Comparison of Methods for Rotating a Point in $\mathbb{R}^3$: A Case Study
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912311538614272 |
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| author | Verhoeff, Tom |
| author_facet | Verhoeff, Tom |
| contents | 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: $P'=\left(\tfrac{4}{3},\tfrac{1}{3},\tfrac{1}{3}\right)^T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_04286 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Comparison of Methods for Rotating a Point in $\mathbb{R}^3$: A Case Study Verhoeff, Tom Metric Geometry 51-08 (Primary) 51-01, 15A66 (Secondary) 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: $P'=\left(\tfrac{4}{3},\tfrac{1}{3},\tfrac{1}{3}\right)^T$. |
| title | Comparison of Methods for Rotating a Point in $\mathbb{R}^3$: A Case Study |
| topic | Metric Geometry 51-08 (Primary) 51-01, 15A66 (Secondary) |
| url | https://arxiv.org/abs/2504.04286 |