Introduction to Modeling and Advanced Modeling

# Introduction to Modeling and Advanced Modeling Techniques

David S. Ebert

September 2000

• Polygonal Models
• Use thousands of small triangle/quadrilaterals to approximate curved objects.

Nice tools available in most modeling packages

Most hardware graphics pipelines geared to polygons (Millions of polygons/second)

• Spline Patch Models
• Use tensor product of 2 spline curves to create a 3D surface patch

Common splines:
• Catmull-Rom (Interpolating)
specified with 4 points (2nd and 3rd interpolated)

• Bezier (interpolating)
specified with 4 points (1st and 4th interpolated)

• B-spline (Approximating)
specified with 2 points and 2 tangents

Most graphics pipelines subdivide spline surfaces into triangles smaller than 1/2 pixel for rendering

NURB = Non-Uniform (placement of control points) Rational (ratio of 2 splines) B-splines

allows to accurately represent quadric surfaces (e.g, sphere)

• Subdivision Surfaces (Catmull-Clark, etc.)
• Described by "polygonal" control mesh

Recursive subdivision of each face converges to the limit surface

at the limit, the rectangular mesh of points define an equivalent surface to a uniform B-spline.

Advantage of patches - non-uniform subdivision

Example approximation of surface by polygonal mesh with recursive subdivision from PRMAN web page:

Reference page at Pixar

### Implicit Surfaces

Surfels Image from MERL

• Authors:Hans-Peter Pfister, Markus Gross, et al. ...
• History:
• Developed in 1999, presented at SIGGRAPH 2000
• Origins: Work by Levoy and Whitted and UNC Chapel Hill in 1980s

• Basic Idea
Use a dense collection of points stored in a multi-resolution framework instead of triangles, patches, volumes, etc.

Fractals:

• Authors: Mandelbrot, Voss, Musgrave, Hart, Peitgen, Jurgens, Saupe, ...
• History:
• Developed in late 19th and early 20th century - considered mathematical monstors - functions that defied normal mathematical principles.

Sierpinski - Start with filled equalateral triangle. Connect midpoints of each side, remove middle triangle, repeat.

von Koch, etc. - Start with line, divide into 1/3s, replace middle 13 with 2 edges of right triangle, repeat.

• 1970s, 1980s - Benoit Mandelbrot - thought these mathematical formulations described nature. They could be imbedded in a geometry describing natural forms

• Characteristics
• self-similarity
• non-unit dimensions

• Often used to describe natural objects (mountains, planets)
• Fractal Dimension:
• Power Law: a = 1/sD
• a = number of pieces
• s = reduction factor
• D = self-similarity dimension (fractal dimension)

• D = log (a) / (log(1/s))

• Example: Koch curve - D = log (4) / log(3) = 1.2619

• Two main classes: Deterministic and Random.
• Examples:

The Mandelbrot set:
Fractal Mountains:
Examples Landscapes from F. Ken Musgrave

fBm landscape

Multi-fractal landscape

Lava Landscape

• See "Chaos and Fractals: New Frontiers of Science" by Peitgen, Jurgens, Saupe.

L-systems

• Authors: Lindemyer & Prusinkiewicz, Alvy Ray Smith, Fowler
• Rule based grammers for describing natural objects
• Great for plants, trees, shells
• Uses turtle graphics for Drawing

• [] denotes a subtree structure

• + - denote positive and negative rotation

• Main difference with traditional grammars: All production rules applied simultaneously.
• Specification: Production ruls plus initial axiom

• Example:
• Simple Example of a another tree created by the following production rule:

• a -> a[+a]a[-a-a]a

with initial axiom a

• Probabilistic and context-sensitive L-systems provide more realistic plants and trees.
• Most recent work is on Open L-systems that react to their environment.
• Examples

Green coneflower. Copyright © 1992 D. Fowler, P. Prusinkiewicz, and J. Battjes.

A model of coniferous trees competing for light. The trees are shown in the position of growth. R. Mech and P. Prusinkiewicz. Copyright © 1996 P. Prusinkiewicz.

A model of coniferous trees competing for light. The trees are moved apart from the position of growth. R. Mech and P. Prusinkiewicz. Copyright © 1996 P. Prusinkiewicz.

Procedural Models

• Authors: Ebert, Perlin, Hart, Musgrave
• Use algorithms, code segments, procedure to define the geometry of your object.
• Add as much physics or art as you want
• Advantages: flexibility, data amplification, procedural abstraction.
• Examples:
• Volume metaball Cloud  This image is a volume rendered and procedural altered metaball. The image has low-albedo gas illumination and atmospheric attenuation.
• Close-up of Cloud

• Genetic Mating of Torus and Sphere

Main Notes Page