Glossary

7.2 Rotor representation of rotations

4 min read•july 30, 2024

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

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  • 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 vv by a rotor RR, the operation is v′=RvR~v' = RvR̃, where R~R̃ is the reverse of RR
    • 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′=RMR~M' = RMR̃, where MM 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 θ\theta is given by R=eIθ/2=cos⁡(θ/2)+Isin⁡(θ/2)R = e^{I\theta/2} = \cos(\theta/2) + I \sin(\theta/2), where II 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 nn by angle θ\theta is given by R=enIθ/2R = e^{nI\theta/2}, where II is the pseudoscalar and nInI represents the plane perpendicular to the rotation axis nn

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 θ\theta about a unit bivector BB is given by the exponential R=eBθ/2=cos⁡(θ/2)+Bsin⁡(θ/2)R = e^{B\theta/2} = \cos(\theta/2) + B \sin(\theta/2)
  • The bivector BB represents the , and the exponential of this bivector generates the rotor

Rotor Construction from Vectors

  • To construct a rotor for a rotation between two unit vectors aa and bb, the rotor is given by R=(1+ba)/2(1+a⋅b)R = (1 + ba)/\sqrt{2(1 + a \cdot b)}, where baba is the geometric product of bb and aa
  • This rotor construction ensures that the rotor RR rotates vector aa to align with vector bb

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 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

Key Terms to Review (15)

Angle of rotation: The angle of rotation is a measure of the amount of rotation needed to turn an object around a specific axis, usually expressed in degrees or radians. This concept is central to understanding how different rotations can be combined and represented in mathematical frameworks, like rotors and quaternions, which help facilitate complex geometric transformations.
Axis of Rotation: The axis of rotation is an imaginary line around which an object rotates. In geometric algebra, it serves as a critical reference point for understanding how rotations occur in space. This concept becomes particularly important when discussing the composition of rotations and the representation of those rotations using rotors, as it allows for the visualization and mathematical manipulation of rotational transformations in a precise manner.
Blade rotor: A blade rotor is a geometric object in the context of Geometric Algebra that represents rotations in a multi-dimensional space. It can be viewed as an extension of the idea of rotating an object around an axis, where the blade rotor encapsulates both the axis of rotation and the angle, providing a compact and elegant way to express rotational transformations. Blade rotors can also be used to generate rotations in a coordinate-free manner, showcasing the power of Geometric Algebra in simplifying complex transformations.
Composition of rotors: The composition of rotors refers to the process of combining multiple rotors to achieve a resultant rotation in geometric algebra. This concept is crucial for understanding how different rotations can be represented and manipulated within a geometric framework, allowing for complex rotations to be simplified into a single rotor representation. It connects various properties of rotors, such as their relationship with blades and their impact on orientation in space.
E^{θ/2}: In geometric algebra, e^{θ/2} represents a rotor, which is a mathematical entity used to perform rotations in a multi-dimensional space. This expression embodies the angle of rotation, θ, divided by 2, and indicates that the rotation takes place in half-angles, providing a unique way to encode rotations that is both efficient and versatile. By employing the exponential map of a bivector, rotors facilitate smooth and continuous transformations in geometric spaces.
Inverse Rotor: An inverse rotor is a mathematical construct in geometric algebra that represents the reversal of a rotation in space. It is essentially the rotor that undoes the action of a given rotor, allowing for the restoration of original orientations after transformations. The inverse rotor plays a crucial role in understanding how to compose and decompose rotations effectively, providing a way to manipulate orientations in three-dimensional space seamlessly.
Orthogonal transformation: An orthogonal transformation is a linear transformation that preserves the inner product, meaning it maintains the angles and lengths of vectors during the transformation. This type of transformation can be represented by orthogonal matrices, which have properties that make them particularly useful for describing rotations and reflections in geometric spaces.
Plane of rotation: The plane of rotation is a geometric concept that refers to the two-dimensional surface where a rotation occurs. This plane is critical in understanding how objects move and orient themselves in space when they are subjected to rotations, especially in the context of rotors which represent these transformations in a compact and efficient way.
Quaternion representation: Quaternion representation is a mathematical framework used to describe rotations in three-dimensional space, utilizing a four-dimensional number system. This approach provides a way to represent complex rotations without the gimbal lock problem often associated with Euler angles, making it particularly useful in computer graphics and robotics. The quaternion consists of one real part and three imaginary parts, which allows for compact representation and efficient computation of rotation transformations.
R(θ): r(θ) represents the rotor, a mathematical entity used to describe rotations in Geometric Algebra. This concept links angles with the geometric representation of rotations, allowing one to encapsulate the action of rotating vectors in space through a single expression. By using r(θ), one can efficiently manage and compute rotations, which are fundamental in various applications such as robotics and computer graphics.
Reflection rotor: A reflection rotor is a mathematical construct used in geometric algebra to represent reflections across a hyperplane. This rotor can be understood as an element that encodes the process of reflecting a vector in space, which is essential for manipulating orientations and transformations within geometric algebra. Reflection rotors are closely tied to the concept of rotations and can be combined with other rotors to achieve complex transformations.
Rotor multiplication: Rotor multiplication is the process of combining two or more rotors to achieve the composition of rotations in geometric algebra. This operation allows for the representation of complex rotational transformations as a single rotor, simplifying calculations and enabling efficient manipulation of rotations in multi-dimensional spaces. Understanding rotor multiplication is essential for expressing and composing rotations without resorting to matrices or other more complex methods.
Rotor representation: Rotor representation is a mathematical construct used in geometric algebra to describe rotations in a space. It utilizes a special type of multivector known as a rotor, which can efficiently encapsulate the notion of rotation around an axis while preserving the geometry of the space. This representation connects algebraic operations with geometric transformations, allowing for a unified approach to understanding rotations.
Spatial Rotation: Spatial rotation refers to the process of rotating an object around a specific axis in three-dimensional space. This concept is essential for understanding how objects can change their orientation without altering their position in space, and it is closely linked to various mathematical representations, such as rotors, that simplify calculations and provide insights into classical mechanics.
Unit rotor: A unit rotor is a geometric algebra element that represents a rotation in a multi-dimensional space, characterized by having a norm of one. This makes it an essential tool for describing rotations without the complications of traditional matrix representations. Unit rotors provide an elegant and efficient means of encapsulating rotational transformations while maintaining geometric properties.
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