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When talking to someone who knows basic mathematics but not really in-depth, how would you explain that parallel lines can touch?

I am referring to Non-Euclidian/ Projective Geometry.

Edit: Why do parallel lines in the Euclidean plane (correspond with lines that) cross in the projective plane?

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    $\begingroup$ You need a definition for parallel lines. $\endgroup$ – Jacky Chong Jan 9 '17 at 0:34
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    $\begingroup$ Show them any Renaissance era drawing and mumble about infinity. $\endgroup$ – Pedro Tamaroff Jan 9 '17 at 5:04
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    $\begingroup$ It is an axiom of the projective plane that no two distinct lines are parallel, and it sounds a bit like you've taken this and garbled it. $\endgroup$ – rschwieb Jan 9 '17 at 5:33
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    $\begingroup$ Draw 2 parallel lines on a piece of paper - then fold it. Parallel lines CAN touch - if you distort the space they occupy. Folding a paper represents distorting a 2D field through a third dimension. $\endgroup$ – Alan Campbell Jan 9 '17 at 12:51
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    $\begingroup$ You cannot convince someone with something which is not true. $\endgroup$ – Vidyanshu Mishra Jan 9 '17 at 13:37

15 Answers 15

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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.

You might be thinking about Projective Geometry where a "point at infinity" is added to every family of parallel lines and the set of all points at infinity is called the "line at infinity".

Addendum


Typically, in formal geometry, points, lines, and planes are not defined. But postulates define their "baseline" behavior. The postulate that every geometry seems to agree on is the one that states

L1: Given two distinct points in a plane, there is exactly one line in that plane that contains them.

The "dual" of that postulate is

DL1: Given two distinct lines in a plane, there is exactly one point in that plane that belongs to both lines.

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.

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    $\begingroup$ Actually there is no infinity in Projective Geometry; all points and all lines are created equal. The line at infinity is an artefact of a certain constructions of projective spaces, namely starting from Euclidean spaces. $\endgroup$ – Marc van Leeuwen Jan 9 '17 at 6:16
  • $\begingroup$ @MarcvanLeeuwen -Yes, it's a model of a projective space. $\endgroup$ – steven gregory Jan 9 '17 at 6:37
  • $\begingroup$ I think that your supposition is false: infinity is a concept and is considered NAN (not a number). Therefore you can not place anything at infinity. In calculus you do have graphs, curves and limits that approach infinity but you can never reach it. Consider the set of All Real Numbers (IR) with the operation applied to each consecutive element {R: Xi, Xi + 1, ... + Xn, Xn+1, ... }. The true statement is that Parallel Lines Converge at an extremely large distance as their limits approach infinity. $\endgroup$ – Francis Cugler Jan 10 '17 at 10:09
  • $\begingroup$ @FrancisCugler - I can't place anything from $\mathbb R^2$ at infinity but I can add new points to $\mathbb R^2$, getting $\mathbb P^2$, projective 2-space. $\endgroup$ – steven gregory Jan 10 '17 at 20:13
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    $\begingroup$ @StevenGregory Isn't that "disturbing" thing something we experience in real life all the time? Our visual field is a projection of 3-space onto a 2-dimensional surface (the retina in the back of our eye) so this is part of how we perceive the real world. (Concrete example: railroad tracks in the desert.) $\endgroup$ – mweiss Jan 18 '17 at 15:30
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I would say if we stand next to each other and both walk due north, we eventually bump into each other near the north pole

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.

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    $\begingroup$ In geodesy, "parallels" are taken to be lines of latitude, which do not meet and indeed are mostly not straight lines or great circles $\endgroup$ – Henry Jan 9 '17 at 0:40
  • $\begingroup$ @Henry you are right, of course. But the spirit of OP's question seemed clear. $\endgroup$ – spaceisdarkgreen Jan 9 '17 at 0:45
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    $\begingroup$ Rather than stand next to each other, start in London and new York, to get your message across better. Oh, and don't forget Boaty McBoatface and suitable clothes, or you are going to be very wet and very cold :-) $\endgroup$ – Mawg Jan 9 '17 at 8:47
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    $\begingroup$ According to what definition are great circles parallel? $\endgroup$ – gerrit Jan 9 '17 at 13:49
  • $\begingroup$ You could use an orange and a knife to make the experiment last slightly less. $\endgroup$ – anderstood Jan 18 '17 at 22:25
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As I said in comments, you can't convince someone with something which is not true.

IMO if someone knows basic geometry then he he must be aware of the fact that :

Parallel lines do not meet at a point.

This section of Wikipedia worth a lot here:

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.

