This is illustrated below. The fact to note here is that the polars of the points on a line form a pencil in a point, which is the polar of that line. The situation is self-dual. In three dimensions we illustrate the same principle but with a sphere and a point. Similarly to the two-dimensional case, if we take the polar planes of all the points in a plane, they all contain a common point which is the pole of that plane.
Lines are now self-polar.
When counter space is studied this property of points and planes is used to conceptualise the idea of a negative space, as we reverse the roles of centre and infinity. In a plane the ideal points form an ideal line, and in space they form an ideal plane or plane at infinity. This is known as affine geometry . A further special case is possible where the volume of objects remains invariant, which is known as special affine geometry . Finally a further specialisation ensures that lengths and angles are invariant, which is metric geometry , so called because measurements remain unaltered by its transformations.