1. Animation of Plant Development Presented by Rich Honhart Paper by Prusinkiewicz, Hammel, and Mjolness

2. Outline Overview of the Paper Current (1993) methods of plant animation  How they work  Their drawbacks The system this paper proposes Examples

3. Overview of the Paper This paper proposes an extension to current (as of 1993) systems used to model plant growth which would make animating the process easier and more elegant.

4. The Standard Technique Lindenmayer Systems (L-Systems) Can be used to describe simple repeated growth processes  Ideal for growing plants

5. L-Systems An L-System describes a structure as a series of productions which replace a predecessor module with successor modules

6. Parametric L-systems Modules are allowed to take additional parameters These parameters can represent such things as age, shape, etc... Allows the structure to go through a more continuous transformation

7. Drawbacks of These Techniques Time progresses in discrete amounts  t.  t becomes part of the model and cannot be easily changed. … so how can we extend L-Systems to fix this?

8. dL-Systems Differential L-Systems Extension of parametric L-Systems which use differential equations to model the growth of the system Allow the user to sample at any resolution of t

9. dL-Systems: Notation A(w) – A module which takes parameter(s) w. A l – The module to the left of A A r – The module to the right of A D A – The domain of values the parameters of A can take on C A – A set of non-intersecting lines representing the borders of D A

10. More Notation The structure is represented as: A module’s growth is described by the diff eq: A production is described as:

11. dL-Systems: Behavior Parameters of a module are allowed to grow until they cross C A When they cross a boundary, the production associated with that boundary is run.

12. Simple Example: The Dragon Curve With Parametric L-Systems:

13. The Dragon Curve with dL-Systems

14. Evaluating dL-Systems Evaluated by a scheduler. Increase t by  t, integrate differential equations. If a production needs to be applied during that interval at a time t’, break the interval down into segments [t, t’) and [t’, t +  t).

15. Growth Functions Sigmoidal for plants (S-shaped)  Velhurst’s logistic function: Zero growth rate at the ends is desirable  This is possible using Hermite curves

16. A Complex Example: Pinnate Leaf

17. Eye Candy

18. Conclusions dL-Systems provide an elegant way to animate plant development. However, in some situations, they can be needlessly complicated.