I have a shapefile with contours and their heights.
Is it possible to make a DEM from contours in QGIS without using GRASS?
I found Creating DEM from contours using ArcGIS Desktop? but the answer is for ArcGIS Desktop.
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Yes, there are several options available in QGIS:
Raster > Grid (Interpolation). GDAL's interpolation is more robust because you can use other interpolation algorithms (IDW, nearest neighbor, moving average, etc.). This tools only works for point data.
r.surf.*). You need convert your shapefile into a GRASS database to use the GRASS modules in QGIS.
GRASS has also many options
available one way or another also trough the QGIS/GRASS plugin.