Fractal landscape


Fractal landscape

A fractal landscape is a surface generated using a stochastic algorithm designed to produce fractal behaviour which mimics the appearance of natural terrain. In other words, the result of the procedure is not a deterministic fractal surface, but rather a random surface which exhibits fractal behaviour. [cite web |url=http://www.fractal-landscapes.co.uk/maths.html |title=The Fractal Geometry of Nature] Because the intended result of the process is to produce a landscape, rather than a mathematical functions, processes are frequently applied to such landscapes which may affect the stationarity and even the overall fractal behaviour of such a surface, in the interests of producing a more convincing landscape.

Behaviour of natural landscapes

Whether or not natural landscapes behave in a generally fractal matter has been the subject of some research. Technically speaking, any surface in three-dimensional space has a topological dimension of 2, and therefore any fractal surface in three-dimensional space has a Hausdorff dimension between 2 and 3 [Lewis] . Real landscapes however, have varying behaviour at different scales. This means that an attempt to calculate the 'overall' fractal dimension of a real landscape can result in measures of negative fractal dimension, or of fractal dimension above 3. In particular, many studies of natural phenomena, even those commonly thought to exhibit fractal behaviour, do not in fact do so over more than a few orders of magnitude. For instance, Richardson's examination of the western coastline of Britain showed fractal behaviour of the coastline over only two orders of magnitude. [Richardson] In general, there is no reason to suppose that the geological processes that shape terrain on large scales (for example plate tectonics) will exhibit the same mathematical behaviour as those which shape terrain on smaller scales (for instance soil creep).

Real landscapes also have varying statistical behaviour from place to place, so for example sandy beaches don't exhibit the same fractal properties as mountain ranges. A fractal function, however, is statistically stationary, meaning that its bulk statistical properties are the same everywhere. Thus, any real approach to modeling landscapes requires the ability to modulate fractal behaviour spatially. Additionally real landscapes have very few natural minima (most of these are lakes), whereas a fractal function has as many minima as maxima, on average. Real landscapes also have features originating with the flow of water and ice over their surface, which simple fractals cannot model [Ken Musgrave, 1993] .

It is because of these considerations that the simple fractal functions are often inappropriate for modeling landscapes. More sophisticated techniques (known as 'multifractal' techniques) use different fractal dimensions for different scales, and thus can better model the frequency spectrum behaviour of real landscapes [Joost van Lawick van Pabst et. al]

Generation of Fractal Landscapes

A way to make such a landscape is to employ the random midpoint displacement algorithm, in which a square is subdivided into four smaller equal squares and the center point is vertically offset by some random amount. The process is repeated on the four new squares, and so on, until the desired level of detail is reached. There are many fractal procedures (such as Perlin noise) capable of creating terrain data, however, the term "fractal landscape" has become more generic.

ee also

*Diamond-square algorithm

Notes

References

*cite web
url=http://www.idiom.com/~zilla/Work/caseagainstfractals.pdf
title=Is the Fractal Model Appropriate for Terrain?
first1=J.P.
last1=Lewis
ref=Lewis

*cite journal
first=L.F.
last=Richardson
title=The Problem of Continuity
journal=General Systems Yearbook. 6
year=1961
pages=pp. 139-187
ref=Richardson

*cite web
url=http://www.lawick.nl/publications/paperft.pdf
title=Dynamic Terrain Generation Based on Multifractal Techniques
first1=Joost
last1=van Lawick van Pabst
first2=Hans
last2=Jense
year=2001
ref=Jense

*cite web
url=http://www.kenmusgrave.com/dissertation.pdf
first=Ken
last=Musgrave
title=Methods for Realisitic Landscape Imaging
year=1993
ref=Musgrave1

External links

* [http://www.fractal-landscapes.co.uk Fractal landscapes]
* [http://ibiblio.org/e-notes/3Dapp/Mount.htm 3D Fractal Mountains in Java]
* [http://landscapestudio.omgames.co.uk/ Landscape Studio Java-based terrain generator]
* [http://www.embege.com/fractals/mdterrain/ MDTerrain Terrain Generator using Midpoint Displacement]
* [http://www.pandromeda.com/ Pandromeda's MojoWorld Generator]
* [http://mrl.nyu.edu/~perlin/experiments/demox/Planet.html A Web-Wide World] by Ken Perlin, 1998; a Java applet showing a sphere with a generated landscape.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Fractal analysis — is the modelling of data by fractals.It consists of methods to assign a fractal dimension and other fractal characteristics to a signal, dataset or object which may be sound, images, molecules, networks or other data.Fractal analysis is now… …   Wikipedia

  • Fractal — A fractal is generally a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced size copy of the whole, [cite book last = Mandelbrot first = B.B. title = The Fractal Geometry of… …   Wikipedia

  • Fractal art — is created by calculating fractal objects and representing the calculation results as still images, animations, music, or other media.Fractal art is usually created indirectly with the assistance of a computer, iterating through three phases:… …   Wikipedia

  • Fractal city — Edward Soja uses the term fractal city to describe the metropolarities and the restructured social mosaic of today s urban landscape or postmetropolis . In his book, Postmetropolis: Critial Studies of Cities and Regions , he discusses how the… …   Wikipedia

  • List of fractal topics — This is a list of fractal topics, by Wikipedia page, See also list of dynamical systems and differential equations topics.*1/f noise *Apollonian gasket *Attractor *Box counting dimension *Cantor distribution *Cantor dust *Cantor function *Cantor… …   Wikipedia

  • Ken Musgrave — Dr. Forest Kenton Ken Musgrave (aka Doc Mojo ), formerly a professor at The George Washington University and currently CEO/CTO of Pandromeda, Inc, is a computer artist, working with fractal images. Education Born on September 16 1955, he obtained …   Wikipedia

  • Procedural generation — is a widely used term in the production of media, indicating the possibility to create content on the fly rather than prior to distribution. This is often related to computer graphics applications and video game level design.OverviewThe term… …   Wikipedia

  • List of mathematics articles (F) — NOTOC F F₄ F algebra F coalgebra F distribution F divergence Fσ set F space F test F theory F. and M. Riesz theorem F1 Score Faà di Bruno s formula Face (geometry) Face configuration Face diagonal Facet (mathematics) Facetting… …   Wikipedia

  • Computer-generated imagery — An example of a computer generated, natural looking, static fractal landscape. Computer generated imagery (CGI) is the application of the field of computer graphics or, more specifically, 3D computer graphics to special effects in art, video… …   Wikipedia

  • Bryce (software) — Infobox Software name = Bryce caption = Bryce screenshot. developer = DAZ 3D latest release version = 6.1 latest release date = March 2007 operating system = Mac OS X, Microsoft Windows genre = 3D computer graphics license = Proprietary website …   Wikipedia