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!