The problem with simply throwing away points is that you can quickly distort the shape of the original polygon. A better approach is to come at it from the other direction; start with a basic approximation of the polygon and then refine it upwards towards your complex shape.

An excellent example of this approach is the Douglas-Puecker algorithm. You start with two vertices drawn from the full polygon. Add a third vertex by selecting the one that lies farthest from an edge drawn between the first two vertices. Continue adding points until you have something that sufficiently resembles your original polygon.

Douglas-Peucker is definitely the right approach. There are some simple ways to access implementations of it in PostGIS and QGIS that I thought I would add here for those who come across this post with a similar question. The goal is to start with something like this:

and end up with something like this:

In PostGIS Douglas-Peucker is implemented as simplify, the syntax, detailed here at bostongis.org, is some variant of:

SELECT transform(simplify(transform(the_geom, 2249), 500),4326) from the_geo_table

This worked very well even on the full national dataset, with some few errors that seem due to bad underlying data. It also turns out that in QGIS the menu item Tools > Geometry Tools > Simplify Geometries will export a simplified shapefile of any geometry and add it as a layer to your current project.

This is a pretty fundamental tool-set and I asked the question at too low a level, though it was nice to learn the underlying math, there is a good explanation of that here: http://www.mappinghacks.com/code/PolyLineReduction/, along with sample code that turns out not to be too necessary!

Instead of QGIS, I suggest using ogr2ogr because it does not delete polygons!

ogr2ogr output.shp input.shp -simplify 0.0001

Here s a simple iterative smoothing algorithm:

for each three sequential points on any path, if the middle point has no intersections and is within some small threshold angle of the direct path between the two outer points, remove it.

Repeat until satisfied.

You could also try Visvalingam’s algorithm, which iteratively removes the least perceptible part of a line. Here is a great explanation of that algorithm:

• http://bost.ocks.org/mike/simplify/

You could also use Simplify.js which uses a combination of Douglas-Peucker and Radial Distance algorithms. There are also links to many ports to other languages listed on the github project

Answer by @unmounted is correct, but I would like to add one more suggestion.

Always use function ST_SimplifyPreserveTopology instead of ST_Simplify in PostGIS. Both use the same underlying algorithm (Douglas-Peucker), but the former avoids any simplifications which would result in invalid geometries. For example, ST_Simplify may result in geometry which intersects itself.