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I am seeking to create an animated map illustrating the simultaneous movement of different objects across a large area over time.

The display should be capable of playing automatically, or by manually selecting a date and/or period.

I would like to retain the interrogable capability of a digital map (zoom and query a database or attributes) while paused, so I do not want to merely create raster images and and incorporate them into a slide show.

Rather I want vector data moving across a base map.

Does anyone know how to perform this trick using QGIS, GeoServer or ArcGIS for Desktop?

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Here are two options:

QGIS Time Manager

ArcGIS Time Slider

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Both of these are good options. I think the QGIS version has a little more functionality, but the ArcGIS option seems more stable. Mind you this is more of a recollection on my part. It's definitely worth trying both to see which is optimal. If you just wanted something very compelling to watch in motion, CartoDB's Torque is hard to beat. – Kevin R Dyke Jun 23 '14 at 4:36

Geoserver has an animated gif option:

but it doesn't give you an interface, to create an interface you should use a javascript library like OpenLayers, check for example the timeslider in GXP:

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