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I need a good quality rasterized world map, Winkel-Tripel projection, in which all countries have different (random?) colors. No borders, text, grid or anything else on the map.

Is GIS the right way to go? I found a lot of map data on naturalearthdata.com. Can I create this map with QGIS? Do I need ArcGIS?

I'm completely new to GIS; it looks like an exciting field. I'd appreciate any help to get started.

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on the winkel tripel part: gis.stackexchange.com/questions/23089/… –  underdark May 16 '13 at 18:27
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The last post on this page might help: anitagraser.com/2012/03/18/… –  nhopton May 16 '13 at 19:20
    
And yes, you can do all of this in QGIS as you've asked. –  Michael Markieta May 16 '13 at 19:52

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up vote 2 down vote accepted

You can find a shapefile (vector) for countries as of 2010 from http://epp.eurostat.ec.europa.eu. It think that it would be easier to change the symbology of a vector file than deal with a raster. Arcmap will project on the fly and you can set the map to a winkel projection (projected->world or world-sphere->winkel).

A quick world map projected in winkel projection

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