Class a Curvature

8/8/2019 Class a Curvature

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Class A CurveReporter: Cao JuanDate: 2006.12.27


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Outline:

Conclusion and future work

Introduction

References

Application

Available definition(2D,3D)


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References:

1111 A shape control fitting method for Bzier curves. CAGD,1998.Yves Mineur, Tony Lichah, Jean Marie Castelain, Henri Giaume.

2222 Class A Bzier curves. CAGD,2006.Gerald Farin

3333Curve fitting for styling application by genetic algorithm.

European Journal of Operational Research,2005. Yves Mineur. Marc

Sevaux.

4444B-spline to Class-A curves adaptation with k-neighbors LOD

processing. CAGD,2006.Giancarlo Amati, Alfredo Liverani.


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Introduction:

Class A surface is a term used in automotive

design to describe a set of freeform surfaces of high

quality. Although, strictly, it is nothing more than

saying the surfaces are aesthetically pleasing, many

people interpret class A surfaces have G2 curvature

continuity to another one.

Class AAussenhaut


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Examples(1):


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Examples(2):


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Introduction:

Class AClass A

How to make class A?

Why it is needed?

mathematical challenges?


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Sectional point from

clay modelFeature curve Direct control

Diagnosis quality Surpport surface

Completed Class A part


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Definition:

2D(typical curve)

by Yves Mineur et al

CAGD (1998)

3D

by Gerald Farin

2003

CAGD (2006)


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Definition (2D):

Bzier,1986


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Definition (2D):


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Definition (3D):

2D 3D


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Conditions of M(1):

Positivedefine:

Geometric view


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Monotonity(1):


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Conditions of M (2):

Geometric view


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Geometry view:

non-class A matrix almost class A matrix


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Subdivision:


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Monotonity(2006):

n=3


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A query:

DO all matrixes satisfy the second condition?

t=0.5


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Application(1.1):

1111 A shape control fitting method for Bzier curves. CAGD,1998.Yves Mineur, Tony Lichah, Jean Marie Castelain, Henri Giaume.

Given a set of 2D date points (P1, , Pn)

Generate a Class-A curve joining the points P1 andPn and pass close to the intermediate points

Problem

description


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Application(1.2)

Constraints (order 3,5):

x

y

Regions for a monotonic curvature variation:


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Application(1.3):

Choose an initial values of anglesStep 1

Determine a modification point

and its displacement vector

Step 2


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Application(1.4):

Assumption :

The new point is

on the starting curve

approximation


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Application(1.5):

Reasoning


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Compare with Least squares (1.1):

Original curve

3rd degree curves


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Compare with Least squares (1.2):

5rd degree curves

4rd degree curves


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Application(2.1):

3333Curve fitting for styling application by genetic algorithm.

European Journal of Operational Research,2005. Yves Mineur. Marc

Sevaux.

Fitting 2D points withG2 Class-A Bzier curves.

Problem

description


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Genetic algorithm(2.1):

GA:

1.Randomly generate an initial population M(0)

2.Compute and save the fitness u(m) for each individual m

in the current population M(t)

3.Define selection probabilities p(m) for each individual m

in M(t) so that p(m) is proportional to u(m)

4.Generate M(t+1) by probabilistically selecting individuals

from M(t) to produce offspring via genetic operators

(crossover & mutate )5.Repeat step 2 until satisfying solution is obtained.


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Underlying shape is curvature monotonic

1111

2222

3333

Assumption:

Fixed number of segments


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Genetic algorithm (2.4)


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3


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result (2.1):


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result (2.2):


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result (2.3):


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Comments:

Non-sequential or uneven point set

Variable segment number


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Application(3.1):

4444B-spline to Class-A curves adaptation with k-neighbors LOD

processing. CAGD,2006.Giancarlo Amati, Alfredo Liverani.

lots of B-spline curves dont hold the property

of Class A, i.e.,

How to transform them to Class A curves as

closely as possible to original curves?

Problem

description


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Application(3.2)

Add your text in hereMulti-resolution analysis

Approximation spaces:

Basis functions:


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Application(3.3):

Add your text in hereMulti-resolution analysis

Direct sum:

Multi-resolution:

Minimal requirement:


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Application(3.4):

Add your text in hereB-spline wavelet

k orderB-spline basis functions defined over:

define a space of piecewise polynomials


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Application(3.5):


k orderB-spline basis functions:

B-spline wavelet:


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Application(3.6):


B-spline curves can be represented as:

Control point relation:


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Class-A adaptation algorithm of

k-neighbors LOD methods:

Setp(1) control point position scanning: extract bad control

point (does not satisfy Class-A properties) at finest level

Click to add Title


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Setp(2) Curve MRA analysis: extract finer lever details

coefficients relate to the bad control point


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Setp(3) Recovery step and details manipulation: all details

that determine the k-neighbors control points position of the

bad point are thresholded.

Case k = 0: may still a non-Class-A curve, increase k

Case k >0:


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Algorithm tests(1):original

curve

result

curve


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Error estimation:


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Algorithm tests(3):

Original

Curve


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Algorithm test(2):

K=3


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Algorithm tests(3):

K=5


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Conclusion and future work:

Class-A B-spline curve (construction)1

More flexible method of construction2

Application (point set, non-sequence)3

Space Class-A (Construction and application)4


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Thank you!

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Class a Curvature