3D Modeling Topics Gerald Farin Computer Science PRISM: Partnership for Research In Spatial Modeling ASU
Dec 18, 2015
3D Modeling Topics
Gerald FarinComputer Science
PRISM: Partnership for Research In Spatial ModelingASU
PRISM’sexpertises
are well-suited tointerdisciplinary
research
Data Acquisition
GeometricModeling
Analysis
Visualization
Data Archiving
RapidPrototyping
Overview
• History• Curvature• Surface analysis• Ruled surfaces• Coons surfaces• Continuity• NURBS• Subdivision surfaces
Lofting
before fairing
after fairing
History
• 1962 C. de Boor: bicubic splines• 1972 J. Ferguson, Boeing: cubic splines• 1974 W. Gordon, R. Riesenfeld: B-splines• 1964 S. Coons: Coons patches, conics• 1975 K. Vesprille: NURBS
History
• 1958 P. de Casteljau, Citroen: de Casteljau algorithm
• 1965 P. Bezier, Renault: Bezier curves
Overview
• History• Curvature• Surface analysis• Ruled surfaces• Coons surfaces• Continuity• NURBS• Subdivision surfaces
Osculating circle
Frenet frame
Interactive demo
Helix curve fitcubic quintic
Helix curvature plotcubic quintic
Curvature plot, cubic B-spline
B-spline curve operations
redundancy creation processes:
• Knot insertion• Degree raising
Knot insertion
quintic
cubic
B-spline curve fairing
“Invert” redundancy creation processes:
• Knot removal• Degree lowering
Cubic spline
Before fairing
After fairing
Curvature & Beauty
Curvature & Cars
Overview
• History• Curvature• Surface analysis• Ruled surfaces• Coons surfaces• Continuity• NURBS• Subdivision surfaces
Surface curvatures
Surface curvatures
Lines of curvature
Surface curvatures
• Gaussian = min curvature x max curvature• Gaussian = 0: developable surface
Gaussian curvature signs
First Gaussian image
Gaussian application
Gaussian curvature
Mean curvature
Max curvature
Reflection lines
Reflection lines
Overview
• History• Curvature• Surface analysis• Ruled surfaces• Coons surfaces• Continuity• NURBS• Subdivision surfaces
Ruled surface
Ruled, bilinear surface
Ruled surfaces
Overview
• History• Curvature• Surface analysis• Ruled surfaces• Coons surfaces• Continuity• NURBS• Subdivision surfaces
Coons surface
+ Ruled in u
+ Ruled in v
- Bilinear
Coons surfaces
Overview
• History• Curvature• Surface analysis• Ruled surfaces• Coons surfaces• Continuity• NURBS• Subdivision surfaces
G1 vs C1 surfaces
• C1: differentiable across common boundary of two surfaces; requires a common domain.
• G1: common tangent plane along common boundary; no common domain required.
Tangent ribbon
G1 condition
T-splines
T-splines
G1 mixed patch types
• History• Curvature• Surface analysis• Ruled surfaces• Coons surfaces• Continuity• NURBS• Subdivision surfaces
NURBS
Conics: Pascal’s theorem
Homogeneous coordinates
Polynomial vs rational
Conics: ellipse; 0<w<1
Conics: parabola; w=1
Conics: hyperbola; w>1
NURB curve, all weights = 1
NURB curve, one weight = 0.2
NURB curve, one weight = 3
B-spline curves
Bicubic NURB surface
One weight changed
NURBS: torus
NURB hierarchies
• Conics: subset of rational Bezier• Rational Bezier: subset of NURB curves• Conics: not subset of splines • Parabolas: subset of splines• Quadrics: subset of NURB surfaces
• Conic lofted surfaces: not subset of NURB surfaces
• History• Curvature• Surface analysis• Ruled surfaces• Coons surfaces• Continuity• NURBS• Subdivision surfaces
Subdivision
• Movie introducing Catmull-Clark subdivision
Viking ships
Symmetry Analysis
Whole body scanner from Cyberware
Art Meets Engineering
Twister, Dan Collins 1995 Rendering in 3D Studio Max
Water Table & Pop Growth (1900 - 2000)