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    $\begingroup$ You know that when you copy a link from Google, you actually copy half your digital footprint with it, right? Look at the link you've pasted here. It's not a link to Wikipedia. It's a link to a Google search. How should I know if you didn't trick me to look at something else instead? (I don't think that you have, but that is something that someone could potentially do.) $\endgroup$ – Asaf Karagila Jan 9 '17 at 8:54
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    $\begingroup$ Click edit, and look at the link that you have put to the Wikipedia article. The link is not to Wikipedia. It is a link to google.co.in, and while I'm sure that you're not a malicious person who is putting links that will direct me to searches that I have no intention of making, this method is bad. If you open the link, and then copy the link directly from that page (or install a userscript which sanitizes these links for you), then you'll see that the link is in fact some address into wikipedia.org instead. $\endgroup$ – Asaf Karagila Jan 9 '17 at 11:17
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    $\begingroup$ Yes. That link is fine. $\endgroup$ – Asaf Karagila Jan 9 '17 at 11:25
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When I was 16 or so I was bored trying to solve my math homework, so I played with a magnifying glass, and I noticed something interesting:

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.)

 

Note, by the way, this might require you to think about the glass as infinite.

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    $\begingroup$ There is a BC comic that I tried to find but couldn't. In it, BC is walking while dragging a two pronged branch behind him. After a while, he stops an looks behind him, He sees two "parallel" lines intersecting in the distance. Puzzled, he walks back to where he started. He looks down and the lines are as far apart as they had always been. He turns around.... $\endgroup$ – steven gregory Jan 9 '17 at 6:51
  • $\begingroup$ Hah, cute. What's BC, though? $\endgroup$ – Asaf Karagila Jan 9 '17 at 7:14
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    $\begingroup$ @AsafKaragila, this $\endgroup$ – Vidyanshu Mishra Jan 9 '17 at 7:30
  • $\begingroup$ eventually that forked stick will wear down and the lines will meet $\endgroup$ – Jasen Jan 9 '17 at 13:13
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    $\begingroup$ @Steven: Were these written in parallel? $\endgroup$ – Asaf Karagila Jan 9 '17 at 22:14
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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".

Geodesic is the most natural for geometry on the sphere. Then in this geometry there are no parallel lines.

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.

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  • $\begingroup$ " The real question is the definition of "line." How about considering the axioms of finite geometry and then drawing the Fano plane. Thereafter, you can consider whether there can be such behavior in a geometry where lines have infinitely many points. $\endgroup$ – Airymouse Jan 9 '17 at 14:37
  • $\begingroup$ It astonishes me that people use the terminology At Infinity. I tend to think that this in itself is Impossible because Infinity is considered a NAN (Not A Number). This is also evident because of the use of Limits in functions as they Approach Infinity. So how is it possible to place a point at infinity? You can't. I think the correct terminology would be to say: Place a point at the point of Convergence; where Lines will Converge at an extremely large distance. $\endgroup$ – Francis Cugler Jan 10 '17 at 10:19
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I would use two meridians on Earth that touch at the poles.

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    $\begingroup$ Who claims meridians (lines of longitude) are parallel? $\endgroup$ – Henry Jan 9 '17 at 0:41
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    $\begingroup$ @Henry I know in geodesy meridians are not considered parallel, but the OP was looking for an example to illustrate non-Euclidian geometry, and lines on a curved surface are usually a first way to get one's intuitions going on something like that. $\endgroup$ – Bram28 Jan 9 '17 at 0:47
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This is a matter of point of view :)

enter image description here

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You can actually speak of trains. Rail ways are pretty suggestive.

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    $\begingroup$ Tracks rather than trains? $\endgroup$ – Henry Jan 9 '17 at 0:45
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    $\begingroup$ railwayinfo.net/wp-content/uploads/2016/03/… $\endgroup$ – math.h Jan 9 '17 at 0:47
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    $\begingroup$ There's even a convenient line at infinity where such things meet: The horizon! $\endgroup$ – Milo Brandt Jan 9 '17 at 1:00
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    $\begingroup$ When my school teacher said this analogy I was not convinced because I thought this is just an optical phenomenon. It didn't make sense until I studied projective geometry at university. $\endgroup$ – Oriol Jan 9 '17 at 1:22
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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.

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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.

See the video here also

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I would draw two parallel lines on a piece of paper. Then I would bring all the edges together into one point and I would show that the parallel lines would touch into that point.

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A sentence said to me by my mother when I was a kid and that I didn't understand at that time was:

Parallel lines are lines that intersect at infinity.

