Update to Functional Equations Reference (version 1.3)

This is a substantial update of this reference document. The most important addition is the chain and substitution rules for arithmetic difference calculus (ADC). Other additions include: more properties of the discrete power function, more properties of ADC operators, definitions of analog functions, and ranges of convergence of (some) z-transforms. I also corrected some errors that were discovered since the last version.

Grab it here.

Region Quadtrees in C++

quadtree

(Original image by GoAwayStupidAI).

Below are four C++ implementations of the region quadtree (the kind used for image compression, for example). The different implementations were made in an attempt to optimise construction of quadtrees. (For a tutorial on implementing region quadtrees, see Issue 26 [6.39 MB zip] of Dev.Mag).

  • NaiveQuadtree is the straightforward implementation.
  • AreaSumTableQuadtree uses a summed area table to perform fast calculations of the mean and variance of regions in the data grid.
  • AugmentedAreaSumTableQuadtree is the same, except that the area sum table has an extra row and column of zeros to prevents if-then logic that slows it down and makes it tricky to understand.
  • SimpleQuadtree is the same as AugmentedAreaSumTableQuadtree , except that no distinction is made (at a class level) between different node types.

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Cellular Automata for Simulation in Games

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A cellular automata system is one of the best demonstrations of emergence. If you do not know what cellular automata (CA) is, then you should go download Conway’s Game of Life immediately:

Conway’s Game of Life

Essentially, CA is a collection of state machines, updated in discrete time intervals. The next state of one of these depends on the current state as well as the states of neighbours. Usually, the state machines correspond to cells in a grid, and the neighbours of a cell are the cells connected to that cell. For a more detailed explanation, see the Wikipedia article.

Even simple update rules can lead to interesting behaviour: patterns that cannot be predicted from the rules except by running them. With suitable rules, CA can simulate many systems:

  • Natural phenomena: weather, fire, plant growth, migration patterns, spread of disease.
  • Socio-economic phenomena: urbanisation, segregation, construction and property development, traffic, spread of news.

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