On Fairingalgorithms And Theirapplication With The Second Discrete Curvature

Fairing of curves and surfaces is one of the most important studies of the computeraided geometric design. It has profound theoretical and practical significance. A lot ofresearches have been done and some effective fairing algorithms have been obtained.Currently, the fairing algorithms of curves can mainly be divided into three sections:curvature-based fairing algorithms, energy-based fairing algorithms and curvature andenergy joined fairing algorithms. Curvature-based fairing algorithms take continuitiesand monotonousness of the curvature as the criteria of smoothing curve. Thesealgorithms can be applied to some simple curves, but when they are applied to moregeneral curves, the expressions of the curvature will be very complicated and can’t beeasily applied. Energy-based fairing algorithms take the physical deformation energy asthe criteria of smoothing curve. These algorithms have good overall smoothing effects,but they not only have abundant calculation and low speed but also modify some pointswhich shouldn’t been changed. Curvature and energy joined fairing algorithms canavoid changing points which shouldn’t be modified, but these algorithms requireadjusting repeatedly to more complicated curve, they not only require abundantcalculations but also be hard to ensure the overall smoothing effects.Therefore, based on the study of the cutting corner methods for planar polygonsand discrete curvatures, a new discrete curvature-the second discrete curvature isintroduced. It is obtained that the curvature of the curve at its joining point isproportional to the second discrete curvature of the corresponding control point at thenodes, and the second discrete curvature and discrete curvature act in a similar way. Thefairing algorithms for spline curves based on the second discrete curvature are given.With these algorithms the curves are faired through adjusting the second discretecurvature of the corresponding control points directly, thus the fairing process is moreconcise and has stronger geometrical intuition. Finally, a discrete fairing algorithm isput forward, and the curve fairing algorithm is extended to the deformation of curves byintroducing a single temporal dimension-‘t’. Numerical examples show that the newalgorithms have better smoothing and morphing effects.This paper is organized as follows:In section1, the important significance and the development course of the fairness of curves and surfaces are introduced, and the advantages and disadvantages of variousmethods of curves and surfaces fairing at home and abroad are analyzed andsummarized in detail, and finally the main contents of the paper are introduced.In section2, on the basis of the study of discrete curvature, the basic concepts ofthe second discrete curvature are presented. Through the comparison with discretecurvature, the properties of second discrete curvature are given. It is concluded that thetwo curvatures both act in a similar way; through the comparison with the curvature, itis obtained that the curvature is proportional to the second discrete curvature at thenodes. Finally, based on the above discussions, the fairing principles and fairingalgorithms based on the second discrete curvature are given.In section3, section4and section5, the properties and the theorems of the discretecurvature for cubic B-spline curve, C-B spline curve and other spline curve are putforward, the properties and the theorems based on the second discrete curvature forcubic B-spline curve, C-B spline curve and other spline curve are obtained, and thecorresponding fairing algorithms based on the second discrete curvature are given indetail.In section6, a discrete fairing algorithm based on the second discrete curvature isput forward with the concept of the second discrete curvature and curvature center and asingle temporal dimension-‘t’ is introduced, so the discrete fairing algorithm isgeneralized to the deformation of curves, thus the application of the proposed algorithmis extended.In section7, the summary of the paper is given and the future research work is putforward.