
- The rotor application preserves the grade of the multivector, so vectors remain vectors, bivectors remain bivectors, and so on
- Applying a rotor to a multivector results in a rotated version of the original multivector, maintaining its geometric properties and relationships
Rotors in Various Dimensions
2D Rotors
- In 2D, rotors are elements of the even subalgebra generated by the pseudoscalar, forming the spin group Spin(2), isomorphic to the complex numbers
- A 2D rotor for a rotation by angle $\theta$ is given by $R = e^{I\theta/2} = \cos(\theta/2) + I \sin(\theta/2)$, where $I$ is the pseudoscalar
3D Rotors
- In 3D, rotors are elements of the even subalgebra generated by bivectors, forming the spin group Spin(3), a double cover of the special orthogonal group SO(3)
- A 3D rotor for a rotation about an arbitrary axis $n$ by angle $\theta$ is given by $R = e^{nI\theta/2}$, where $I$ is the pseudoscalar and $nI$ represents the plane perpendicular to the rotation axis $n$
Higher-Dimensional Rotors
- In higher dimensions, rotors are elements of the even subalgebra generated by bivectors, forming the spin group Spin(n) for n-dimensional space
- Rotors provide a unified framework for rotations in any dimension without the need for specialized representations like quaternions
Rotors for Arbitrary Axes
Rotor Construction from Bivectors
- A rotor for a rotation by angle $\theta$ about a unit bivector $B$ is given by the exponential $R = e^{B\theta/2} = \cos(\theta/2) + B \sin(\theta/2)$
- The bivector $B$ represents the plane of rotation, and the exponential of this bivector generates the rotor
Rotor Construction from Vectors
- To construct a rotor for a rotation between two unit vectors $a$ and $b$, the rotor is given by $R = (1 + ba)/\sqrt{2(1 + a \cdot b)}$, where $ba$ is the geometric product of $b$ and $a$
- This rotor construction ensures that the rotor $R$ rotates vector $a$ to align with vector $b$
Composing Rotations with Rotors
- Rotors can be composed by multiplying them together, allowing for the representation of a sequence of rotations as a single rotor
- Rotor composition is associative, so the order of multiplication matters when combining rotations
- Composing rotors is more efficient and numerically stable compared to composing rotation matrices
Rotors vs Matrices for Rotations
Advantages of Rotors
- Rotors provide a singularity-free representation of rotations, avoiding the gimbal lock problem associated with Euler angles
- Rotors are more compact than rotation matrices, requiring fewer parameters to represent a rotation (4 parameters for a 3D rotor compared to 9 for a rotation matrix)
- The rotor representation is geometrically intuitive, as it directly encodes the plane of rotation (bivector) and the rotation angle, making it easier to reason about and manipulate rotations
Comparison to Other Rotation Representations
- Rotors offer advantages over other rotation representations (Euler angles, quaternions, rotation matrices) in terms of compactness, efficiency, and geometric interpretability
- Rotors can be easily converted to and from other rotation representations when necessary for compatibility with existing systems or algorithms
- Rotors provide a unified and consistent framework for representing rotations in any dimension, simplifying the implementation and understanding of rotational transformations in geometric algebra