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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.

MSC:

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

Citations:

Zbl 1078.41014
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References:

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