PARALLEL LINES INTERSECT AT INFINITY

Parallel Lines Intersect At Infinity

Do Parallel lines intersect at infinity?

Introduction

The answer to the question depends on whether the geometry of a deal, and that the "point" and "lines" mean. If we talk about conventional lines and conventional geometry, parallel lines do not intersect (Coxeter 2003). For example, the line x = 1 and the line x = 2 does not match at any time, as the X coordinates of the point, there can be no 1 and 2 at the same time. In this context, there is no such thing as "infinity" and parallel lines do not intersect.

Nevertheless, we can construct other forms of geometry, the so-called non-Euclidean geometry. For example, you can take the usual points of the plane and give them an extra item called "infinity" and consider all lines to include this additional item (Silvester 2001). In this context, there is one "infinity" where all lines converge. In geometry, as it is, all lines intersect at infinity, in addition to any end point, where they might happen to meet.

Or, you can attach not only one extra point, but a whole set of additional points, one for each direction. Then we can consider two parallel lines meet at an additional point corresponding to their general direction, while two non-Parallel lines intersect at infinity, but meet only at the usual end point of intersection. This is called projective geometry, and described in more detail in the answer to another question.

Hyperbolic Geometry

Hyperbolic Geometry is a form of non-Euclidean geometry. It supports all the principles of Euclid, except the parallel postulate, which says that if the line [PIC] and point [PIC] not [PIC], there is exactly one line through the [PIC], which does not intersect [PIC] (Coxeter 2003 ). Hyperbolic geometryinstead the following modified postulate: ...