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I'm building a web application that will be transferring polygons from server to client. Initially, the shapefile is coming from an external source that I have no control over.

Often, I find polygons with way too many points to describe a straight line. The result is, the size of the payload increases and therefore the fetch takes longer. This obviously isn't an issue with only a few polygons, but it will add up as more shapes are added to my dataset and I'd like to optimize bandwidth.

too many points - qgis

Is there a plugin, or native QGIS function that will take some kind of tolerance/variance as input, and produce (approximately) the same polygon with much fewer actual points?

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In this case the ultra-classic douglas–peucker algorithm is probably the best solution for you,

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There is a Simplify features tool in Advanced Digitizing toolbar that does exactly what you're describing.

http://docs.qgis.org/2.0/html/en/docs/user_manual/working_with_vector/editing_geometry_attributes.html?highlight=editing#simplify-feature

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you could also use the online tool mapshaper http://mapshaper.com/test/MapShaper.swf to "produce (approximately) the same polygon with much fewer actual points" give it a try to see if it fits your needs

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