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I am trying to use GEE to make maps intended for use on trails with compasses. I have already managed to place the contour lines but I couldn't put the grids with UTM coordinates.

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    Apologies if this is not very helpful but have you considered using other tools? GEE excels at raster analysis, not sure if it is the best suited for this task. – Nick Apr 3 at 20:36
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You can access the coordinates for any projection using ee.Image.pixelCoordinates. Then with a little math, you can turn that into a grid. Set the projection scale to select how close together you want them. These are at 1km.

var proj = ee.Projection("EPSG:32612").atScale(1000)
var coords = ee.Image.pixelCoordinates(proj)
Map.addLayer(coords.subtract(coords.round()).zeroCrossing().reduce('sum').selfMask())

enter image description here

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