Simulating Trees with Fractals and L-Systems

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Simulating Trees with Fractals and L-Systems

• Eric M. Upchurch
• CS 579

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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!)

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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

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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

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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

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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

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Iterated Function Systems

• If the system is made up by k functions (or
  transformations) fi(x) 1 lt i lt 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
  lt ij lt k, j (1, 2, .. , k)
• In the limit that n becomes infinite, an IFS
  becomes a fractal.

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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

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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

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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.

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Axial Trees

• 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 n1.
• The order of a branch is equal to the order of
  its lowest-order or main axis

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Example Axial Tree
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Hondas 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 a between consecutively
  issued lateral segments is maintained

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Hondas Geometry
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Hondas Trees
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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 Hondas work
• Drawing infrastructure borrowed from a Julia Set
  generator

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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

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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

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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

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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.

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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!