📐Geometric Algebra Unit 7 – Rotors and Rotations

What Are Rotors?

• Rotors are elements of geometric algebra that represent rotations in a space
• Consist of a scalar part and a bivector part, which encodes the plane and angle of rotation
• Can be thought of as a generalization of complex numbers, with the bivector part playing the role of the imaginary unit
• Provide a compact and efficient way to represent and manipulate rotations in any number of dimensions
• Rotors form a group under the geometric product, which means they can be composed and inverted
• Enable the representation of rotations without the need for matrices or Euler angles
• Rotors are unit magnitude elements, ensuring that they preserve the length of the vectors they rotate

Geometric Product and Bivectors

• The geometric product is a fundamental operation in geometric algebra that combines the inner and outer products of vectors
• For two vectors $a$ and $b$, their geometric product is given by $ab = a \cdot b + a \wedge b$, where $a \cdot b$ is the inner product and $a \wedge b$ is the outer product
• The outer product of two vectors results in a bivector, which represents a directed plane segment
  • For example, in 3D, the outer product of two orthogonal unit vectors $e_1$ and $e_2$ gives the bivector $e_1 \wedge e_2$, which represents the plane spanned by these vectors
• Bivectors are grade-2 elements of geometric algebra and form a subspace of the algebra
• The geometric product of two bivectors results in a combination of a scalar and a grade-4 element (a quadrivector)
• Bivectors can be used to represent rotations in a plane, with the magnitude of the bivector determining the angle of rotation

Rotor Algebra Basics

• Rotors are elements of the even subalgebra of geometric algebra, which means they are composed of scalars and bivectors
• The general form of a rotor in 3D is $R = \cos(\frac{\theta}{2}) - \sin(\frac{\theta}{2})B$, where $\theta$ is the angle of rotation and $B$ is a unit bivector representing the plane of rotation
• Rotors are invertible, with the inverse of a rotor $R$ being its reverse $\tilde{R}$, which is obtained by reversing the order of the factors in the geometric product
• The geometric product of a rotor and its reverse gives the scalar unity, $R\tilde{R} = 1$
• Rotations are applied to vectors using the sandwich product, $v' = RvR^{-1}$, where $v$ is the vector being rotated and $v'$ is the rotated vector
• Rotors can be interpolated using the geometric product and exponential map, enabling smooth interpolation between rotations
  • For example, given two rotors $R_1$ and $R_2$, an interpolated rotor $R(t)$ can be obtained as $R(t) = R_1 \exp(t \log(R_1^{-1}R_2))$, where $t \in [0, 1]$ is the interpolation parameter

Rotations in 2D and 3D Spaces

• In 2D, rotations are represented by rotors of the form $R = \cos(\frac{\theta}{2}) - \sin(\frac{\theta}{2})e_{12}$, where $e_{12}$ is the bivector representing the 2D plane
• 2D rotations can be visualized as rotations in the complex plane, with the bivector $e_{12}$ playing the role of the imaginary unit
• In 3D, rotations are represented by rotors involving the three basis bivectors $e_{23}$, $e_{31}$, and $e_{12}$, which correspond to rotations about the $x$, $y$, and $z$ axes, respectively
• A general 3D rotor can be written as $R = \cos(\frac{\theta}{2}) - \sin(\frac{\theta}{2})(n_1e_{23} + n_2e_{31} + n_3e_{12})$, where $(n_1, n_2, n_3)$ is a unit vector representing the axis of rotation
• 3D rotations can be composed by multiplying the corresponding rotors using the geometric product
• Rotors provide a singularity-free representation of 3D rotations, avoiding the gimbal lock problem associated with Euler angles

