Fourier approximation of symmetric ideal knots. (English) Zbl 1273.49048
Summary: Enforcing a specific symmetry group on a curve, knotted or not, is not trivial using standard interpolations such as polygons or splines. For a prescribed symmetry group we present a symmetrization process based on a Fourier description of a knot. The presence of symmetry groups implies a characteristic pattern in the Fourier coefficients. The relations between the coefficients are shown for five ideal knot shapes with their proposed symmetry groups.
MSC:
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 53A04 | Curves in Euclidean and related spaces |
| 42B05 | Fourier series and coefficients in several variables |
| 58D19 | Group actions and symmetry properties |
References:
| [1] | DOI: 10.1080/10586458.2011.544581 · Zbl 1261.49007 · doi:10.1080/10586458.2011.544581 |
| [2] | DOI: 10.1142/9789812703460_0016 · doi:10.1142/9789812703460_0016 |
| [3] | DOI: 10.1140/epjb/e2008-00443-y · Zbl 1188.82155 · doi:10.1140/epjb/e2008-00443-y |
| [4] | DOI: 10.1142/9789812703460 · doi:10.1142/9789812703460 |
| [5] | DOI: 10.1007/s00222-002-0234-y · Zbl 1036.57001 · doi:10.1007/s00222-002-0234-y |
| [6] | DOI: 10.1142/9789812703460_0005 · doi:10.1142/9789812703460_0005 |
| [7] | DOI: 10.1007/s00205-010-0390-y · Zbl 1268.49050 · doi:10.1007/s00205-010-0390-y |
| [8] | DOI: 10.1142/S0218216503002354 · Zbl 1028.57008 · doi:10.1142/S0218216503002354 |
| [9] | DOI: 10.1073/pnas.96.9.4769 · Zbl 1057.57500 · doi:10.1073/pnas.96.9.4769 |
| [10] | DOI: 10.1007/s005260100089 · Zbl 1006.49001 · doi:10.1007/s005260100089 |
| [11] | DOI: 10.1142/9789812796073_0019 · doi:10.1142/9789812796073_0019 |
| [12] | DOI: 10.1142/9789812796073_0003 · doi:10.1142/9789812796073_0003 |
| [13] | Oppenheim A. V., Signals and Systems (1997) |
| [14] | DOI: 10.1142/9789812796073_0002 · doi:10.1142/9789812796073_0002 |
| [15] | DOI: 10.1142/9789812796073 · doi:10.1142/9789812796073 |
| [16] | DOI: 10.1142/9789812796073_0018 · doi:10.1142/9789812796073_0018 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
