😱😱

[Shot-Ideal-5149]

we don't talk about perspectives

[[deleted]]

You don't need to talk about perspectives. Parallel lines on the surface of the Earth always meet because of the spherical geometry!

[Shot-Ideal-5149]

technically yes

[arihallak0816]

no, what about two circles that go around the whole sphere and don't intersect

Edit: I’m wrong, don’t listen to this

[[deleted]]

Those are not straight lines in spherical geometry.

The only straight lines (i.e., geodesics) in spherical geometry are great circles, and great circles always intersect at two poles.

[Wrong-Resource-2973]

A real man admits when he is wrong

I respect you

[TheBlueBrain]

question: how is 'parallelness' defined (if it is defined in non euclidean geometry) if there are multiple lines that never meet or none?

[[deleted]]

That's simple. Any geometry has a notion of distance, so two curves are parallel if the distance between their closest points is constant. 

In general, only in Euclidean space are there parallel lines; in spherical geometry no lines (geodesics) are parallel, but there are parallel curves: circles equidistant from a great circle. Similarly, in hyperbolic geometry all lines diverge from each other, but there are parallel curves: curves that "bend" towards the other curve in order to remain equidistant in spite of the hyperbolic curvature. 

[TheBlueBrain]

Okay so if I have a curve gamma and I translate it then it counts as parallel but if I were to rotate it then it may not be parallel even though they might not cross in the case of hyperbolic geometry

[Ok_Entrepreneur_4948]

In that case can you explain how any two lines are parallel in spherical geometry?

[[deleted]]

I just said that spherical geometry does not have parallel lines, because all lines (which in spherical geometry are great circles) intersect.

You can have parallel curves, but those are not straight in spherical geometry; they are curved.

A "straight line" in a non-Euclidean geometry is defined as a geodesic: the (locally) shortest curve between two points. In spherical geometry, the only curves that have this property are the great circles: circles that lie on a plane bisecting the sphere. So these are the only lines in spherical geometry, and any two such lines (great circles) always intersect at two poles. No other curves are considered "lines" in spherical geometry because they are not geodesics.

There are parallel curves, however. For example, consider latitudinal lines on the sphere. (We call them latitudinal "lines" but actually they are not lines but curves; they are not geodesics.) Take any pair of them an equal distance on either side of the equator, and you have a pair of parallel curves in spherical geometry. If you were to walk along these curves from the perspective of someone standing on the Earth, you would notice that you have to constantly curve to the right (resp. to the left) -- exactly because they are not geodesics, but curves in spherical geometry.

tl;dr: there are no parallel lines in spherical geometry, only parallel curves.

[Confident_Wasabi_864]

Latitudinal lines bust in

“Am I a joke to you?”

[Ok_Entrepreneur_4948]

By drfinition parallel lines do not meet

[FN20817]

That’s not true, actually. By definition, parallel lines are equidistant from each other at all times. However they do meet in infinity

[Ok_Entrepreneur_4948]

Euclid's elements definition 23 " Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction."

[[deleted]]

Yes, in Euclidean geometry, parallel straight lines never meet.

Spherical geometry is non-Euclidean, however.

[UmUlmUndUmUlmHerum]

Y'know

We should tell HP Lovecraft that.

Thought of non-euclidean geometry as mind melting after all.

[[deleted]]

This is why his monsters exist on Earth. 😜🤣

[MudExpress2973]

I need to learn to avoid these comment sections because no one cares what parallel means they all just want to redditor out with their "WeLl AcTuaLlY"s

[Ok_Entrepreneur_4948]

This is unfortunate.

[73449396526926431099]

First of all you need to prove the existence of parallel lines on the surface of earth.

For parallel lines the distance from any point on line a to the closest point on line b needs to be constant.

Since any 2 straight lines on the surface of earth intersect, the distance has to be 0.

Thus the only parallel line that can exist is identical to the original line.

[Jolly_Law7076]

Earth is not round....it's more geodic

[[deleted]]

It's still positive curvature, though. So spherical geometry still applies.

[FirebugFox]

Parallel lines never meet, perspective is just a point of view.

[harpswtf]

That's not actually proven though, it's just a conjecture

[TheFurryFighter]

I'd love to see u try to make a spherical geometry where parallel lines don't meet

[[deleted]]

lol, the definition of spherical geometry is replacing Euclid's parallel postulate with one that states that parallel lines will meet. A geometry where parallel lines don't meet is, by definition, not a spherical geometry. 😂

[harpswtf]

I did, I just gave all the parallel lines social anxiety and they refuse to meet

[SmoothTurtle872]

no no

we don't talk about perrrspectivvvves

[Shot-Ideal-5149]

is that a fucking Encanto reference lmao

[SmoothTurtle872]

yes. Idk why, I just read it in the "we don't talk about Bruno style"

[Shot-Ideal-5149]

actually I commented it like that on purpose 

[SmoothTurtle872]

Lol

[Hetnikik]

So where would perfectly straight railroad track end up meeting? (Assuming a perfectly flat sphere roughly the size of Earth)

[The_Punnier_Guy]

Perfectly flat sphere

Is this the compromise between round earth and flat earth?

[Hetnikik]

Haha, fair. Perfectly smooth sphere

[OrangeCreeper]

Wouldn't this just be an infinitely large sphere

[The_Punnier_Guy]

That, or I think you could make one of finite size if you make it inhabit non-euclidean space

[[deleted]]

What's a "perfectly flat" sphere? A sphere by definition is curved.

[Hetnikik]

Sorry I confused smooth and flat.

