Re: Parallel Lines Sometimes Meet
Vinnie Breidenthal
There is a person that takes a calculus course with us, and every time we ask him for something he answers us with I'll do it when two parallel lines meet each other. So I decided to give him a proof of this so that he won't say it anymore (it is annoying).
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 where lines cannot be parallel. These settings help regularize the geometry. For example, spherical geometry takes place on the surface of a sphere. The "lines" in spherical geometry are the "great circles": the circles which have the diameter of the sphere. Note then that two lines always intersect in a "point" (which in spherical geometry is defined as the two points opposite each other on the sphere).
Spherical geometry regularizes plane geometry in several ways. First, it elminates parallel lines: now every two lines intersect in a point, and every two points define a line (exercise!). Second, it unifies the treatment of lines and circles: everything is now a circle, in effect.
So "parallel" does strictly mean two lines that do not meet, but there are ways to eliminate the concept with a suitable geometry. Projective geometry is another very useful but more complex way to do this.
Lines are parallel if they lie in the same plane and they don't intersect. In other geometries, there may be no parallel lines, lines may not have a common point but they may have a common limit point at infinity, or they may just not intersect.
Since Euclidean geometry contains parallel lines, DL2 is false. But Projective geometry accepts DL2 as a postulate. The big question is, "Does there exists a geometry that satisfies the postulates of Projective geometry?" Yes there does.
The creation of such a geometry is really quite clever. You start with a Euclidean plane and you add points to it as follows. Pick any line in the plane. To that line and all lines parallel to it, you add one extra point, a point at infinity. This is a set thing. We are treating a Euclidean line, $l$, as a set of points and we are adding a non Euclidean point $p$ to that set, $l' = l \cup \p\$.
Adding this point to those lines means that those lines are no longer parallel.Define the set of all points at infinity to be the line at infinity. The Projective plane is the Euclidian plane with all points at infinity and the line at infinity added to it. This particular Projective plane can be proved to satisfy DL2.
In the other direction, pick any line, $l$, in the projective plane, $\mathbb P^2$, and remove it. What you end up with is the Euclidean plane, $\mathbb E^2 \cong \mathbb P^2 - \l\$. Some lines will still intersect. Those that intersected at a point on the line that was removed will now be parallel.
Note, as several have commented, your premise is slightly off. What you can convince your friend of is that any two 'straight lines' i.e. geodesics cross on a sphere (not parallel lines), whereas on a plane, for any line there is a special family of lines that never touch it.
In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel.
Though if you are convincing/arguing with someone naive in field of geometry that Parallel line do not meet, ask him why people call him by his/her name and not by Justin Bieber/Selena Gomez? The expected answer would be that Because it is my name and that's it. You got him/her. Actually parallel lines cannot meet at a point or intersect because they are defined that way, if two lines will intersect then they will not remain parallel lines.
If you look at the grid paper with a magnifying glass the lines remain parallel (this is evident when they are more or less on the "top of the glass"), but all the lines meet at the edge of the glass.
(I went to my older brother, who was an engineering freshman at the time, and I told him that parallel lines can meet; but he replied that they cannot because it's an axiom. Some years later I learned that there is a thing called non-Euclidean geometry.)
In his comment @JackyChong has identified the preliminary problem. The definition of "parallel" is clear: lines that don't meet, so there are no parallel lines that meet. The real question is the definition of "line".
For projective geometry one definition is to add a "point at infinity" on each line, and then make the added points into a "line at infinity". With these extra points and lines there are no parallel lines. Two that are parallel in the Euclidean plane share their point at infinity. The railroad track analogy helps here.
In hyperbolic geometry there are multiple lines parallel to a given line through a point not on that line. If you get that far with the "someone who knows no mathematics" you can show him or her the Poincare model.
You cannot convince any mathematician let alone your friend as it is false. They can however grapple with the idea that for purely theoretical calculations we assume that 2 parallel lines can meet at a point in infinity. Should you wish to travel to infinity to prove this point then please send a postcard when you get there.
Someone who knows basic mathematics can understand it easily since according to the definition of parallel lines
Parallel lines: The lines those distance is constant are called parallel lines[see the topmost definition]
Now, as we know the definition the coinciding lines are also parallel since the distance between them is constant or $0$. Hence, as coinciding lines touch each other implies parallel lines also touch.
This statement and its explanation though art (as suggested by Pedro Tamaroff in the comments) and by showing how many statements and proofs are simplified (if the person knows more math) is a good introduction to the idea that thinking on parallel lines as lines meeting at infinity makes sense and can be useful.
In usual geometry (i.e. affine and Euclidean geometry), the above definition or statement does not make sense as parallel lines don't intersect. This can be the definition in the plane, but it is generally (in higher dimensions) a result derived from the definition of parallel lines as lines with the same direction.
However, the reason why we can still make sense of the above statement about intersection at infinity is because affine geometry can be put inside projective geometry. When doing this, the points outside the affine space are called "points at infinity", parallel lines intersect at them and become the same as intersecting lines simplifying enormously many statements and proofs by permitting one to not distinguish cases. An example can be Pappus's hexagon theorem.
In conclusion, don't try to convince or show that parallel lines touch. Just try to explain the usefulness of thinking of parallel lines as lines that intersect at infinity. In the Renaissance, you have many examples of why this a useful statement from the point of view of representing reality and perspectives well; in mathematics, there are many examples of how this is really useful for simplifying statements and proofs in geometry.
In the flat Poincare disk model circle segment geodesic parallels meet tangentially only at infinitely distant points on the boundary of the "horizon" or boundary circle. The lines can be seen to touch and move on the boundary seen in the Wiki link.
This dictionary is not quite 1-to-1, because there are lines (and one plane) through the origin in $\mathbbR^3$ that are parallel to the picture plane, and therefore do not correspond to any points or to a line in the picture plane. These are points and this line can be considered "at infinity", but really all this means is that they do not have images in the picture plane.
Now that we have this dictionary set up, let's think about what parallel lines in the picture plane are. As the image below shows, parallel lines in the picture plane correspond to two planes in $\mathbbR^3$. (More precisely, lines $f$ and $g$ are "parallel" in the blue picture plane, and correspond to the orange and yellow planes through the origin.)
Of course if you choose a different picture plane, then the lines will no longer appear parallel, and their intersection will be plainly visible as a point in the new picture plane. The final image below shows this. In that image, the orange and yellow planes are the same ones as before, but the blue picture plane has been moved and reoriented; the images of the two planes are now non-parallel lines.
Visually or conceptually parallel lines converge over an extremely large distance, they NEVER intersect. I don't care how many dimensions you are working in or what geometrical space you are working in, if you have Lines: L1 and L2 you can always abstract a corresponding vector from each L1:v1 and L2:v2 and if you apply the arcos of the cosine angle between two vectors where the cosine angle is the dot product of v1 & v2 divided by the product of their magnitudes you will then know if they are parallel or not.
I think that the better or correct statement for this supposition would be: "If one is to place a point at the point of Convergence". I say this because parallel lines will always be parallel and always have the same amount of separation at any given point on those lines. Now because of how our eyes work and how we interpret light signals visually parallel lines will Converge at an Extremely Large Distance. If one was to say place a point on the lines at a place that is approaching Infinity it would be a better statement than placing a point at infinity but even this is still incorrect. Why?
If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
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