Topology and Projections: 21st Century Cartography

maps splash walters gallery baltimore

The traveling map exhibit MAPS: FINDING OUR PLACE IN THE WORLD at the Walter’s in Baltimore, Maryland (via the Field Museum, Chicago) wrapped up last weekend. While looking at John Adam’s Road distance map of England and Wales I was put in mind of how map projections try to preserve several of:

  • Area
  • Distance
  • Shape, and
  • Direction/angle

Then the question arose: Might information (thematic) topology now be interchangable with purely geographical topology?

John Adam’s map from 1680 places towns in relative (not absolute) geographic lat-long coordinates in a rough framework that preserves their orientation to one another and in the rough-shape of their original geography. But the primary purpose of this map is to emphasize the relationship between towns and intra-town distances. Below by Royal Geographic Society.

john adams road map 1680

This topological focus (of NODES and EDGES in math-speak) is perfectly represented in Adam’s map. Circles (nodes) are inscribed with town names and straight lines (edges) connecting town circles are annotated with road distance (not straight-line geographic distance).

Modern scientific cartography, with an emphasis on visualization, might finally be loosening the geographic straight jacket to the point where purely lat-long geography doesn’t matter so much but the inter-connection (edges) of said features (nodes) gains emphases and is preserved.

I believe such thematic topology maps are “geographically” accurate and employ projection just like conventional cartography but these “projections” are as of now ad hoc and not properly defined or formalized and are often created manually. Efficient and effective mathematical formulas should be devised and listed along conventional map projections in publications like Snyder’s (USGS) Map Projections: A Working Manual.

The nearest we come to topologic maps are subway maps and cartograms. More on cartograms below.

new york subway map slice

This subway map of New York City is a topological map where the island area of Manhattan is relatively small geographically but is significantly exaggerated to accommodate the “accurate” display of the topological nodes and edges of subway stations and subway lines. (Dorling cartogram example below by Zach Johnson.)

zach johnson cartogram

Cartograms are a good example of topological maps:

  • Area of symbol represents the NODE weight alone.
  • Distance is based on EDGE weight first and and geographic distance second (trying to approximate the “relatedness” between each, eg close countries close, far countries far).
  • Direction is approximated.
  • Shape is approximated.

Zach Johnson has a good post on this topic on his blog (cartograms are the focus of his Masters Thesis).

Below a New York Times map showing the weighted electoral votes of the 48 contiguous states as the topological area of each state.

ny time cartogram example

Let us examine a map of water flow in a stream network (Kelso and Araya):

six rivers streamflow

One usually sees these maps with a conic projection to preserve equal-land-area. But the river segments are drawn exaggerated to their geographic width to represent the EDGE weight between nodes in the true geography space.

The map is dispensing with equal-land-area between nodes (the overall area and shape are preserved) and instead focusing on DISTANCE and DIRECTION between each node. The edges are “preserved” by exaggerating the stream centerlines to preserve the thematic variable. Overall SHAPE is preserved, but local land AREA is not.

Such topological maps are not diagrams because they are still rooted in land-geography; the placement of the nodes is guided by land geography but shift accordingly to best show the interrelationships between nodes. Ignoring the land-geography by listing the nodes and edges in a chart or table is not a map. A topological map takes a complex n-dimensioned space and represents that topology in a 2d dimensioned space.

precip swiss atlas

Some precipitation maps use “gridded” tightly spaced, regularized nodes and edges (above: Swiss Atlas, 2.0). The “weight” of rain and snow fall is indicated by color. Because of the spacing of the nodes and the hyper-localness of the mapped theme, this “topological map” manages to preserve both the topology and the geography.

nat geo 8th world atlas human chapter opener

The above example from the 8th Edition National Geographic Atlas of the World focuses on the quantity difference between nodes and represents that with height spikes (3d). If this were a topologic projection that needed to show the contents of each node (not the inter-relationship between nodes) the spike height would be flattened out into area alone (2.5d), leading to a grossly exaggerated land-area map but correct population-area cartogram such as:

mworlds_zj.png

Above from the Dutch company Mapping Worlds via Zach Johnson.

tom patterson relief example

Tom Patterson (above) uses this 2.5d term to talk about relief shading of land elevation. But I think it can be used to represent any map that is a representation of more than simply 2 variables (lat and long). Really, much of thematic cartography is 2.5d when it tries to represent complex datasets (like precipitation) with color and other visual variables.

