Tile maps

Mapping a set of random points to a uniform lattice

Problem

Given a set of N polygons corresponding to countries/regions/municipalities find a regular tessellation of N triangles, squares, or hexagons where each tile corresponds to a country such that the original spatial arrangement is preserved as much as possible.

More generally, find a mapping from a set of N arbitrary/random points whilst preserving the original arrangement.

The code in this repository is experimental. For an excellent/stable R package solution to this problem, see Joseph Bailey's geogrid.

Integer Linear Programming (ILP) approach

My first approach to this problem makes use of Linear Programming.

Installing COIN-OR

Cbc (Coin-or branch and cut) is an open-source mixed integer programming solver. It is faster than the GNU Linear Programming Kit (GLPK).

git clone --branch=stable/2.9 https://github.com/coin-or/Cbc Cbc-2.9
cd Cbc-2.9
git clone --branch=stable/0.8 https://github.com/coin-or-tools/BuildTools/
chmod 0700 BuildTools/get.dependencies.sh
BuildTools/get.dependencies.sh fetch
./configure --prefix=/usr/local --enable-cbc-parallel
make
sudo make install

verify parallel enabled

cbc -threads 8 -unitTest -dirMiplib=_MIPLIB3DIR_ -miplib

Simulated Annealing (SA) approach

My second approach to this problem makes use of the popular Simulated Annealing metaheuristic.

References

• Formulating Linear and Integer Linear Programs: A Rogues' Gallery

• Area Cartograms: Their Use and Creation - Daniel Dorling. Appendix lists code for a circular force-directed/gravity method for cartogram generation. (see section 9 in book)

• Let’s Tesselate: Hexagons For Tile Grid Maps - NPR visuals.

• Integer Linear Programming Model approach) - interesting formulation.

• Area Cartograms: Their Use and Creation - Daniel Dorling, 1996.

• US as an hexagonal map - Formalisation of problem as an Integer Programming Model.

• Hexagonal Grids - Distances, path-finding and geometry.

• Animated mapping is bringing elections to life

• ONS - Visualising your constituency - Geographic cartogram of UK.

• Something About The Automated Stippling Drawing - Very interesting.

• Conformal Projections

• Conceptualisation of a D3 linked view with a hexagonal cartogram

• Crystallographic defect - Clues from defective lattice? Perform Annealing?

• Molecular dynamics

• Crystallisation -- Simulate

• New metastable form of ice and its role in the homogeneous crystallisation of water

• Linear Assignment - mapping a set of size N to a a grid of size N. ***