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Simulating Trees with Fractals and L-Systems

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Simulating Trees with Fractals and L-Systems

Eric M. Upchurch

CS 579

- 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!)

- 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

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

- 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

- 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

- 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.

- 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

- 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

- 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.

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

- 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

- 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

- 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

- 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

- 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”

- 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.

- 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.

- 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!