2010 asian conference on design & digital engineering fairing spline curves: a thorough and...

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


Outline

Background

Fairness criterion

Algorithm

Results

Conclusion


Shape Design

• Modeling tools– B-spline, NURBS, …– Subdivision surfaces– Implicit surfaces

• Focus on continuity: Ck, Gk


Fair Design

• Practical criteria– Less small bumps– A curve is even not fair practically

• Focus on fairness and eye pleasingC


Geometric Design

Shape Design- Well studied

Fair Design- Far from solved


◇ Shoe sole ◇ Cam Profile ◇ Car Profile ◇ Ship hull ◇ Plane profile ◇ …

Fair design is important !


Fair lofting in curves design

Plan Drawing Shape design

Examine by eyeAdjusti

ng weight

s


(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


Fairing in mathematics

(1) Input: Output:

(2) Interpolating using spline.

(3) Make curvature uniform.

( 1, ... , )iP i n

x x

k k

( 1, ... , )iP i n


Curvature plot

Magnifier of fairness.

k


Previous Criterion

Global Criterion

- C 2 continuous - Minimizes integral of the squared curvature

2 0.1589C

k ds

2 0.1330C

k ds

Not well-recognized:

(1) Local bumps.

(2) Loftman fairs curves locally.


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


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


Observation 1

C◊ .

Unfair reason: Too many vibrations(inflections).


Observation 2

◊ .

◊ Winds along the red parabola curve.

C

Unfair reason: Curvature plot has too many vibrations.

◊ No inflections.


Observation 3

◊ Discontinuous.◊ Not fair if k1 and k2 are much different.

1k

2k

1O

2O

Unfair reason: Curvature plot has large amplitude.


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.


Novel Criterion

Continuous Inflection Vibrationnumber

Vibrationamplitude

(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 !

√√√

√ √ √√

√××√

√×× ×√


Algorithm

Goal:

- Eliminate unnecessary zeros. (criteria (ii) )

- Eliminate unnecessary vibrations. (criteria (iii) )

- Reduce vibration amplitudes. (criteria (iv) )


K is Non-linear

2 3/2

''( )

(1 )

yk x

y

3

2

'( ) ''( ) '( ) ''( )( )

[ '( ) '( )]

x t y t y t x tk t

x t y t

Difficult to define zeros, vibrations and amplitudes

Difficult to control curvature


Key ideas

Cubic spline function

Approximately linear curvature

Easily define zeros, vibrations and amplitudesManipulate curvature directly


Cubic spline function

P1

P2

Pi

Cubic

C 2


Polyline approximation

1P

x

y

2P

iP

1iP1

i

| | 60i

( ) ''( )k x y x

(polyline)

Red: y’’ Black: kCurve


Polyline approximation


Fairness Indicator

x

c ( , )i ix c

1 1( , )i ix c

1 1( , )i ix c

2 2( , )i ix c

1 11

1 1

tan( ) tan( )i i i ii i i

i i i i

c c c ce

x x x x

i 1i

2i

1 2 1tan( ) tan( )i i ie

Vibration: If , it has a vibration.

Vibration amplitude: If , denote as the amplitude.

Inflection: If , the curve has an inflection in .

1 0i ie e

1 0i ie e 1( ) | |i iAm i e e

1 0i ic c 1[ , ]i ix x


Curvature plot adjusting

• Adjust curvature

• Adjust curvature variation

''( ) / 6c y x

1 11

1 1

tan( ) tan( )i i i ii i i

i i i i

c c c ce

x x x x

x

c ( , )i ix c

1 1( , )i ix c

1 1( , )i ix c

2 2( , )i ix c

i 1i

2i


Manipulate ci/ei by adjusting data point

( , )i ix c

i

1 1( , )i ix c

xx

cy


3-steps fairing

Step 1: Initial fairing - reducing vibration amplitudes (Criterion (iv))

1 10 0i i i ie e e e

Curve Curvature plot Curve after fairing


Step 2: Basic fairing - reducing unnecessary inflections (Criterion (ii))

(1) Small wave

3-steps fairing

CiCi+1<0,CiCi-1<0 CiCi+1>0,CiCi-1>0

Curve Curvature plot Curve after fairing


3-steps fairing

(2) Medium wave

Curve Curvature plot


3-steps fairing

Step 3: Fine fairing - reducing unnecessary vibrations(Criterion (iii))


3-steps fairing

Original All 3 steps

Step 1 Step 1+ Step 2


1P

x

y

2P

iP

1iP1

i

Segmentation

| | 60i


Results - Section line of ship hull

Original Our 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)


Turbine

Original Our methodE=3.9e+5 E=1.7e+5

E=1.4e+5 E=2.4e+5

MST FS


Results

Car

Original Ours MST FS


Results

Mouse Mat Section



Results (curvature bar)



Conclusion

• A thorough and precise criteria for curve fairness

• A practical algorithm for curve fairness• From practical lofting experience


Thanks!

2010 asian conference on design & digital engineering fairing spline curves: a thorough and...

Documents

asian conference

difficult slide

solved slide

curvature plot continuous

curvature uniform

acddecurve fairing slide

eye pleasing slide

squared curvature