In a pappian projective plane ( one defined over a field ), all conic sections are equivalent to each other, and the different types of conic sections are determined by how they intersect the line at infinity, denoted by ?.
42.
The pencil of circles of equations a ( x ^ 2 + y ^ 2-1 )-bx = 0 degenerates for a = 0 into two lines, the line at infinity and the line of equation x = 0.
43.
In order to turn this process into a correlation, the Euclidean plane ( which is not a projective plane ) needs to be expanded to the extended euclidean plane by adding a line at infinity and points at infinity which lie on this line.
44.
When extending the concept of line to the line at infinity, a set of "'parallel planes "'can be seen as a " sheaf of planes " intersecting in a " line at infinity ".
45.
When extending the concept of line to the line at infinity, a set of "'parallel planes "'can be seen as a " sheaf of planes " intersecting in a " line at infinity ".
46.
There are advantages in being able to think of a hyperbola and an ellipse as distinguished only by the way the hyperbola " lies across the line at infinity "; and that a parabola is distinguished only by being tangent to the same line.
47.
Geometrically, the line at infinity is not special, so while some conics intersect the line at infinity differently, this can be changed by a projective transformation pulling an ellipse out to infinity or pushing a parabola off infinity to an ellipse or a hyperbola.
48.
Geometrically, the line at infinity is not special, so while some conics intersect the line at infinity differently, this can be changed by a projective transformation pulling an ellipse out to infinity or pushing a parabola off infinity to an ellipse or a hyperbola.
49.
If two pairs of opposite sides are parallel, then all three pairs of opposite sides form pairs of parallel lines and there is no Pascal line in the Euclidean plane ( in this case, the line at infinity of the extended Euclidean plane is the Pascal line of the hexagon ).
50.
If the conic is non-degenerate, the conjugates of a point always form a line and the polarity defined by the conic is a bijection between the points and lines of the extended plane containing the conic ( that is, the plane together with the points and line at infinity ).
