Introduction to Modeling and Advanced Modeling

Introduction to Modeling and Advanced Modeling Techniques

David S. Ebert

September 2000

Traditional Geometric Representations

Advanced Modeling Techniques

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.


  • 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/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.


  • 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

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

    A Horsechestnut tree model with branches competing for access to light, R. Mech and P. Prusinkiewicz. Copyright © 1996 P. Prusinkiewicz.

    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

    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 (

Main Notes Page