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from vector to raster (2nd try) with R or GRASS (GRASS prefered...)

Good morning,

I try to be clearer than I was in my precedent post and also copy this 
to the R SIG GEO.
On the one hand, I have a shape of lines representing roads
On the other, a GRASS region with a given resolution (250 x 250 meters 
pixel)

I would like to create a new raster map identical to my region settings 
(same resolution, extent, etc.) with the length of roads summed in each 
pixel.
Being a beginner, I tried different ways but got stuck quite rapidely.

Any help appreciated.


Alex