Unifying C-curves and H-curves by extending the calculation to complex numbers |
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Authors: | Jiwen Zhang Frank-L Krause Huaiyu Zhang |
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Affiliation: | aState Key Lab. of Cad&CG, Mechanical Engineering Department, Zhejiang University, PR China bIWF Institute, Mechanical Engineering Department, Technical University Berlin, Germany cElectrical Engineering Department, Technical University Berlin, Germany |
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Abstract: | Recently, we found that the CB-splines that use basis {sint,cost,t,1} and the HB-splines that use basis {sinht,cosht,t,1} could be unified into a complete curve family, named FB-splines (Zhang and Krause, 2005). 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 ∞. 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. |
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Keywords: | F-curves FB-splines C-curves CB-splines C-Bézier curves B-splines Bézier curves Surface modeling CAD/CAM |
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