The empty set may be the line at infinity considered as a double line, a ( real ) point is the intersection of two complex conjugate lines and the other cases as previously mentioned.
32.
Hyperbolas intersect the line at infinity in two distinct points and the polar lines of these points are the asymptotes of the hyperbola and are the tangent lines to the hyperbola at these points of infinity.
33.
Just as the Riemann sphere needs a north pole point at infinity to close up the complex projective line, so a line at infinity succeeds in closing up the plane of dual numbers to a cylinder.
34.
In the case of a parabola, that is, when, there is no center since the above denominators become zero ( or, interpreted projectively, the center is on the line at infinity .)
35.
The Euclidean plane is embedded in the real projective plane by adjoining a line at infinity ( and its corresponding points at infinity ) so that all the lines of a parallel class meet on this line.
36.
Historically there was a process by which projective geometry added more points ( " e . g . " complex points, line at infinity ) to simplify the geometry by refining the basic objects.
37.
To obtain the extended Euclidean plane, the absolute line is chosen to be the line at infinity of the Euclidean plane and the absolute points are two special points on that line called the circular points at infinity.
38.
Somewhat less intuitively, over the complex numbers, an ellipse intersects the line at infinity in a " pair " of points while a parabola intersects the line at infinity in a " single " point.
39.
Somewhat less intuitively, over the complex numbers, an ellipse intersects the line at infinity in a " pair " of points while a parabola intersects the line at infinity in a " single " point.
40.
However, if one were to consider the line at infinity as the directrix, then by taking the eccentricity to be a circle will have the focus-directrix property, but it is still not defined by that property.