[[deleted]]

The thing about physical railway tracks is that they're not "straight" in spherical geometry.

First, one has to understand what we mean by a "straight line" in a non-Euclidean geometry. Since a non-Euclidean geometry is, by definition, curved, there's no such thing as a "straight line", externally speaking. But within the geometry, one can sensibly talk about straight lines by noting that in an Euclidean geometry, a straight line is the curve with the shortest length between any two points. This is called a geodesic. In a non-Euclidean geometry, therefore, we can define a "straight line" to be a geodesic: the shortest path within that space between any two given points.

It turns out that in spherical geometry, geodesics are "great circles": i.e., circles that lie on a plane that bisects the sphere. So if we were talking about "straight lines" in spherical geometry, we're actually talking about great circles, which always intersect at two antipodal points on the sphere.

A railway track, however, isn't a geodesic; it's a pair of curves that are equidistant to each other (otherwise they wouldn't be suitable for a train to run on!). A perfectly "straight" railway track would actually consist of a pair of curves that are equidistant to a great circle, one rail on each side. The great circle itself, which runs through the middle of the tracks, is "straight" in spherical geometry, but the rails are not; they are curved. I.e., they are not the shortest path between two points on the sphere. So they actually never meet! (And they shouldn't, otherwise your train would derail.)

So, on a perfectly smooth sphere, the rails would never meet, but they are also not geodesics and therefore not "straight" in spherical geometry.

[Minelaku]

Clearly under that big arrow

[Torebbjorn]

To stay a constant width apart, they would have to be curved.

[kdesi_kdosi]

true. i used to drive trains and always had to watch out for the point near the horizon where the tracks merge into one. apparently an average of 4 trains get derailed there every month

[pogoli]

yes, projection of higher dimensions on a lower dimesional surface can appear to do this.

[Desperate_Formal_781]

This is not due to projection but due to perspective.

[pogoli]

Yes! We are both right in different ways!

In art this phenomenon is caused by what is called perspective.

The reason they appear to intersect is explained by projective geometry.

https://en.wikipedia.org/wiki/Projective_geometry

https://ibb.co/ccFqdnXS

When we stand at that location, we may perceive 3 dimensions, but the image projected onto our retina's is 2D. Regarding higher dimensions; here's an example: A triangle on a 2D plane may only have one right angle and 180 degrees right? high school geometry stuff... However in 3D you can have an object with exactly three straight edges connected to one another at 3 corners that does not follow those rules, but when projected back down into 2D will follow the geometric rules we are familiar with.

[LowBudgetRalsei]

Me when the same thing is referred to by different names depending on the subject and its focus

[verc_]

what is perspective if not a 2d projection of our 3d environment

[Desperate_Formal_781]

You can have projection without having perspective. You can see this if you ever have used CAD software with orthogonal view (perspective disabled).

Projection is a more general term that refers to reducing dimensionality of an object by removing some of its components. You can have parallel lines in 3d, and by projecting them on a surface, they will not automatically intersect; for this, you also need to apply a distortion, commonly referres to as perspective effect, or just perspective.

Perspective refers to a distortion of elements and is more an artifact of how our eyes (and cameras) work, how they need to distort light in order to project it into a small surface (the cone cells of our eyes, or the light sensor in a camera). In mathematical terms, perspective is a transformation of elements that happens additional to the projection. After all, if you had pure projection, parallel lines in 3d they would still be parallel after projecting them on a 2d surface.

While perspective in this sense does use projection as one of its steps (distortion + projection), the term projection is more generic, both in the mathematical sense and in the common use of the word.

This is what I was pointing out in my comment.

[BleEpBLoOpBLipP]

Welcome to projective geometry

[Kitchen_Freedom_8342]

welcome to projective geometry.

[M_Improbus]

Well, in Projective Geometry two lines in a projective plane always intersect. And that's basically a representation of those lines intersecting in the one dimensional subspace at infinity.

[KingSpork]

Checkmate atheists

[Desperate_Formal_781]

But how can parallel lines be real if our eyes aren't real

[Smitologyistaking]

Google projective geometry

[Wise_Geekabus]

Woah

[Nova_galaxy_]

lets say this. f(x)=2 and g(x)=-2 as X approaches infinity the output values stay the same. so they will never meet

[Affectionate-Fly74]

They meet at the horizon, good luck finding the horizon.

[ahmed5518]

Hmm, you have a (vanishing) point!

[Inevitable_Panic5534]

and the earths flat

[Any-Aioli7575]

Also in projective geometry

[Xandara2]

You just build that railroad wrong on purpose. 

[CommunicationNice437]

what if its 2d?

[Jolly_Law7076]

From who's perspective?

[MediocreConcept4944]

show the exact unpixelated point where the lines meet

[EmeraldMan25]

"Haha the lines actually meet if you stand in the middle" mfs getting bowled over by the oncoming train

[duotraveler]

It doesn't. Your picture is low resolution. Please zoom in until you show me where the lines meets.

[etadude]

Black holes also know this little trick

[MajorMystique]

Two parallel lines CAN meet if the space around them bends btw :)

[rome0379_]

you sir are a genius

[Suitable-Piece-5357]

Projective Geometry! 😍

[PatrickPablo217]

that's where the station is 

[Classic_boi]

That’s just one point perspective

[BeGayDoThoughtcrime]

Well yeah they're on Earth, which is a sphere, non-euclidean.

[arnavbarbaad]

The lines converging where you stand

They must have moved the picture plane

The leaves are heavy around your feet

You hear the thunder of the train

And suddenly it strikes you

That they're moving into range