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# Simulating Trees with Fractals and L-Systems - PowerPoint PPT Presentation

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

• 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

Simple L-System

• 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.

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

• 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:

• 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”

• 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!