There is one caveat though. However, internally, elements of a pencil are indistinguishable. By definition, Projective Plane is a pencil of straight lines and a bundle of planes through the same point. When modeled with a projective mapping as above, the plane of the bundle parallel to the observation plane, is called the line at infinity . Each line in the pencil parallel to the observation plane defines a point at infinity .
Parallel lines define the same point at infinity which , naturally, belongs to the line at infinity.
Lines in the projective plane all pass through a point at infinity defined by their direction. Therefore, in Projective Geometry, any two lines intersect. Analytically, Projective Plane is defined with homogeneous coordinates . This is done in the following way. Assume O is placed at the origin of some coordinate system consisting of three vectors e , f , and g .