Simulating Trees with Fractals and L-Systems

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

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### Simulating Trees with Fractals and L-Systems

Eric M. Upchurch

CS 579

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!)
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
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
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
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
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.
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
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
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.
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
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
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
My Tree Simulator
• A tree is specified by:
• 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
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
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”
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.
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!