| Title:
|
Rational Bézier curves with infinitely many integral points (English) |
| Author:
|
Dospra, Petroula |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
59 |
| Issue:
|
4 |
| Year:
|
2023 |
| Pages:
|
339-349 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we consider rational Bézier curves with control points having rational coordinates and rational weights, and we give necessary and sufficient conditions for such a curve to have infinitely many points with integer coefficients. Furthermore, we give algorithms for the construction of these curves and the computation of theirs points with integer coefficients. (English) |
| Keyword:
|
Bézier curve |
| Keyword:
|
rational Bézier curve |
| Keyword:
|
curve of genus 0 |
| Keyword:
|
integral point |
| MSC:
|
14H25 |
| MSC:
|
14H45 |
| MSC:
|
14H50 |
| MSC:
|
14Q05 |
| MSC:
|
65D17 |
| idZBL:
|
Zbl 07790551 |
| idMR:
|
MR4641950 |
| DOI:
|
10.5817/AM2023-4-339 |
| . |
| Date available:
|
2023-08-15T13:32:48Z |
| Last updated:
|
2024-02-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151791 |
| . |
| Reference:
|
[1] Farine, G.: Curves and Surfaces for CAGD. A Practical Guide.fifth ed., Academic Press, 2002. |
| Reference:
|
[2] Hoschek, J., Lasser, D.: Fundamentals of Computer Aided Geometric Design.AK Peters, 1993. MR 1258308 |
| Reference:
|
[3] Mortenson, M.E.: Geometric Modelling.Industrial Press Inc., 2006. MR 0794672 |
| Reference:
|
[4] Poulakis, D.: Affine curves with infinitely many integral points.Proc. Amer. Math. Soc. 131 (2) (2002), 1357–1359. MR 1949864, 10.1090/S0002-9939-02-06841-7 |
| Reference:
|
[5] Poulakis, D., Voskos, E.: On the practical solution of genus zero Diophantine equations.J. Symbolic Comput. 30 (2000), 573–582. MR 1797269, 10.1006/jsco.2000.0420 |
| Reference:
|
[6] Poulakis, D., Voskos, E.: Solving genus zero Diophantine equations with at most two infinite valuations.J. Symbolic Comput. 33 (2002), 479–491. MR 1890582, 10.1006/jsco.2001.0515 |
| Reference:
|
[7] Ramanantoanina, A., Hormann, K.: New shape control tools for rational Bézier curve design.Comput. Aided Geom. Design 88 (2021), 11 pp., 102003. MR 4263538, 10.1016/j.cagd.2021.102003 |
| . |
