surface to surface intersection n. m. patrikalakis, t. maekawa, k. h. ko, h. mukundan may 25, 2004
Post on 19-Dec-2015
223 views
TRANSCRIPT
Surface to Surface Intersection
N. M. Patrikalakis, T. Maekawa, K. H. Ko, H. Mukundan
May 25, 2004
Slide No. 2
Introduction Motivation
Surface to surface intersection (SSI) is needed in:
Solid modeling Contouring Numerically controlled machining
Slide No. 3
Introduction Background
Intersection of two parametric surfaces defined in parametric spaces andcan have multiple components [4].
An intersection curve segment is represented by a continuous trajectory in parametric space.
),(),( vut QP 1,0 vu1,0 t
10 10 u0
1
t
0
1
v
Parametric space of ),( vuQ
Parametric space of ),( tP3D Model Space
),( tP
),( vuQ
Slide No. 4
IntroductionPossible Approaches
3 Major Methods
Lattice Methods Subdivision Based Methods
Marching Scheme (Our Choice) Intersection curve segment is represented as an IVP
Topology of the curve segments is maintained. Even small loops can be traced, given initial conditions.
roots.extraneoustopology,
roots. missing e,approximattopology,
torelatedIssues
Slide No. 5
Introduction Background
It is thus a challenge to:
Identify all components,
Obtaining a strict starting point in each component,
Trace the given intersection correctly.
Assumption:
If we are given an intersection curve segment,
No singularities in the intersection curve segment.
Based on a topological configuration
Slide No. 6
IntroductionObjective
Given an error bound on the starting point in both parametric spaces, obtain a bound for the entire intersection curve segment in 3D model space.
Strict Error Bound on Starting Point (Given) Strict Error Bound on the Entire Intersection Curve
Segment (Goal)
Slide No. 7
Outline
Problem Formulation
Error Bounds in Parametric Space
Error Bounds in 3D Model Space
Results and Examples
Conclusions
Slide No. 8
Problem FormulationTransversal Intersection
Intersection formulated as a system of ordinary differential equations (ODEs) in the parametric space [4].
,),(),(
)],(,,[
,),(),(
)],(,,[
,),(),(
)],(,,[
,),(),(
)],(,,[
vuvu
vuu
Det
ds
dv
vuvu
vuv
Det
ds
du
tt
tDet
ds
dt
tt
tt
Det
ds
d
Q
Q
PP
P
PP
P
NN
NcQ
NN
NQc
NN
NcP
NN
NPc
:obtainwe),,(),(From vut QP
sderivativepartialarevut Q,Q,P,P
andparameter length arc theis where s
Slide No. 9
Problem FormulationTangential Intersection
ODEs are very similar to transversal intersection case
From the condition of equal normal curvatures we obtain an equation
Where are functions of the first and second fundamental form coefficients of the surfaces.For a unique marching direction, and
Thus if: or if:
surfaces. theof sderivative second theusing determined is
),(),(
,),(),(
)],(,,[',
),(),(
)],(,,[',
),(),(
)],(,,[',
),(),(
)],(,,['
c
QQQNPPPN
NN
NcQ
NN
NQc
NN
NcP
NN
NPc
QP
Q
Q
PP
P
PP
P
vutoNormalvu
andttoNormalt
vuvu
vuu
Detv
vuvu
vuv
Detu
tt
tDett
tt
tt
Det
0)'()')('(2)'( 2
2212
2
11 tbtbb
221211 ,, bbb
0)( 2211
2
12 bbb 0)( 2
22
2
11
2
12 bbb
0,0 2211 bb,011 b
Slide No. 10
Problem FormulationVector IVP for ODE
Given a starting point (initial condition) corresponding to an intersection curve segment, we can integrate the system of ODEs.
The system of ODEs with the starting point represents an initial value problem (IVP).
Written in vector notation as:
Thus obtaining the intersection reduces to solving an IVP.0)(),( y0yyf
y
ds
d
),,,(
),,,(
),,,(
),,,(
4
3
2
1
vutf
vutf
vutf
vutf
ds
dvds
duds
dtds
d
Slide No. 11
Outline
Problem Formulation
Error Bounds in Parametric Space Review of Standard Schemes
Interval Arithmetic
Validated Interval Scheme
Error Bounds in 3D Model Space
Results and Examples
Conclusions
Slide No. 12
Error Bounds in Parametric SpaceReview of Standard Schemes
Famous Standard Schemes: Runge-Kutta Method Adams-Bashforth Method Taylor Series Method
Properties of Standard Schemes: They are approximation schemes and introduce a truncation
error They do not consider uncertainty in initial conditions They are prone to rounding errors They suffer from straying or looping near closely spaced
features
Slide No. 13
Error Bounds in Parametric SpaceInterval Arithmetic (Introduction)
Intervals are defined by [2]:
Example:
Basic interval arithmetic operations defined by:
][3.142 , 3.141
4635897932383.14159265
Slide No. 14
Error Bounds in Parametric Space Interval Arithmetic (Solution of IVPs)
For strict bounds for IVPs in parametric space, we employ validated interval scheme for ODEs [3].
The error in starting point is bounded by an initial interval.
Interval solution represents a family of solutions passing through the initial interval satisfying the governing ODEs.
Slide No. 15
Error Bounds in Parametric Space Validated Interval Scheme (Introduction)
Every step of a validated interval scheme involves [3]:
Computing an interval valued function such that:
and The width of the is below a given tolerance
And the scheme verifies the existence and uniqueness of the solution in
the scheme notifies if the IVP has no solution or if it has more than one solution in .
