Introduction to Modeling and Advanced Modeling

Introduction to Modeling and Advanced Modeling Techniques

David S. Ebert

September 2000

Traditional Geometric Representations

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

Catmull-Clark subdivides using quads =>

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

Advanced Modeling Techniques

Surfels

Fractals

L-Systems

Procedural Models

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.
Cantor - start with unit interval, remove middle 1/3, repeat.
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/s^D
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.

See also the work by Prusinkiewicz & Hammel

Examples

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

Implicit Models

Authors: Wyvills, Hart, Blinn

Describe objects with implicit equations: F(x,y,z)=0

Blend objects together using smooth function:

R = distance where function has a value of 0.

Graph of Blending Function:

Create iso-surface of blended density fields.

Example: 10 Years of Implicit Surfaces by Brian Wyvill

Common Example effects: liquid metal creature from Terminator 2, flubber, klingon blood

Images courtesy of Brian Wyvill (blob@cpsc.ucalgary.ca).

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