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 art

This, as you can also see in Yves Daoust's answer, can be seen whenever you look a straight road going to the horizon. For example, see Wikimedia Commons for the source, Source: Wikimedia Commons
The above image shows the reason why this conception was in the beginning a concept originated in art as it was really useful for representations of how we see things. An striking application of this can be seen in the Santa Maria presso San Satiro, where projective geometry was used emulating an absent space in a church. However, there are many other examples.

In geometry

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

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.

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  • $\begingroup$ Regardless if it is art or not, but looking at that road, the point at an extremely large distance that appears to be the Convergence Point might only be a few miles, yet if you travel down that road to that point that Convergence Point travels with you as you move. It has to do with our Depth Perception and how we visualize things. However the two sides of the roads are for the most part Parallel. $\endgroup$ – Francis Cugler Jan 10 '17 at 10:24
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In Elliptic geometry the supposition is false.

In Euclidean geometry parallel lines "meet" and touch at infinity as their slope is same.

In flat Hyperbolic geometry parallel lines can also touch but only at at infinity.

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.

Poincare Disk_MathWorld

So also in the half-plane model semi-circular geodesics meet at point at $\infty$, on the x-axis. An animation can be also found elsewhere semi-circles touch on x-axis while changing semi-circle size during movement.

PoincareHalfPlane

Both examples are touching parallel lines, touching tangentially /asymptotically at point infinity.

http://math.etsu.edu/multicalc/prealpha/Chap3/Chap3-8/part4.htm

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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.

v1<> v2<> mag1 mag2 // where mag1 and mag2 are the magnitudes of v1 and v2
cosAngle = (v1 dot v2) / (mag1 * mag2) 
Determining Value = arcos( cosAngle )
If Determining Value == 0, 180, 360 if in Degrees 
 or 0, PI or 2PI in radians 
then the vectors Are Parallel
Otherwise they are not 

Edit

The same thing can be said about ARCs if two arcs belong to two different circles with different radius lengths where both radii have the same starting position; another words concentric circles as in a bulls eye in a dart board or an archery board or if they are separated at a distance greater than the sum of their radii else if either of their radius are different lengths and the do not have the same center point then they can have an intersection point. There may be cases where they don't have the same center point and still don't intersect.

Of the next few images only the concentric circles and ellipses would be considered parallel, the rest either intersect or don't.

These do not intersect

Concentric Circles Non Concentric Internal Circles Non Concentric External Circles

These do intersect

Outer Tangent Circles Inner Tangent Circles Non Tangent Intersecting Circles

The only one of these that can be considered parallel are the Concentric Circles since each arc has the same distances of separation from another circle.

The Same can be applied for ellipses - Except they have two radii of different measure, one for the x-axis and one for the y-axis.

Concentric Ellipses

Edit

I've noticed that many have used the terminology or the statement saying: Place a Point At Infinity either in their comments or in their answers. To me this is bothersome and I'm astonished by it. Why? I think that it is impossible to place a point at infinity because Infinity is a concept and by definition it is considered to be a NAN (Not A Number). This concept of Infinity is applied in mathematics with the properties of functions and their limits. This is why we say that such a function or equation has a limit that approaches infinity.

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?

Consider this: Imagine being out in the flat plains of the Midwestern part of the United states with very minimal hills where you can practically see the full horizon without being obscured: Now look down a railroad track that happens to be practically straight with very minimal bends. With our eyes on an extremely clear day (less moisture in the air) and we can see down those tracks for approximately 20 - 40 miles. The point of convergence is at or just shy of what we see as being the Horizon, yet we know that these tracks can go on for hundreds or even thousands of miles. This is far from the approach of infinity. We can perceptually conceive the idea of 100,000s of thousands of miles, even millions or billions of miles and yet these are but a fraction of a fraction of the approach of infinity.

