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I have an old map in aln, 1 aln=0.5934 meters. I would like to draw lines using directly this measure unit without having to convert in meters each time to draw the lines. Is there any way to customize the unit type in arcmap 10.1?

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If you know the detailed coordinate system of your map, I'd suggest creating custom projected coordinate system with custom units and reprojecting / georeferencing map directly into this coordinate system. You can define "meters per unit" for alns, e.g.:

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