So my teacher told my class today that, in an spherical plane, "lines" parallel at one point eventually converge.
It blew my mind, I can't understand why. Lines sorted like latitudes on earth are actually parallel at all points and never converge. Can someone explain this to me? I know my teacher isn't wrong cause I googled it as soon as I got home but I have no clue what I'm missing.
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it's because the lines have to be greater circles
Latitude lines are constantly curving to be what they do. They're sorta parallel or at least their tangents are but those tangents would intersect
There are no lines on a spherical plane. At best the tiniest portion of a line would touch the plane. Remember, lines are straight, if they are not straight, they are something other than lines.
Lmao
>lines are straight
not necessarily
Longitude lines are parallel on a flat plane. Latitude lines are only parallel when veiwed from outside the sphere, when transformed to a flat plane they are heavily curved.
Okay, but they still don't converge.
Yeah, because they are not lines.
You have to remember that the meaning of straight line in a curved setting gets upgraded to the notion of a geodesic, on a sphere these geodesics happen to be the great circles (circles formed from the intersection of the sphere and a plane passing through the origin). It is because these are what are defined as straight lines that two initially parallel "straight lines" (read geodesic) on a sphere will converge. Going further, in the hyperbolic plane if you consider a geodesic and point not on the geodesic then there are an infinite number of geodesics going through that point that are parallel to it. There's actually a lot of amazing facts and theorems in the field of classical geometry (we haven't even gotten to projective spaces, flag manifolds, decomposition of spaces, and much much more) If you want I can suggest some online notes and books to read up on these topics, classical geometry really is a beautiful subject.
I'm not OP but I'd be interested
Longitude lines DO converge as they are parallel. Lines of latitude on a flat plane would be concentric circles on opposite sides of one straight line(the equator).
Correction, they would be concentric circles that have been slightly lopsidedly indented.
>If you want I can suggest some online notes and books to read up on these topics, classical geometry really is a beautiful subject.
Yes please.
Also interestingly enough, a straight line on a spherical plane will always cut the sphere in two equal hemispheres.
You need to use an adjective with "line" in order for it to be curved. Like, a "curved" line or "latitude" line, like this user, but you can't simply use "line" because all "lines" are straight.
>I have no clue what a line is
...
Some books that personally like are geometry revealed and geometry 1-2 (berger he also has a very good riemannian geometry book), elements of projective geometry (cremona), methods of AG (hodge), lectures on curves, surfaces and projective varieties (beltrametti), elementary geometry (moise), euclidean geometries (greenberg), geometry (hartshorne not the ag book), pretty much any geometry book by coxeter is fantastic, just check the prereqs and dive in. Some online notes can be found here ams.org
big things intersect on a finite surface. wow, who knew?
>all "lines" are straight
With respect to what user? A straight line in SPHERICAL SPACE is not going to be straight in rectilinear space.
ask her to show you her light cones
>plane
>spherical
Man fuck Wodzicki tho
>an spherical plane
wat
The fundamental property of a straight line in a Euclidean space is that it's the shortest curve connecting two points. This observation is used to generalize the notion of "straight line" on a curved surface. Such curves are called geodesics and they are the unique curves minimizing distances between pairs of points. (the actual definition is very different but this is the intuitive explanation). If you take a sphere for example, you can convince yourself that the geodesics are precisely great circles - intersections of the sphere with planes passing through the origin. Or equivalently circles with center at the center of the sphere. And of course, any two distinct great circles intersect at two antipodal points - this is what you call that they "converge". Also, latitudes are NOT "lines" in this context. They are just arbitrary curves on a sphere without any special property. They are not geodesics.
I needed that picture to see why it's true.