2010 Asian Conference on Design & Digital Engineering Fairing spline curves: a thorough and precise criteria and practical algorithm Xiaoguang Han, Ligang Liu, Guangchang Dong Zhejiang University 2010-ACDDE-Curve Fairing

2010 Asian Conference on Design & Digital Engineering Outline Background Fairness criterion Algorithm Results Conclusion

2010 Asian Conference on Design & Digital Engineering Shape Design Modeling tools – B-spline, NURBS, … – Subdivision surfaces – Implicit surfaces Focus on continuity: C k, G k

2010 Asian Conference on Design & Digital Engineering Fair Design Practical criteria – Less small bumps – A curve is even not fair practically Focus on fairness and eye pleasing

2010 Asian Conference on Design & Digital Engineering Geometric Design Shape Design - Well studied Fair Design - Far from solved

2010 Asian Conference on Design & Digital Engineering ◇ Shoe sole ◇ Cam Profile ◇ Car Profile ◇ Ship hull ◇ Plane profile ◇ … Fair design is important !

2010 Asian Conference on Design & Digital Engineering Fair lofting in curves design Plan Drawing Shape design Examine by eye Adjusting weights

2010 Asian Conference on Design & Digital Engineering (1)No mathematical definition. (2)Experience based. (3)Time consuming. We need a precise criterion of fairness and an automatic algorithm ! Fair lofting is difficult

2010 Asian Conference on Design & Digital Engineering Fairing in mathematics (1) Input: Output: (2) Interpolating using spline. (3) Make curvature uniform.

2010 Asian Conference on Design & Digital Engineering Curvature plot Magnifier of fairness.

2010 Asian Conference on Design & Digital Engineering Previous Criterion Global Criterion - C 2 continuous - Minimizes integral of the squared curvature Not well-recognized: (1) Local bumps. (2) Loftman fairs curves locally.

2010 Asian Conference on Design & Digital Engineering Previous Criterion Local Criterion (Su and Liu, 1989) : The curve - C 2 continuous - uniform curvature variation - few unnecessary inflection points (Sapidis and Farin, 1990): Curvature plot - curvature plot is continuous, - appropriate sign - few monotone pieces

2010 Asian Conference on Design & Digital Engineering Previous Criterion (Pigounakis et al., 1996): Curvature plot - be free of unnecessary variation. - distribute as uniform as possible. (Farin, 2002): Curvature plot - continuous - few monotone pieces

2010 Asian Conference on Design & Digital Engineering Observation 1 ◊. Unfair reason: Too many vibrations(inflections).

2010 Asian Conference on Design & Digital Engineering Observation 2 ◊. ◊ Winds along the red parabola curve. Unfair reason: Curvature plot has too many vibrations. ◊ No inflections.

2010 Asian Conference on Design & Digital Engineering Observation 3 ◊ Discontinuous. ◊ Not fair if k 1 and k 2 are much different. Unfair reason: Curvature plot has large amplitude.

2010 Asian Conference on Design & Digital Engineering Novel Criterion A curve is fair if Curve is: (i) C l+1 ; - C 1 but C 2 almost everywhere, - Second derivative has bounded variation. Curvature plot has: (ii) Few zeros(inflections); (iii) Few vibration numbers; (iv) Small vibration amplitudes.

2010 Asian Conference on Design & Digital Engineering Novel Criterion ContinuousInflectionVibration number Vibration amplitude (Su and Liu, 1989) C 2 (Sapidis and Farin, 1990) C 2 (Pignouak et.al, 1996) (Farin, 2002) C 2 Our C l+1 Our new criteria is thorough and precise ! √√√ √√ √ √ √ × × √ √ × × × √

2010 Asian Conference on Design & Digital Engineering Algorithm Goal: - Eliminate unnecessary zeros. (criteria (ii) ) - Eliminate unnecessary vibrations. (criteria (iii) ) - Reduce vibration amplitudes. (criteria (iv) )

2010 Asian Conference on Design & Digital Engineering K is Non-linear Difficult to define zeros, vibrations and amplitudes Difficult to control curvature

2010 Asian Conference on Design & Digital Engineering Key ideas Cubic spline function Approximately linear curvature Easily define zeros, vibrations and amplitudes Manipulate curvature directly

2010 Asian Conference on Design & Digital Engineering Cubic spline function P1P1 P2P2 PiPi Cubic C 2

2010 Asian Conference on Design & Digital Engineering Polyline approximation (polyline) Red: y’’ Black: kCurve

2010 Asian Conference on Design & Digital Engineering Polyline approximation

2010 Asian Conference on Design & Digital Engineering Fairness Indicator Vibration: If, it has a vibration. Vibration amplitude: If, denote as the amplitude. Inflection: If, the curve has an inflection in.

2010 Asian Conference on Design & Digital Engineering Curvature plot adjusting Adjust curvature Adjust curvature variation

2010 Asian Conference on Design & Digital Engineering Manipulate c i /e i by adjusting data point

2010 Asian Conference on Design & Digital Engineering 3-steps fairing Step 1: Initial fairing - reducing vibration amplitudes (Criterion (iv)) CurveCurvature plotCurve after fairing

2010 Asian Conference on Design & Digital Engineering Step 2: Basic fairing - reducing unnecessary inflections (Criterion (ii)) (1) Small wave 3-steps fairing C i C i+1 <0,C i C i-1 <0 C i C i+1 >0,C i C i-1 >0 CurveCurvature plotCurve after fairing

2010 Asian Conference on Design & Digital Engineering 3-steps fairing (2) Medium wave CurveCurvature plot

2010 Asian Conference on Design & Digital Engineering 3-steps fairing Step 3: Fine fairing - reducing unnecessary vibrations(Criterion (iii))

2010 Asian Conference on Design & Digital Engineering 3-steps fairing Original All 3 steps Step 1Step 1+ Step 2

2010 Asian Conference on Design & Digital Engineering Segmentation

2010 Asian Conference on Design & Digital Engineering Results - Section line of ship hull OriginalOur method MST(Matlab Spline Toolbox)FS(Farin and Sapidis) (3.8e-6, 4, 19) (6.7e-8, 2, 5) (8.6e-8, 2, 7)(8.2e-7, 2, 13)

2010 Asian Conference on Design & Digital Engineering Turbine OriginalOur method E=3.9e+5 E=1.7e+5 E=1.4e+5 E=2.4e+5 MST FS

2010 Asian Conference on Design & Digital Engineering Results Car Original Ours MST FS

2010 Asian Conference on Design & Digital Engineering Results Mouse Mat Section Original Ours MST FS

2010 Asian Conference on Design & Digital Engineering Results (curvature bar) Original Ours MST FS

2010 Asian Conference on Design & Digital Engineering Conclusion A thorough and precise criteria for curve fairness A practical algorithm for curve fairness From practical lofting experience

2010 Asian Conference on Design & Digital Engineering