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ftr
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I read somewhere that a unique line connecting two points does not exist in higher than 3D, is this correct?
jambaugh said:No. It isn't higher dimensions but curvature that makes non-unique (locally) shortest paths. What is the shortest path from the north pole to the south? (on the surface please!)
ftr said:Thanks for the reply. I am new to differential Geometry. I have quoted this post because it is a good example to further explain my question. Now, if you run a straight line from NP to SP that is unique. these other Longitude lines that run on the surface are not unique. So my question is (I guess) can we have a unique STRAIGHT line in higher than 3D.
also, I have read that the concept of line(I guess they mean straight line) in higher than 3D does not make sense or something to that effect.
If my question is not making sense please tell me.
Hi Tiny! If by "straight line" you mean geodesics then you can in fact define geodesics without any riemannian metric as long as you have an affine connection ##\nabla## endowed on your smooth manifold. The curves satisfying ##\nabla_{\dot{\gamma}}\dot{\gamma} = 0## are called affine geodesics and intuitively these are the curves that parallel transport their own tangent vectors. This definition is much more intuitive geometrically in my opinion than the notion of a geodesic as something that has vanishing first variations of arc length or energy. If you don't have a smooth manifold (i.e. just a topological manifold) then yeah you're perfectly right. However if we are talking about smooth manifolds then you can always give the manifold a Riemannian structure so it really isn't much of an issue. I hope that bit of trivia was worthwhile, cheers!Tinyboss said:And in some spaces, the notion of "straight line" is not meaningful, for instance a manifold without a defined metric.
ftr said:Thanks micro and tiny ( what coincidence of names). Please bear with my ignorance. Now, a line element length can be described as
s^2=dx1^2 +dx2^2 +dx3^2 +dx4^2 + ...
for 3D dx=1 line element length =sqrt(3)
for 4D dx=1 line element length =sqrt(4)
for 3D that is clear but for 4D do I actually walk that much if I go into that space, sounds very strange.
yenchin said:Do you mean straight line or geodesic [as the previous posters pointed out?] Assuming you really do mean straight line, then I am still not exactly sure what you mean: let's back down a bit, how about line in 1D [just line by itself], line in 2D [line in a plane], and line in 3D? You see that any line in 3D space can really be contained in a plane, so there is really no difference between a line in 2D or 3D or 4D -- they are one-dimensional entity, what exactly is confusing you?
In higher dimensions, lines are defined by more than two points and can exist in multiple planes. In 3D, lines are defined by two points and exist in a single plane.
In mathematics, lines in higher dimensions are represented using vector equations, parametric equations, or Cartesian equations.
Yes, lines in higher dimensions can intersect at a single point, just like lines in 3D can intersect at a single point.
The slope of a line in higher dimensions can be calculated using the coordinates of two points on the line and the formula (y2 - y1) / (x2 - x1).
Yes, lines in higher dimensions are used in computer graphics, physics, and engineering to represent objects and their movements in 3D space.