Piecewise rational rotation-minimizing motions via data stream interpolation
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910942422368256 |
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| author | Giannelli, Carlotta Sacco, Lorenzo Sestini, Alessandra Šír, Zbyněk |
| author_facet | Giannelli, Carlotta Sacco, Lorenzo Sestini, Alessandra Šír, Zbyněk |
| contents | When a moving frame defined along a space curve is required to keep an axis aligned with the tangent direction of motion, the use of rotation-minimizing frames (RMF) avoids unnecessary rotations in the normal plane. The construction of rigid body motions using a specific subset of quintic curves with rational RMFs (RRMFs) is here considered. In particular, a novel geometric characterization of such subset enables the design of a local algorithm to interpolate an assigned stream of positions, together with an initial frame orientation. To achieve this, the translational part of the motion is described by a parametric $G^1$ spline curve whose segments are quintic RRMFs, with a globally continuous piecewise rational rotation-minimizing frame. A selection of numerical experiments illustrates the performances of the proposed method on synthetic and arbitrary data streams. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14229 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Piecewise rational rotation-minimizing motions via data stream interpolation Giannelli, Carlotta Sacco, Lorenzo Sestini, Alessandra Šír, Zbyněk Numerical Analysis 65D05 65D07 65D17 (Primary) When a moving frame defined along a space curve is required to keep an axis aligned with the tangent direction of motion, the use of rotation-minimizing frames (RMF) avoids unnecessary rotations in the normal plane. The construction of rigid body motions using a specific subset of quintic curves with rational RMFs (RRMFs) is here considered. In particular, a novel geometric characterization of such subset enables the design of a local algorithm to interpolate an assigned stream of positions, together with an initial frame orientation. To achieve this, the translational part of the motion is described by a parametric $G^1$ spline curve whose segments are quintic RRMFs, with a globally continuous piecewise rational rotation-minimizing frame. A selection of numerical experiments illustrates the performances of the proposed method on synthetic and arbitrary data streams. |
| title | Piecewise rational rotation-minimizing motions via data stream interpolation |
| topic | Numerical Analysis 65D05 65D07 65D17 (Primary) |
| url | https://arxiv.org/abs/2405.14229 |