Contiguous area cartograms Ingeborg Groeneweg

Introduction What are cartograms Difficulties creating cartograms History: previous approaches Current approach in-depth Summary

Cartograms Resizing regions of map by geographically related parameter Other way of representing the same: –3D-map –choropleth

Cartograms Area resizing:  x  y = h  u  v Shape preservation Topology preservation Contiguous

Example

2 parameter example

Difficulties Resizing area, preserving shape, preserving angle

Error measure Optimization problem Error measure –Area error –Shape error

History: Rubber map method Tobler, 1973 One of the first cartogram algorithms Used for population districting Idea: –put a dot for every person on a rubber map –Stretch the map until every dot is at equal distance Problem: –poor performance –Large area error –Overlapping shapes

Pseudo cartogram Tobler, 1986 Reduce area error Starting point for rubber map method

Rubber sheet distortion Dougenik, Chrisman, Niemeyer, 1985 Improvement on Tobler Difference: computing “force” on one polygon per iteration Overlapping shapes occur infrequently

DEMP Selvin et al.,1984 Density Equalized Map Projection (DEMP) Used to detect non-random distributions of disease Calculate spatial magnificent factor Radial transformation projected on selected area

Line integral Gusein-zade and Tikunov, 1995 Stokes theorem and line integrals

Forced-based Kocmoud and House, 1998 Alternately optimize shape and area error Superior to former methods

Cartodraw Keim, north, panse, 2004 Goal: –creating cartograms on the fly –Small error Cartodraw: –Simplify shape –Define error functions –Scanline –Main algorithm

Cartodraw: simplify shape Few vertices important for defining shape Vertices almost no noticeable difference: –angle near 180 degree –With short edges Different reduction algorithm for global shape and inner vertices

Global polygon reduction Only look at vertices v with d( v )> f( v ) –v at outer polygon –v do not belong tomultiple polygons Determine least important vertex w Finding polygon p where w is part of Counting difference d between p before and after removing w Remove w if d < constant

Least important vertex

Inner polygon reduction Remove all interior vertices v with d( v ) = 2 Reintroduce few vertices

Cartodraw Area error function Shape similarity function Scanline algorithm Main algorithm

Area error function Relative area error of polygon p j Area error for set of polygons P

Shape error function Translation, scale and partially rotation invariant Euclidean distance in Fourier space useful for shape similarity measure Use of differential geometric curvature of polygons Curvature will be square wave function

curvature

Example curvature

Fourier transformation Approximate function by summing sine and cosine Fourier approximation is defined as:

Fourier of square wave

Scanline Scanline sl = Line segment of arbitrary position and length Incrementally reposition vertices along scanline

Scanline Scaling factor

Cartodraw: main algorithm Transformation applied for each scanline –If E rel and shape distortion below certain threshold changes are retained Test improvement of area error

Automatic vs interactive Automatic generation of scanlines: –Fixed grid of horizontal and vertical scanlines –Resolution can be varied Interactive position of scanlines –Better results

Summary No ideal solution Several approaches reviewed Poor performance Handmade solutions superior