class a curvature
TRANSCRIPT
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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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Genetic algorithm(2.2):
Underlying shape is curvature monotonic
1111
2222
3333
Assumption:
Fixed number of segments
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Genetic algorithm(2.3):
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Genetic algorithm (2.4)
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Genetic algorithm (2.5)
3
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Genetic algorithm (2.6)
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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):
Add your text in hereB-spline wavelet
k orderB-spline basis functions:
B-spline wavelet:
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Application(3.6):
Add your text in hereB-spline wavelet
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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Class-A adaptation algorithm of
k-neighbors LOD methods:
Setp(2) Curve MRA analysis: extract finer lever details
coefficients relate to the bad control point
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Class-A adaptation algorithm of
k-neighbors LOD methods:
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!