Math Forum - Ask Dr. Math
As Daniel Rust notes, the definition of parallel is that two lines don't meet. What some people are trying to point out as examples are situations. Do parallel lines intersect at infinity? Is this in projective space?. Renzo Cavalieri. Where parallel lines meet. Q: How many points of intersection do two distinct lines in the plane have? . three-dimensional space. P. 2 = R. 3.
At first glance it would seem that the parallel postulate ought to be a theorem deducible from the other more basic postulates, rather than something that has to be assumed separately. For centuries mathematicians tried to prove it, but always failed. Then you can draw a unique line through P that is perpendicular to that line segment.
In some sense this is parallel to L because the two angles in the picture below are right angles, but how can one prove from this that this means the lines never intersect? We know they don't in our familiar mental picture of what an infinite flat plane looks like, but is that fact a logical necessity deducible from the other postulates?
Eventually it was discovered that the parallel postulate is logically independent of the other postulates, and you get a perfectly consistent system even if you assume that parallel postulate is false. This means that it is possible to assign meanings to the terms "point" and "line" in such a way that they satisfy the first four postulates but not the parallel postulate.
These are called non-Euclidean geometries. Projective geometry is not really a typical non-Euclidean geometry, but it can still be treated as such. In this axiomatic approach, projective geometry means any collection of things called "points" and things called "lines" that obey the same first four basic properties that points and lines in a familiar flat plane do, but which, instead of the parallel postulate, satisfy the following opposite property instead: Any two lines intersect in exactly one point.
Depending on how one words the other axioms, they may need some slight modification too. Using only this statement, together with the other basic axioms of geometry, one can prove theorems about projective geometry. Many of them are the same as ordinary geometry; the big difference is that there is no such thing as a pair of parallel, non-intersecting lines in projective geometry.
One interesting fact is worth mentioning: That is, any statement about points and lines would still be true even if you replaced all occurrences of the word "point" with the word "line", and vice versa. For instance, the basic axiom that "for any two points, there is a unique line that intersects both those points", when turned around, becomes "for any two lines, there is a unique point that intersects i. There is a complete duality between points and lines in projective geometry. Now, if this approach were all there was to projective geometry, it would be little more than an intellectual curiosity.
All it means is that it logically consistent for there to be concepts called "points" and "lines" that satisfy the axioms of geometry with the projective axiom in place of the parallel postulate. It says nothing about whether such concepts would be interesting, relevant, or have any relation whatsoever to the normal concepts of lines and planes in Euclidean geometry. However, there are other approaches that reveal the connection: Take each line of ordinary Euclidean geometry and add to it one extra object called a "point at infinity".
Do this in such a way that the same extra object is added to parallel lines so that the extended lines now intersectwhile different extra objects are added to non-parallel lines so that the extended lines don't intersect more than once. For example, you could let f l be the slope of l a real number, or the symbol " infinity " if l is vertical.
Question Corner -- Do Parallel Lines Meet At Infinity?
Alternatively, you could let f l be the counterclockwise angle from some fixed reference line to l. The precise method you use is unimportant.
The lines of projective space are lines l in Euclidean space together with the extra object f l attached. In addition, the collection of all the extra objects together is also called a line in projective space called the line at infinity. This definition satisfies all the axioms of projective geometry.
For example, here's a proof that any two of these "lines" L and L' intersect in exactly one "point": If one of L and L' say, L is the line at infinity and the other L' is not, then they intersect at exactly one point because by definition L' contains exactly one point at infinity.
If neither L nor L' is the line at infinity, then each of them consists of an ordinary Euclidean line together with one point at infinity.
Parallel lines do not meet at a point.
That is, we can write and where l and l' are Euclidean lines. If l and l' intersect at a point p then f l does not equal f l' since f l only equals f l' when l and l' are parallelso p is the one and only intersection point of L and L'. This view of projective geometry makes it relatively easy to answer questions of concurrence and collinearity. For example, what does a collection of concurrent lines in projective space look like?
It is one of three things: Also, the points at infinity are all collinear in projective space. Although this view of projective geometry helps answer your question, it's still a little artificial, with all this talk of just "adding extra objects at infinity". There are two other, much more natural, ways of looking at it.
Lines In Space Projective geometry can be thought of as the collection of all lines through the origin in three-dimensional space. That is, each point of projective geometry is actually a line through the origin in three-dimensional space. The distance between two points can be thought of as the angle between the corresponding lines. A line in projective geometry is really a family of lines through the origin in three-dimensional space. To see how this ties in with the previous view of projective geometry, let P be a horizontal plane in space that does not pass through the origin.
As can be seen in the picture below, every line through the origin passes through exactly one point on P, except for the horizontal lines. So there is a one-to-one correspondence between the points on the ordinary plane P, and some of the points in projective space namely, all non-horizontal lines through the origin in 3-d space. If you are talking about ordinary lines and ordinary geometry, then parallel lines do not meet. 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.
Line at infinity - Wikipedia
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. There is no such thing as infinity, and it is wrong to say that parallel lines meet at infinity.