So visualization / modern scientific cartography is focused on examining and preserving / projecting topological relationships. Often these are closely related to geographic space, but not always. That is why I am so fascinated by cartograms :)

How do we measure the “error” and “conformal”-ness in a topological map?

  • Area: Does this “view” of the topology preserve the node and edge weights?
  • Distance: Does this “view” preserve the inter-relations between nodes?
  • Direction: Both topological between nodes and geographically.
  • Shape: Purely geographical. This is what sets some cartograms above others.

For topologic shape:

Projecting a n-dimensioned topology onto a 2d surface has one or more points tangent to the 2d surface. An ideal solution shows all nodes and edges shown flattened out but this would likely require using an interrupted projection with dashed linkage lines between like-lobes content (I have seen this somewhere, need example).

For geographic shape and direction:

We are concerned with local shape (direct neighbors in the topology) and global shape. In the England example above for the Dorling cartogram the north-south direction axis tilts left in the topology. A “best” solution preserves this geographic orientation by rotating the topology network until it “conforms” more to the geography.

Finally, we can visualize this with a modified cartography cube from Zach Johnson:

cart cube zach johnson

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6 Responses to “Topology and Projections: 21st Century Cartography”

  1. daan Strebe says:

    These are good examples of how to improve map imagery. There will be hurdles. Mapmakers rely heavily on automated tools. It’s hard to get paid for labor-intensive processes. Many of these techniques will remain labor-intensive because they defy algorithmic construction. Most cartograms and other topological mapping devices introduce discontinuities in the mapping function, an inconvenience which shuts down analytic solutions. There are ways to assist the mapmaker analytically, but ironically it’s usually easier either to work completely manually or completely automatically when it comes to a single step of a process, foregoing mixed human/machine processes. Mixing automated assistance with manual work requires the mapmaker to thoroughly understand the entire process, not just its results. It also requires purveyors of the tools to understand how a mapmaker might approach a partial solution and to believe a mapmaker would even be willing to. Hence, while papers get written on automated cartogram algorithms, few of them see their way into real maps. Lack of tools, lack of deep understanding of the process, lack of reward for the effort, and lack of a culture that encourages such rich imagery — formidable obstacles.

    Kudos for pulling together a compelling editorial.

    – daan Strebe

  2. nathaniel says:

    Thanks for your comment, daan!

    I wonder if we’re coming to a point where the raster distortion grids and “bump” mapping techniques of 3d can get graphed back into the 2d cartographic main stream?

    The computer processing power seems to be here. Now we just want good algorithms and tools.

    I see tools like Bernhard Jenny’s MapAnalyst (http://mapanalyst.cartography.ch/) which evaluates the “inaccuracy” in old maps and think this could be turned into a forward projection algorithm.

    Indeed, Bernhard’s other tool, Flex Projector (http://www.flexprojector.com/) allows cartographers to “mashup” their own projection based on mathematical formulas.

    Photoshop’s mesh-distort free transformation tool makes it so I don’t even need to know the original projection to “deproject” the hurricane maps I get from NOAA before projecting them onto a new projection. Amazing!

    The next step is to represent such novel projections not as formulas but as distortion grids and store the final positional distortions resulting from cartogram creation logic ala Josh’s Dorling cartogram programming in this format (http://indiemaps.com/blog/2008/02/i-got-better/). This can then be the filter geographic long-lats can be projected thru.

    For an example showing the London Underground map in this format with diagrams, see: http://jenny.cartography.ch/pdf/2006_Jenny_DistortedNetworkMaps.pdf.

    Having implemented an interrupted map projection (look, 6 projections in 1!) for a forthcoming project for The Washington Post I totally hear you on the time and resources needed to accomplish this.

    I remember when ArcMap couldn’t deal with the 180° seam. Hopefully we’ll be able to deal with interruptions in a like fashion in the future.

    Let’s hope people like Bernie and Josh keep advancing this area of cartography.

    _nathaniel

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