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is it possible to display markers with UTM coordinates (zone 32) with leaflet and OSM? How can I do this? If I use UTM Coordinates like latlng, the position is totally wrong:

        L.marker([5698015.694, 312400.093]).addTo(map)
        .bindPopup("<b>Hello world!</b><br />I am a popup.").openPopup();

Did I have to translate them first, or is it possible to display UTM coordinates directly with leaflet?

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Leaflet does not do coordinate translation (except the implicit one between LatLon and Web mercator).

Check out https://github.com/kartena/Proj4Leaflet, this should enable you to work with UTM coordnates (WGS84 UTM Zone 32 is EPSG:32632, proj4js def here: http://spatialreference.org/ref/epsg/32632/proj4js/)

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Now (released in July 2017) you can use this plugin https://github.com/jjimenezshaw/Leaflet.UTM, much simpler than Proj4Leaflet if you just want to convert to and from UTM coordinates. It works for any zone.

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