Spiral Exponential Function— Chapter 8: Derivation of exp(Bθ) Using Clifford Algebra
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| Format: | Recurso digital |
| Sprache: | Englisch |
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2025
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| _version_ | 1866901175725457408 |
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| author | Abe, Kohei |
| author_facet | Abe, Kohei |
| contents | <p><span lang="EN-US">This chapter demonstrates how the conserved term </span><span lang="EN-US"> in the spiral exponential function can be naturally derived from the geometric structure of Clifford algebra.</span></p> <p><span lang="EN-US">It begins with examples of divergent and rotational vector fields on the xy-plane and introduces a method for continuously unifying them through the mediating angle θ.</span></p> <p><span lang="EN-US">The structure is then extended to three-dimensional space, describing rotations across the xy, yz, and zx planes using a rotational generator </span></p> <p> </p> <p><span lang="EN-US">By normalizing this generator as </span><span lang="EN-US">, the Clifford exponential function exp(</span><span lang="EN-US">θ) = cosθ + </span><span lang="EN-US">·sinθ is derived.</span></p> <p><span lang="EN-US">For this expansion to hold, </span><span lang="EN-US"> must always span a single plane (a pure blade), which is guaranteed in this construction since the norm of </span><span lang="EN-US"> is always unity.</span></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_16319505 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Spiral Exponential Function— Chapter 8: Derivation of exp(Bθ) Using Clifford Algebra Abe, Kohei Spiral exponential function Clifford algebra rotation bivector pure blade complex exponential expansion wedge product integration of divergence and curl spatial generalization geometric algebra exp(Bθ) <p><span lang="EN-US">This chapter demonstrates how the conserved term </span><span lang="EN-US"> in the spiral exponential function can be naturally derived from the geometric structure of Clifford algebra.</span></p> <p><span lang="EN-US">It begins with examples of divergent and rotational vector fields on the xy-plane and introduces a method for continuously unifying them through the mediating angle θ.</span></p> <p><span lang="EN-US">The structure is then extended to three-dimensional space, describing rotations across the xy, yz, and zx planes using a rotational generator </span></p> <p> </p> <p><span lang="EN-US">By normalizing this generator as </span><span lang="EN-US">, the Clifford exponential function exp(</span><span lang="EN-US">θ) = cosθ + </span><span lang="EN-US">·sinθ is derived.</span></p> <p><span lang="EN-US">For this expansion to hold, </span><span lang="EN-US"> must always span a single plane (a pure blade), which is guaranteed in this construction since the norm of </span><span lang="EN-US"> is always unity.</span></p> |
| title | Spiral Exponential Function— Chapter 8: Derivation of exp(Bθ) Using Clifford Algebra |
| topic | Spiral exponential function Clifford algebra rotation bivector pure blade complex exponential expansion wedge product integration of divergence and curl spatial generalization geometric algebra exp(Bθ) |
| url | https://doi.org/10.5281/zenodo.16319505 |