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Do Parallel Lines Meet At Infinity?

Asked by a student at St-Joseph Secondary School on October 5, 1997:
Could you help me prove that parallel lines meet at infinity or that infinity begins where parallel lines meet.

I am curious. Could this ever happen?

The answer to the question depends on exactly what kind of geometry you are dealing with and what "points" and "lines" mean.

If you are talking about ordinary lines and ordinary geometry, then parallel lines do not meet. For example, the line x=1 and the line x=2 do not meet at any point, since the x coordinate of a point cannot be both 1 and 2 at the same time.

In this context, there is no such thing as "infinity" and parallel lines do not meet.

However, you can construct other forms of geometry, so-called non-Euclidean geometries. For example, you can take the usual points of the plane and attach to them an additional point called "infinity" and consider all lines to also include this additional point. In this context, there is a single "infinity" location where all lines meet. In a geometry like this, all lines intersect at infinity, in addition to any finite point where they might happen to meet.

Or, you could attach not just one additional point, but a whole collection of additional points, one for each direction. Then you can consider two parallel lines to meet at the extra point corresponding to their common direction, whereas two non-parellel lines do not intersect at infinity but intersect only at the usual finite intersection point. This is called projective geometry, and is described in more detail in the answer to another question.

In summary, then: in usual geometry, parallel lines do not meet. There is no such thing as infinity, and it is wrong to say that parallel lines meet at infinity.

However, you can construct other geometric systems, whose "points" include not only the points of familiar geometry (describable as coordinate pairs (x,y)), but also other objects. These other objects can be constructed in various ways, as described in the discussion of projective geometry. In these other geometric systems, parallel lines may meet at a "point at infinity". Whether this is one single point or different points for different classes of parallel lines, depends on the particular geometric system you are considering.

You may also be interested in our answers and explanations page, which contains a discussion of the question does infinity exist?

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