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  • $\begingroup$ I'm not new to the stack exchange communities for I've been a member of Stackoverflow for a few years, but I'm new to here and just letting others know that I do not know the syntax for the math parsing tags. I will learn them as I go so please take into consideration that this is my first answer here. $\endgroup$ – Francis Cugler Jan 9 '17 at 20:10
  • $\begingroup$ Have you checked out Homogeneous Coordinates? $\endgroup$ – steven gregory Jan 18 '17 at 19:12
  • $\begingroup$ Geometry is also a concept. There was a time when all of the "smart" mathematicians agreed on a set of rules (postulates) that must be true about geometry, wrote them down, and started proving things that were consequences of those rules. Yes infinity is a concept. But most of the controversy about geometry has to do with what happens when lines "wander around" out there. Turns out there is more than one possible answer. $\endgroup$ – steven gregory Jan 20 '17 at 5:55
  • $\begingroup$ @StevenGregory Yes I know about Homogeneous Coordinates with Projective Geometry. I work with 3D Graphics Programming. It makes it easier to do affine transformations using matrices such as doing a MVP (Model to View to Projection) Matrix multiplication. Which takes the coordinates space vertices, texture coordinates, normals etc., and translates them into View Space (sometimes called world space) and then finally to Projection Space which is also called Screen Space. And you also have to account for the perspective divide otherwise everything will be skewed looking. (more...) $\endgroup$ – Francis Cugler Jan 21 '17 at 5:48
  • $\begingroup$ @StevenGregory It is the process of how a 3D model can be translated to a 2D plane and still look 3D. My argument is even though you have a "vanishing point" or what is called the point at "infinity" in projective geometry with homogeneous coordinates, in truth Parallel lines never intersect. They do however Converge. It is the appearance or illusion that they come together and through some combinations of mathematical inductions we can translate and map the points accordingly and the points converge to a single point, but in truth it is the limits of those functions that approach that point. $\endgroup$ – Francis Cugler Jan 21 '17 at 5:53
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One way to explain this is to think of the Euclidean plane as a picture of (part of) an ambient 3-dimensional world. This corresponds to how our vision actually works: light from the three-dimensional world is projected through the lens of an eyes onto the surface at the back of our eye.

In more detail: Place a plane anywhere in three-dimensional space (as long as it doesn't pass through the origin). Call this the "picture plane". Now any line through the origin in $\mathbb{R}^3$ pierces the picture plane in exactly one point (as long as the line it is not parallel to the picture plane). Any plane through the origin pierces the picture plane in exactly one line (again, as long as the plane is not parallel to the picture plane). See below (in which the picture plane is shown in blue).

enter image description here enter image description here

Now we have a dictionary: $$\textrm{point in picture plane} \longleftrightarrow \textrm{line through origin in 3-space}$$ $$\textrm{line in picture plane} \longleftrightarrow \textrm{plane through origin in 3-space}$$

This dictionary is not quite 1-to-1, because there are lines (and one plane) through the origin in $\mathbb{R}^3$ 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.

You can experience this visually by holding your finger at arm's length in front of and a few inches above. As you draw your arm closer (while maintaining its height) the location of your fingertip seems to get "higher" or "farther away" in your field of view, as your eyes have to strain more and more to see it. Eventually when the fingertip is directly above the eye, the finger "disappears". It's not really gone, of course, but the "line of sight" from the fingertip to the lens of your eye is parallel to the retina, so that the fingertip cannot be seen. Perceptually, the fingertip has receded "infinitely far away" in the picture plane -- but it is actually just a couple of inches away in 3-space.

However, we all know what to do in that circumstance: turn your head! This corresponds to choosing a new picture plane with a different orientation. Now the point that was "at infinity" snaps into view and is revealed as an ordinary point in the new picture plane (and at the same time points that were previously in view have no disappeared).

Note that a statement like

given any two distinct points in the picture plane, there exists a unique line in the picture plane through the given points

which describes a true fact about the geometry of the picture plane corresponds, via this dictionary, to

given any two distinct lines through the origin in 3-space, there exists a unique plane through the origin containing both lines

which is a true statement about 3-space.

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 $\mathbb{R}^3$. (More precisely, lines $f$ and $g$ are "parallel" in the blue picture plane, and correspond to the orange and yellow planes through the origin.)

enter image description here

Those planes are not parallel; they intersect, but the intersection is a line that does not pierce the picture plane. (In this illustration, the line of intersection happens to be the $y$-axis.) That line in $\mathbb{R}^3$ corresponds to a point "at infinity".

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.

enter image description here

In summary:

  • If the 2-dimensional plane is understood as an image of an ambient 3-dimensional space, then a "point" is just the image of a line through the origin and a "line" is just the image of a plane through the origin
  • Some "points" (and the "line" containing that point) will have no image in the picture plane, which we describe by saying that they are "at infinity"
  • However being "at infinity" is an artifact of which picture plane we use; changing picture planes changes which points are at infinity
  • Furthermore whether two "lines" are parallel is an artifact of the picture plane we use to view them; lines that seem parallel from one perspective are non-parallel from a different perspective
  • The connection between the last two bullets is that "lines" that appear parallel in one picture plane do intersect, but the point of intersection is a point "at infinity" relative to the picture plane; change perspectives and the point is no longer at infinity, and the lines are no longer parallel.
  • All of the above corresponds more-or-less exactly with how the human eye works, which is why perspective art can do such a good job at fooling the eye and creating the illusion of depth on a flat surface.
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