,)]([)( ss yy ],[ 1
jj sss
],[ 1jj ss
)]([ sy
)]([ sy
],[ 1jj ss
Slide No. 16
Error Bounds in Parametric Space Validated Interval Scheme (Overview)
One step of a validated interval scheme done in two phases:
Phase I Algorithm
A step size
An a priori enclosure such that:
Phase II Algorithm
Using compute a tighter bound
at .
]~[ jy
]~[ jy ][ 1jy
],[],~[)( 1 jjj sssysy
jjj ssh 1
Phase 1:
1js
Slide No. 17
Error Bounds in Parametric Space Validated Interval Scheme (Phase I : Validation)
A pair of and satisfying the relation:
assures existence and uniqueness of the solution.
This method is called a constant enclosure method [3].
The a priori enclosure bounds the true solution in the parametric space .
Numerical implementation Choosing a and, iterating to find a corresponding .
jjjj h])~([][]~[ yfyy
jh]~[ jy
jh]~[ jy
]~[ jy],[ 1 jj sss
Slide No. 18
Error Bounds in Parametric Space Validated Interval Scheme (Phase II : Tighter Bound)
Using the a priori enclosure wefind a tighter bound at [3].
This phase helps in the propagation of the solution by providing a small initial interval for the successive step.
The key idea is to use:Interval version of Taylor’s formula [3].
1js
tcoefficienTayloritherepresentswhere
hh
thj
i
jkk
jji
k
i
ijjj
])([
])~([])([][][
][
][][1
11
yf
yfyfyy
][ 1jy
Slide No. 19
Error Bounds in Parametric Space Validated Interval Scheme (Application to SSI)
We represent the surfaces as interval surfaces. Interval surfaces have interval coefficients and are written
as:
We obtain a vector interval ODE system :
With an interval initial condition :
)])(([ sds
d
ds
dv
ds
du
ds
dt
ds
dT
yfy
),(and),( vut QP
Tvut ][][][][][ 00000 y
Slide No. 20
Error Bounds in Parametric Space
Validated ODE solver produces a priori enclosures in parametric space of each surface guaranteed to contain the true intersection curve segment.
The union of a priori enclosures bounds the true intersection curve segment in parametric space.
Slide No. 21
Outline
Problem Formulation
Error Bounds in Parametric Space
Error Bounds in 3D Model Space
Results and Examples
Conclusions
Slide No. 22
Error Bounds in 3D Model Space Mapping into 3D Model Space
Mapping from parametric space to 3D model space using corresponding surfaces coupled with interval arithmetic evaluation
Ensures continuous error bounds in 3D model space [1] guaranteed to contain the true curve of intersection.
),(o),( vurt QP
Slide No. 23
Outline
Problem Formulation
Error Bounds in Parametric Space
Error Bounds in 3D Model Space
Results and Examples
Conclusions
Slide No. 24
Results & Examples Error Bounds in 3D Model Space (Transversal)
The method was implemented using C++ and extensive tests were performed.
Torus and cylinder Two bi-cubic surfaces
0.020.001
Self intersection of a bi-cubic surface
Slide No. 25
Tangential intersection of two parametric surfaces
Results & Examples Error Bounds in 3D Model Space (Tangential)
Slide No. 26
Validated ODE solver can correctly trace the intersection curve segment even through closely spaced features, where standard methods fail.
Results & Examples Preventing Straying or Looping
Adams-Bashforth Runge-Kutta
Result from a validated interval
scheme
Perturbation Steps Required bythe Method
+0.000003 1139
0.0 Singularity Reported
-0.000003 1303
t
t
t
Slide No. 27
Outline
Problem Formulation
Error Bounds in Parametric Space
Error Bounds in 3D Model Space
Results and Examples
Conclusions
Slide No. 28
ConclusionsMerits
We realize validated error bounds in 3D model space which enclose the true curve of intersection.
The scheme can prevent the phenomenon of straying or looping.
Scheme can accommodate the errors in: initial condition perturbation in the surface itself rounding during digital computation
Validated error bounds for surface intersection is essential in interval boundary representation for consistent solid models [5].
Slide No. 29
ConclusionsLimitations and Future Work
Limitations We assume that we have a
Identifying each of the segment strict error bound on the starting point
Increasing width of the interval solutions due to Rounding Phenomenon of wrapping
Scope for future work Accurate evaluation of starting points Cases of tangential intersections and surface overlaps
Slide No. 30
References 1. Tracing surface intersections with a validated ODE system solver,
Mukundan, H., Ko, K. H., Maekawa, T. Sakkalis, T., and Patrikalakis, N. M., Proceedings of the Ninth EG/ACM Symposium on Solid Modeling and Applications, G. Elber and G. Taubin, editors. Genova, Italy, June 2004. Eurographics Press.
2. R. E. Moore. Interval Analysis. Prentice-Hall, Englewood Cliffs, 1966.
3. N. S. Nedialkov. Computing the Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation. PhD thesis, University of Toronto, Toronto, Canada, 1999.
4. N. M. Patrikalakis and T. Maekawa. Shape Interrogation for Computer Aided Design and Manufacturing. Springer-Verlag, Heidelberg, 2002.
5. T. Sakkalis, G. Shen and N.M. Patrikalakis, Topological and Geometric Properties of Interval Solid Models, Graphical Models, 2001.