PreprintarXiv licencearXiv
New Explicit Examples of Mannheim Pairs in ℝ3
Abstract
In this paper, we construct new explicit parametrizations of Mannheim pairs in Euclidean 3-space. Unlike the known examples in the literature, which are essentially slant helices or general helices, the curve presented here is neither a helix nor a slant helix. We provide the full Frenet apparatus, the explicit Cartesian coordinates expressed in terms of elementary functions, and the corresponding Mannheim partner curve. The construction is based on a specific choice of the angular function that governs the curve's tangent indicatrix, leading to integrable expressions.
Related
- New explicit examples of Mannheim pair in 𝕃3
- Examples of descendent generating series for Pandharipande--Thomas stable pairs on smooth projective Fano threefolds via one-dimensional wall-crossing
- Explicit equations of three pairs of fake projective planes
- Explicit construction of optimal binary linear complementary pairs via Belov codes
- Those Confounded Examples
BibTeX
@misc{alan2026explicit,
title = {{New Explicit Examples of Mannheim Pairs in $\mathbbR^3$}},
author = {José Alan and Renato Teixeira and Thiago Mendonça},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38111},
eprint = {2610.38111},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38111},
}
RIS
TY - UNPB
TI - New Explicit Examples of Mannheim Pairs in ℝ^3
AU - José Alan
AU - Renato Teixeira
AU - Thiago Mendonça
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38111
UR - https://arxiv.org/abs/2610.38111
AB - In this paper, we construct new explicit parametrizations of Mannheim pairs in Euclidean 3-space. Unlike the known examples in the literature, which are essentially slant helices or general helices, the curve presented here is neither a helix nor a slant helix. We provide the full Frenet apparatus, the explicit Cartesian coordinates expressed in terms of elementary functions, and the corresponding Mannheim partner curve. The construction is based on a specific choice of the angular function that governs the curve's tangent indicatrix, leading to integrable expressions.
ER -