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)