lines of the projective plane will be isomorphic if and only if the removed lines are in the same orbit of the collineation group of the projective plane. These statements hold for infinite projective planes as well. Construction of projective planes from affine planes The affine plane K2 over K embeds into KP2 via the map which sends affine (non-homogeneous) coordinates to homogeneous coordinates, The complement of the image is the set of points of the form . From the point of view of the embedding just given, these points are the points at infinity. They constitute a line in KP2—namely, the line arising from the plane in K3—called the line at infinity. The points at infinity are the "extra" points where parallel lines intersect in the construction of the extended real plane; the point (0, x1, x2) is where all lines of slope x2 / x1 intersect. Consider for example the two lines in the affine plane K2. These lines have slope 0 and do not intersect. They can be regarded as subsets of KP2 via the embedding above, but these subsets are not lines in KP2. Add the point to each subset; that is, let These are lines in KP2; ū arises from the plane in K3, while ȳ arises from the plane The projective lines ū and ȳ intersect at . In fact, all lines in K2 of slope 0, when projectivized in this manner, intersect at in KP2. The embedding of K2 into KP2 given above is not unique. Each embedding produces its own notion of points at infinity. For example, the embedding has as its complement those points of the form , which are then regarded as points at infinity. When an affine plane does not have the form of K2 with K a division ring, it can still be embedded in a projective plane, but the construction used above does not work. A commonly used method for carrying out the embedding in this case involves expanding the set of affine coordinates and working in a more general "algebra". Generalized coordinates One can construct a coordinate "ring"—a so-called planar ternary ring (not a genuine ring)—corresponding to any projective plane. A planar ternary ring need not be a field or division ring, and there are many projective planes that are not constructed from a division ring. They are called non-Desarguesian projective planes and are an active