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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background fractals
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
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
simple l system
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
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
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 systems1
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
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
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
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.
axial trees1
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
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
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 simulator1
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
my tree simulator2
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 simulator3
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 simulator4
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
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!