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I have a shapefile which I want to reproject to lambert conformal conic and set the projection parameters such as the following:

False_Easting, False_Northing, Central_Meridian, Standard_Parallel_1, Standard_Parallel_2, Latitude_Of_Origin

I used to do this easily with ArcMap, but I do not have any access to ArcMap anymore. Also I can't seem to do this with reproject under the vector toolbox in QGIS.

How could I do this in geopandas?

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The solution is easy with geopandas

import geopandas as gpd

Then we need to save a dictionary containing the necessary projection parameters:

dict = {'proj': 'lcc', 'lat_1': 5, 'lat_2': 20, 'lat_0': 14.60004, 'lon_0': 120.98, 'x_0': 0, 'y_0': 0, 'datum': 'WGS84', 'units': 'm', 'no_defs': True}

Then we pass this dictionary to the to_crs() method:

new_proj = polygon.to_crs(dict)

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