##
**Unifying C-curves and H-curves by extending the calculation to complex numbers.**
*(English)*
Zbl 1087.65516

Summary: Recently, we found that the CB-splines that use basis \(\{\sin t, \cos t, t, 1\}\) and the HB-splines that use basis \(\{\sinh t,\cosh t, t,1\}\) could be unified into a complete curve family, named FB-splines [J. Zhang and F.-L. Krause, Graph Models 67, 104–119 (2005; Zbl 1078.41014)]. FB-splines are a scheme of what we call here F-curves. This paper explains that in the domain of complex numbers, the extended C-curves and extended H-curves are the same curves. Therefore, F-curves can be constructed in two identical styles, C and H. The C style is an extension of C-curves that uses sin and cos, and the H style is an extension of H-curves that uses sinh and cosh.

Here the representations of F-curves are clearer and simpler. For real applications, the definitions, equations and main properties for the F-curves in different schemes (FB-splines, F-Bézier and F-Ferguson schemes) are introduced in details. F-curves are shape adjustable, and their curvatures on terminals can be any expected value between 0 and \(\infty\). They can represent the circular (or elliptical) arc, the cylinder, the helix, the cycloid, the hyperbola, the catenary, etc. precisely. Therefore, F-curves are more useful than C-curves or H-curves for the surface modeling in engineering.

Here the representations of F-curves are clearer and simpler. For real applications, the definitions, equations and main properties for the F-curves in different schemes (FB-splines, F-Bézier and F-Ferguson schemes) are introduced in details. F-curves are shape adjustable, and their curvatures on terminals can be any expected value between 0 and \(\infty\). They can represent the circular (or elliptical) arc, the cylinder, the helix, the cycloid, the hyperbola, the catenary, etc. precisely. Therefore, F-curves are more useful than C-curves or H-curves for the surface modeling in engineering.

### MSC:

65D17 | Computer-aided design (modeling of curves and surfaces) |

### Citations:

Zbl 1078.41014
PDF
BibTeX
XML
Cite

\textit{J. Zhang} et al., Comput. Aided Geom. Des. 22, No. 9, 865--883 (2005; Zbl 1087.65516)

Full Text:
DOI

### References:

[1] | Chen, Q.; Wang, G., A class of Bézier-like curves, Computer aided geometric design, 20, 29-39, (2003) · Zbl 1069.65514 |

[2] | Farin, G., Curves and surfaces for computer aided geometric design, (1997), Academic Press San Diego, CA · Zbl 0919.68120 |

[3] | Lu, Y.; Wang, G.; Yang, X., Uniform hyperbolic polynomial B-spline curves, Computer aided geometric design, 19, 379-393, (2002) |

[4] | Mainar, E.; Peña, J.M.; Sánchez-Reyes, J., Shape preserving alternatives to the rational Bézier model, Computer aided geometric design, 18, 37-60, (2001) · Zbl 0972.68157 |

[5] | Mainar, E.; Peña, J.M., A basis of C-Bézier splines with optimal properties, Computer aided geometric design, 19, 291-295, (2002) · Zbl 0995.68135 |

[6] | Morin, G.; Warren, J.; Weimer, H., A subdivision scheme for surfaces of revolution, Computer aided geometric design, 18, 483-502, (2001) · Zbl 0970.68177 |

[7] | Pottmann, H., The geometry of Tchebycheffian spines, Computer aided geometric design, 10, 181-210, (1993) · Zbl 0777.41016 |

[8] | Pottmann, H.; Wagner, M.G., Helix splines as example of affine Tchebycheffian splines, Adv. comput. math., 2, 123-142, (1994) · Zbl 0832.65008 |

[9] | Zhang, J.W., C-curves, an extension of cubic curves, Computer aided geometric design, 13, 199-217, (1996) · Zbl 0900.68405 |

[10] | Zhang, J.W., Two different forms of CB-splines, Computer aided geometric design, 14, 31-41, (1997) · Zbl 0900.68418 |

[11] | Zhang, J.W., C-Bézier curves and surfaces, Graph. models image process., 61, 2-15, (1999) · Zbl 0979.68578 |

[12] | Zhang, J.W.; Krause, F.-L., Extend cubic uniform B-splines by unified trigonometric and hyberolic basis, Graph. models, 67, 2, 100-119, (2005) · Zbl 1078.41014 |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.