Q: Geometry vs geography for line-to-polygon intersection I have two WGS84 NAD83 shapefiles, one polygonal, the other containing lines:
Geometry: Polygon
Feature Count: 1294
Extent: (-79.761059, 44.991143) - (-57.105486, 62.582865)
Layer SRS WKT:
GEOGCS["GCS_North_American_1983",
    DATUM["North_American_Datum_1983",
        SPHEROID["GRS_1980",6378137.0,298.257222101]],
    PRIMEM["Greenwich",0.0],
    UNIT["Degree",0.0174532925199433]]

Geometry: Line String
Feature Count: 427227
Extent: (-79.882212, 44.929534) - (-55.834430, 62.427856)
Layer SRS WKT:
GEOGCS["GCS_North_American_1983",
    DATUM["North_American_Datum_1983",
        SPHEROID["GRS_1980",6378137.0,298.257222101]],
    PRIMEM["Greenwich",0.0],
    UNIT["Degree",0.0174532925199433]]

which I export to PostGIS 2 using the -G flag, to obtain geographies:
$ shp2pgsql -G -I -s 4326 polygons | psql ...
$ shp2pgsql -G -I -s 4326 lines | psql ...

However when I perform a query to obtain their intersection:
select * from polygons p, lines l
where p.geog && l.geog 
and st_intersects(p.geog, l.geog)

it runs for a very, very long time, and doesn't yield the same results as the one performed over the equivalent tables, but with geometries instead (i.e. shp2pgsql without the -G flag), which runs very rapidly. According to the docs, st_intersects should play well with both types, so why this difference?

A: The results may very legitimately be different.
Interpretations on a plane are not the same as interpretations on a sphere, and that's what you're seeing when you compare geometry to geography calculations.

A: From the documentation:

For geography, this function has a distance tolerance of about 0.00001
  meters and uses the sphere rather than spheroid calculation.

Unless the CRS you are using for your geometries also uses a sphere, the result will always be different.

A: The Postgis documentation notes that SRID 4326 is mandatory for geography, and calculations will take longer because they are done on a sphere instead of a plane as would be done for projected coordinates.
