Gauss' hemisphere is another compactification of the plane R2. The Riemann sphere is much better-known, and probably more useful, but Gauss' hemisphere is useful for studying some aspects of projective geometry.
To perform the compactification, place a hemisphere on the plane, much like you'd place a grapefruit on a plate. From every point x on the plane, draw a line to the centre of the
grapefruithemisphere. The line intersects the hemisphere at one point -- that point is the image of x on the hemisphere.
Lines on the plane are transformed to appropriate halves of great circles on the hemisphere. In particular, they remain geodesics! Even nicer is that angles are kept.
Points on the "rim" of the hemisphere correspond to no points of the plane (they are the points added to compactify the plane). Parallel lines in the plane indicate a "direction". On the hemisphere, their corresponding great circles all pass through the same 2 antipodal points on the rim. Great circles corresponding to parallel lines intersect at 2 points on the hemisphere; great circles corresponding to intersecting lines intersect at 1 point on the hemisphere.
Since parallel lines indicate direction, and since points on the rim correspond to such sets of parallel lines, we can consider points on the rim (sometimes called "infinite points" or "points at infinity", but such terminology is often misleading, as the plane doesn't contain these points!) to indicate direction.
It can be interesting to consider what happens when the hemisphere is tilted in some fashion prior to being set on the plane, or the hemisphere's relationship to the projective plane.