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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.19376 |
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| _version_ | 1866914169409765376 |
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| author | Nurmatov, A. Skopenkov, M. Rist, F. Klein, J. Michels, D. L. |
| author_facet | Nurmatov, A. Skopenkov, M. Rist, F. Klein, J. Michels, D. L. |
| contents | This work investigates flexible Kokotsakis polyhedra with a quadrangular base of equimodular elliptic type, filling a significant gap in the literature by providing the first explicit constructions of this type together with an explicit algebraic characterization in terms of flat and dihedral angles. A straightforwardly constructible class of polyhedra - called quasi-symmetric nets (QS-nets) - is introduced, characterized by a symmetry relation among flat angles. It is shown that every elliptic QS-net has equimodular elliptic type and is flexible in real three-dimensional Euclidean space (rather than only in complex configuration spaces), except for a few exceptional choices of dihedral angles, and that its flexion admits a closed-form parameterization. Examples are constructed that are non-self-intersecting and belong exclusively to the equimodular elliptic type. To support applications in computational geometry, a numerical pipeline is developed that searches for candidate solutions, verifies them using the explicit algebraic characterization, and constructs and visualizes the resulting polyhedra; numerical validations achieve high precision. Taken together, these results provide constructive criteria, algorithms, and validated examples for the equimodular elliptic type, enabling the design of a broad range of flexible Kokotsakis mechanisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19376 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasi-symmetric nets: a constructive approach to the equimodular elliptic type of Kokotsakis polyhedra Nurmatov, A. Skopenkov, M. Rist, F. Klein, J. Michels, D. L. Metric Geometry Computational Geometry 52C25 This work investigates flexible Kokotsakis polyhedra with a quadrangular base of equimodular elliptic type, filling a significant gap in the literature by providing the first explicit constructions of this type together with an explicit algebraic characterization in terms of flat and dihedral angles. A straightforwardly constructible class of polyhedra - called quasi-symmetric nets (QS-nets) - is introduced, characterized by a symmetry relation among flat angles. It is shown that every elliptic QS-net has equimodular elliptic type and is flexible in real three-dimensional Euclidean space (rather than only in complex configuration spaces), except for a few exceptional choices of dihedral angles, and that its flexion admits a closed-form parameterization. Examples are constructed that are non-self-intersecting and belong exclusively to the equimodular elliptic type. To support applications in computational geometry, a numerical pipeline is developed that searches for candidate solutions, verifies them using the explicit algebraic characterization, and constructs and visualizes the resulting polyhedra; numerical validations achieve high precision. Taken together, these results provide constructive criteria, algorithms, and validated examples for the equimodular elliptic type, enabling the design of a broad range of flexible Kokotsakis mechanisms. |
| title | Quasi-symmetric nets: a constructive approach to the equimodular elliptic type of Kokotsakis polyhedra |
| topic | Metric Geometry Computational Geometry 52C25 |
| url | https://arxiv.org/abs/2511.19376 |