Comparison of Methods for Rotating a Point in $\mathbb{R}^3$: A Case Study

TL;DR

The paper compares four hand-rotation methods in $\mathbb{R}^3$—an ad hoc symmetry approach, a projection-based coordinate setup, a rotation-matrix construction from rotated basis vectors, and a Geometric (Clifford) Algebra rotor method—for rotating a vector about an axis by an angle $\theta$. Using $P=(1,0,1)^T$, $a=(1,1,1)^T$, and $\theta=60^\circ$, all methods yield the same result $P' = \left(\tfrac{4}{3}, \tfrac{1}{3}, \tfrac{1}{3}\right)^T$. The work discusses the generality, computational trade-offs, and conceptual unification across approaches, including how GA rotations can be viewed as double reflections via rotors and how the rotation matrix aligns with basis rotation. The findings highlight guidance for method selection based on context, ranging from symmetry exploitation to dimension-independent GA formulations and quaternion-equivalent structures in 3D.

Abstract

This article presents and compares four approaches for computing the rotation of a point about an axis by an angle in $\mathbb{R}^3$. We illustrate these methods by computing, by hand, the rotation of point $P=(1,0,1)^T$ about axis $\mathbf{a}=(1,1,1)^T$ by angle $θ=60^\circ$ (following the right-hand rule). The four methods considered are: (1) an ad hoc geometric method exploiting a symmetry in the situation; (2) a projection method that sets up a new coordinate system using the dot and cross products; (3) a matrix method which rotates the standard basis and uses matrix-vector multiplication; (4) a Geometric (Clifford) Algebra method that represents the rotation as a double reflection via a rotor. All methods yield the same exact result: $P'=\left(\tfrac{4}{3},\tfrac{1}{3},\tfrac{1}{3}\right)^T$.

Figures (5)

• Figure 1: Rotation problem: Rotate given point $P$ about given axis $\mathbf{a}$ by given angle $\theta$.
• Figure 2: Method 2: Ad hoc approach exploiting symmetry.
• Figure 3: Method 2: Projection and new coordinate system (the latter is not to scale).
• Figure 4: Method 3: Using a rotation matrix based on rotated basis vectors.
• Figure 5: Method 4: Geometric (Clifford) Algebra (unit bivector $\mathbf{U}$ is not to scale).