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I've been looking through all of the GRASS commands in the QGIS 2.10 processing toolbox and it seems like one of these must be able to do this but I haven't found one yet. There is 'v.sample' which can sample a DEM at a point location. Is there a way to do something similar to sample a DEM when calculating the length of a line segment?

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Yes and you can automate this process with python grass.script or bash script (bat script in Windows). I test it in my system and the individual commands are (for my line vector named route and raster dem named utah_demUTM2):

v.to.rast input=route value=1 output=route_raster 
r.thin input=route_raster output=route_raster_thin
r.mapcalc route_raster_dem="if(route_raster_thin, utah_demUTM2)" #in GRASS console

to produce route_raster_dem with the raster values (see next image where I used the plugin Value Tool).

enter image description here

Afterward, you can use the command r.to.vect.points to create points (distance between points equal to raster resolution) along input line route_raster_dem for calculating the 3D length of the line segment by using the adequate algorithm (with code or in a spreadsheet).

At the next image you can see some selected features at the start of route_vector_dem (obtained by using r.to.vect.points) and its attributes table.

enter image description here

3D distance is the sum of the length of the hypotenuse given by the difference in height between two adjacent segments along the line (the individual steps are the same as the raster resolution).

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