I hope I am using the right terms here. But as I recall from math class (and bear with me, it’s been a while), asymptotic lines get closer and closer, but never really touch.
But is that true? Do they really never touch, or do they some day? Seems to depend on who you ask. My HS geometry teacher said No. But some people say they eventually do. Guess it depends on how you define infinity.
Also, I don’t mean to answer my own question (partly at least). But when I took Calculus, we sometimes used an integral to find the area under a curve. If it was asymtotic, the area eventually converged. If it wasn’t, no area could be found. I just thought I’d toss that out.
Also, a famous man once said, infinity is where parallel lines cross. Now, unlike asymptotic lines, parallel lines don’t bend into each other. So what on earth was he talking about? Does anyone know what quote I am talking about? If I had a cite, I’d include it. But really, was he just talking about asymptotic lines? I know some people say ALL space curves. So the shortest distance between two points is a curve. Is that what he meant?
Depending on the context, this could refer to extending the Euclidean plane into a projective plane by adding a line at infinity, in such a way that two originally parallel lines now meet at one of these new points. The projective plane now has the property that any two distinct lines meet at exactly one point.
This is not merely science fiction (mathematical fiction?) but an important object in geometry; in fact, one learns that we can generalize it and study arbitrary projective varieties; these all have some extremely nice properties such as being complete and not missing any points at infinity.
Any way of dealing with infinity breaks some standard assumptions about numbers. Sometimes you can pick and choose which things it breaks, but there’s always something that breaks.
Well, hopefully nobody has to break any numbers (but if you do, it sounds mathematically interesting
The usual way to put coordinates on the projective plane is to use homogeneous coordinates [x : y : z]. Your ordinary affine plane may be regarded as the subset of points [u : v : 1]. Now if you have two ordinary parallel lines au + bv + c = 0 and au + bv + d = 0 (here c is not equal to d so that these are distinct parallel lines), then solving the system of equations ax + by + cz = ax + by + dz = 0 we get a single point of intersection [-b : a : 0].
Well, if you stand on a long straight road and look to the horizon, it appears the two sides meet. And “parallel” lines of longitude on the globe meet at the north pole. I think Einstein had some exotic spacetime geometry idea where “parallel” lines meet (can anyone confirm or deny, or was it another physicist), and lines of longitude on the globe is the analogy to help you think about it. Also, I think there is some axiom in Euclids geometry which cannot be proven or disproven from the others, but which is considered the most dubious, or was long considered provable but just not yet, which involves parallel lines, but I am not sure if it is equivalent to “parallel lines do not meet at infinity”. Again, someone else research or recall it and write up a report with citations for us. I may have just created 2 (false) urban legends: about Einstein and Euclid. And if so, I will take pride in it to my grave.
Lines of longitude are not parallel, they are geodesics, which can be thought of as a straight line on curved surfaces.
A similar concept is in curved space time, like the orbit of the earth moon system around the sun is a straight line, that the earth travels without acceleration but it isn’t parallel.
The North and South pole are coordinate singularities. Maybe you were thinking of parallel transport? That is more about detecting properties about the surface you are working on, and it is the tangent vector that stays parallel, but does not indicate anything about convergence of a line or a geodesic.
You are thinking of the Poincaré hyperbolic disk, or the Klein hyperbolic disk. Points on the boundary of the disk correspond to “ideal points at infinity”.
I think this is getting a bit esoteric for the OP, which was, as I read it, a question about simple asymptotic curves in normal Euclidean space.
Consider the curve y=1/x. As x increases the curve asymptotically approaches the x axis. Do they ever meet? If they did, it would be at a point where y is zero – that is, the point (x,0) would be on the curve for some x. Plugging that into the equation, it says 0=1/x at that point. Clearly there is no such value of x, so the curve never meets the x axis. This is how all asymptotic curves work (that’s the definition of asymptotic). The curve approaches, but never meets, some other curve.
I thought the whole point was to throw out a bunch of esoteric ideas making sense of “parallel lines meet”. As to your “no such value of x”: not if we use hyperreal numbers:
Which are employed in “nonstandard analysis” to provide alternate proofs of theorems true in standard analysis (which is restricted to only the usual “real numbers”).
We can also treat infinity and infinitesimals as members of the “Surreal Numbers” {Wikipedia} … I searched on Amazon for “Complexier Numbers”, like ∞ + ∞i, but only got hits for acne medicine …
Surreal numbers will not make the equation ax = 0, for a non-zero constant a, have any nonzero solutions. That is, parallel lines will not meet at any point even if you extend the field of scalars.
First place, in the ordinary Euclidean space, asymptotic curves never meet and parallel lines never meet. In the projective plane, there is a line at infinity and parallel lines meet there. If the asymptotic curves can be extended to the projective plane, I assume they will meet there as well.
Finally, to correct a claim made in the OP, there area between two asymptotic curves does not necessarily converge. For example, the integral from 1 to t of 1/x dx is ln t, which does not converge as t --> infinity. But the integral from 1 to t of 1/x^2 is 1 - 1/t which does. This gives rise to a well-known paradox. If you rotate the curve about the x-axis, you get a figure that you cannot paint with a finite amount of paint, but you can fill that solid figure with a finite amount of paint.
The figure formed by rotating y=1/x (not y=1/x[sup]2[/sup]) around the x axis is called Gabriel’s Horn. It’s a pretty cool object. It’s certainly not intuitively obvious that an object can have an infinite surface area but a finite volume.
Eh, there are figures in a plane (such as the Koch snowflake) that have a finite area but infinite perimeter, and you could take an extrusion of any of those to make a figure with finite volume but infinite surface area.
Or a practical example, the Coastline paradox. Coastlines have a fractal dimension which really invalidates the notion of length, but for many reasons people want to quantify the length of coastlines.
This is a common concern as it continues to cause cartographic conundrums in my corner of creation.
I was going to mention this one. Note the “painter’s paradox”. It can hold a finite amount of paint but requires an infinite amount of paint to paint it. The Wikipedia explanation is less that satisfying. If really painting the inside, the thickness of a coat soon becomes too fat to fit into the narrowness of the horn. From then on only a finite amount of paint is needed (to go with the finite amount before that point).
If painting the outside ~all bets are off. It’s an entirely different calculation than the volume of the interior so infinite paint for a fixed thickness is a-ok.