Composing Rotations

• Rotations represented by rotors can be easily composed using the geometric product
• Given two rotors $R_1$ and $R_2$, their composition $R_3$ is obtained by multiplying them in the desired order, $R_3 = R_2R_1$
  • This means that applying $R_3$ to a vector is equivalent to first applying $R_1$ and then $R_2$
• The composition of rotors is associative, meaning that $(R_3R_2)R_1 = R_3(R_2R_1)$
• The order of composition matters, as rotor multiplication is generally not commutative
• Composing rotors allows for the efficient representation and manipulation of sequences of rotations
  • For example, a sequence of rotations about different axes can be combined into a single rotor, simplifying the application of the rotations to vectors or other geometric objects
• The inverse of a composed rotor is the reverse of the composition in the opposite order, $(R_2R_1)^{-1} = R_1^{-1}R_2^{-1}$

Applications in Physics and Engineering

• Rotors find applications in various fields of physics and engineering where rotations play a crucial role
• In classical mechanics, rotors can be used to represent the orientation and motion of rigid bodies
  • The dynamics of a rigid body can be described using rotors, with the angular velocity and angular momentum being represented as bivectors
• In quantum mechanics, rotors are used to represent the spin states of particles and the rotational symmetries of systems
  • The Pauli matrices, which describe the spin of fermions, can be represented as bivectors in geometric algebra
• In computer graphics and robotics, rotors are used for efficient and stable rotation computations
  • Rotors provide a compact and numerically stable representation of rotations, avoiding the singularities and gimbal lock issues associated with other rotation representations
• In aerospace engineering, rotors are used for attitude control and navigation of spacecraft and aircraft
  • Rotors enable the efficient computation of spacecraft orientations and the planning of attitude maneuvers
• Rotors also find applications in other areas such as computer vision, crystallography, and electromagnetic theory, where the manipulation of rotations is essential

Comparison with Other Rotation Methods

• Rotors offer several advantages over other rotation representations, such as Euler angles, quaternions, and rotation matrices
• Compared to Euler angles, rotors provide a singularity-free representation of rotations, avoiding the gimbal lock problem
  • Euler angles suffer from singularities at certain orientations, which can lead to numerical instabilities and discontinuities in the rotation representation
• Rotors are more compact than rotation matrices, requiring only four components in 3D compared to the nine components of a 3x3 rotation matrix
  • This compactness leads to reduced memory usage and improved computational efficiency
• Rotors are geometrically intuitive, as they directly encode the plane and angle of rotation
  • In contrast, the geometric interpretation of quaternions is less straightforward, as they involve a scalar and a vector part
• Rotor algebra is a natural extension of vector algebra, allowing for the seamless integration of rotations with other geometric operations
  • Quaternions, on the other hand, require a separate algebraic framework and lack the geometric interpretability of rotors
• Rotors can be easily interpolated using the geometric product and exponential map, enabling smooth and efficient interpolation between rotations
  • Interpolating quaternions requires normalization to ensure unit magnitude, which can be computationally expensive

Practical Examples and Problem Solving

• Example 1: Given a vector $v = (1, 0, 0)$ and a rotor $R = \cos(\frac{\pi}{4}) - \sin(\frac{\pi}{4})e_{12}$, find the rotated vector $v'$.
  • Solution: $v' = RvR^{-1} = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, 0)$
• Example 2: Compose two rotations in 3D, one about the $x$-axis by an angle $\alpha$ and another about the $y$-axis by an angle $\beta$, and find the equivalent single rotor.
  • Solution: $R_x = \cos(\frac{\alpha}{2}) - \sin(\frac{\alpha}{2})e_{23}$, $R_y = \cos(\frac{\beta}{2}) - \sin(\frac{\beta}{2})e_{31}$, and the composed rotor is $R = R_yR_x$
• Example 3: Interpolate between two rotors $R_1$ and $R_2$ at a parameter value of $t = 0.5$.
  • Solution: $R(0.5) = R_1 \exp(0.5 \log(R_1^{-1}R_2))$
• Problem 1: Find the rotor that represents a rotation of 60° about the axis $(1, 1, 1)$.
• Problem 2: Given two vectors $u$ and $v$, find the rotor that rotates $u$ to $v$ by the smallest angle.
• Problem 3: Implement a function that composes a sequence of rotors and returns the equivalent single rotor.

These examples and problems demonstrate the practical application of rotors in solving rotation-related tasks and highlight the efficiency and elegance of rotor algebra in geometric computations.