Simulating Trees with Fractals and L-Systems

1. Simulating Trees with Fractals and L-Systems Eric M. Upchurch CS 579

2. Background - Fractals • Fractals are recursive, self-similar structures • Infinitely detailed – zooming in reveals more detail • Similar, though not necessarily identical, at any level of magnification • Generated using a variety of methods, such as IFS and L-Systems • Many natural forms display fractal geometry (like trees!)

3. Background – L-Systems • Formal grammar developed by Aristid Lindenmayer as a theoretical framework for studying development of simple multicellular organisms • Subsequently applied to investigate higher plants • Uses rewriting rules (productions) to grow a string or system • Productions are applied in parallel (unlike Chomsky grammars) • This is motivated by biological considerations – this is how living organisms grow

4. Strings built of letters a and b Each letter is associated with a rewriting/production rule: a  ab b  a Rewriting starts from an axiom, a starting string (b in this case) Each production is applied simultaneously in each step Simple L-System

5. L-Systems – Properties • Can be context-free or context-sensitive • Different production rules for the same symbol based upon neighboring symbols • If run repeatedly and interpreted as images, can produce fractal geometry • Can describe many “traditional” fractal patterns, such as Koch curves and constructions • Can describe & produce very complex fractal patterns

6. Iterated Function Systems • An IFS is any system which recursively iterates a function or a collection of arbitrary functions on some base object • An IFS can be used to generate a fractal pattern • The IFS fractal is made up of the union of several copies of itself, each copy being transformed by a function • No restriction on transformations, though they are usually affine • Can use deterministic or stochastic processes

7. Iterated Function Systems • If the system is made up by k functions (or transformations) {fi(x) | 1 < i < k} and iterated n times on the base set b, then the IFS is defined as the set: • { fi1(fi2 ( .. fik(b) .. )) | for all ij, with 1 < ij < k, j = (1, 2, .. , k) } • In the limit that n becomes infinite, an IFS becomes a fractal.

8. Relationship of IFS to L-Systems • Both are recursive in nature • Both can be used to produce fractals • L-Systems are, arguably, a specialized form of IFS, whose functions are specified by the production rules of the grammar • Replace the formal grammar of an L-System with functions/transformations, and you have an IFS

9. Describing Trees • A rooted tree has edges that are directed and labeled • Edge sequences form paths from a distinguished node, called the root or base, to terminal nodes • In biological sense, these edges are branch segments • A segment followed by at least one more segment in some path is called an internode • A terminal segment (with no succeeding edges) is called an apex

10. Axial Trees • Special type of rooted tree • At each node, at most one outgoing straight segment is distinguished • All remaining edges are called lateral or side segments. • A sequence of segments is called an axis if: • the first segment in the sequence originates at the root of the tree or as a lateral segment at some node • each subsequent segment is a straight segment • the last segment is not followed by any straight segment in the tree. • Together with all its descendants, an axis constitutes a branch. A branch is itself an axial (sub)tree.

11. Axes and branches are ordered The root axis (trunk) has order zero. Axis originating as a lateral segment of an n-order parent axis has order n+1. The order of a branch is equal to the order of its lowest-order or main axis Axial Trees

12. Example Axial Tree

13. Honda’s Model • Limited model with following assumptions: • Tree segments are straight and their girth is not considered • A mother segment produces two daughter segments through one branching process • Lengths of the two daughter segments are shortened by constant ratios, r1 and r2, with respect to the mother segment • Mother and daughter segments are contained in the same branch plane. The daughter segments form constant branching angles, a1 and a2, with respect to the mother branch • Branch plane is fixed with respect to the direction of gravity so as to be closest to a horizontal plane. An exception is made for branches attached to the main trunk, where a constant divergence angle αbetween consecutively issued lateral segments is maintained

14. Honda’s Geometry

15. Honda’s Trees

16. My Tree Simulator • Written in C++, using the DirectDraw API • Draws 2D trees by writing to a bitmap • Does simple lighting and shading • Produces a fair mix of completely ugly trees and good looking/accurate trees • Uses stochastic IFS transformations, inspired from L-Systems and extends Honda’s work • Drawing infrastructure borrowed from a Julia Set generator

17. My Tree Simulator • A tree is specified by: • Radius (thickness of trunk) • Height • Some number of branches, each having: • Scaling factor (all branches are scaled according to order) • Height/length of branch • Lean angle (from higher order branch) • Rotation angle (around higher order branch) • Number of branches is used in recursively creating branches

18. My Tree Simulator • IFS is performed by transformations specified in each branch. • Transformations: scale, lean, rotation, are stochastically determined • Foliage is done using the same method but using different colors to give the illusion of leaves • Z-buffer utilized for determining which pixels are drawn

19. My Tree Simulator • A tree contains several flags that define which IFS transformations are applied: • Use Branch Heights  If true, places branches randomly up trunk. If false, places all branches coming out of trunk (and recursive branches) at same point. • False tends to make trees more irregular. • Global Scaling  If true, scales branches and foliage based on a single scale (stored in the trunk branch). Otherwise, scales each branch separately based on a scale stored per branch. • True makes trees more regular, and “tighter”. • False makes trees more irregular, but can cause “puffiness”

20. My Tree Simulator • Scale By Height  If true, branches and foliage decrease (more dramatically) as their height increases. • True keeps some trees from getting too “puffy”. • False makes trees more top-heavy, fuller, like elms. • True makes trees slimmer and decreasing, like a willow bush. • True will give a younger looking tree, false will give an older looking tree with the same structure.

21. Future Work • Convert to 3D • 2D representation is easier, but very limiting • Allow model exportation for placement into virtual worlds • Use foliage models in 3D • More realistic effect, scales well, could provide physics-based modeling effects (wind, sway, etc) • Enhance parameterization of trees • Currently, everything is random!