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I have two lines in ArcMap 10.2.2 and was wondering if it is possible to create a raster showing the Euclidean Distance between them.

How do I map each cell on one line to its nearest cell on the other line?

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You could rasterize (Polyline to Raster) the lines and use the Euclidean Distance tool as suggested. If the lines are in the same feature class, note that ED cell values will be distance from a line, so values will increase away from the two lines until the halfway point. If you want the distance from one line to another, you'd need to run ED on only one line (select one or the other before Polyline to Raster to run the tool only on selected feature) and then extract the cell values where the other intersects the result raster. One way to do so would be rasterize the other line and use that raster as a mask for Extract by Mask.

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