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Geometric Modeling with -Complexes
SPEAKER:
Bart H.M. Gerritsen (TNO)
Klaas van der Werff (DUT)
Remco Veltkamp (UU)
Overview
• Modeling natural objects
• Modeling with alpha complexes
• Weighting
• Case studies
• Conclusions & further research
Natural object features
Natural object
• Complex geometry and topology
• Scale dependent geometry, topology and material
• Fuzzy boundaries
• Embedded in “background”
• Holes, separations
• Heterogeneous, an-isotropic material
• Ruled by natural evolutionary processes
Engineering object
• Moderately complex geometry and low complexity topology
• Virtually scale independent geometry and topology
• Well-established boundaries
• assemblies of monolithic parts
• no separations
• homogenous, isotropic or ortho-tropic material
• Demands-driven features and functions
Modeling with alpha complexes
Preprocess data set
Design weight set
Visualize and inspect
Triangulate
Compute -family
export
Modeling: distances
triangulation nearest-neighbor
furthest neighbor
Un
eq
ual
weig
hts
Eq
ual
weig
hts
Modeling: weights
• Euclidean distance: d(x1,x2 ) = | x1 - x2 |
• Laguerre distance: L(x1,x2 ) = d2 (x1,x2 ) - ( w1+w2 )
x2
x1x1
r1
r = w
Scapula: data analysis
• Anatomic landmarks
• Point processes
• Local distances
• Position in body
• Curvatures
• Mathematical landmarks
• Pseudo-landmarks
• Geometric/topological
constraints
Scapula: finding the best alpha
• Fitting physical constraints,
e.g., volume, curvature
• Absence / presence of holes
• A-priori knowledge and
expertise
• Subjective matters
Conclusions
+ Working by example + observed ‘landmarks’
+ Intuitive
+ Can cope with roughness and vagueness of natural objects
+ hooks up well with knowledge-based and variational geometry
- Weighting can be (overly) complicated
- Limited control
- Large models require heavy computing
- Serious lack of data on natural objects
Modeling: level of abstraction
Knowledge-based modeling
Parametric modeling Variational geometry
Current geometric modeling
Lev
el o
f ab
stra
ctio
n
0% 100%Relative effort
Progress
abstract modeling geometric modeling
Alpha complex
Further research
• Knowledge-frameworks for natural objects
• Hyper-spatial modeling
• Further weighting strategies
• An-isotropic weighting
• Interactive tools for weighting
• Improved numerical modeling with alpha complexes
• Data collection on natural objects