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(We are assuming euclidean properties).

For example, we can take the distance between (3,2) and the origin (0,0) and draw a line between the 2 points. For calculating distance we use d = sqr{(3-0)^2 + (2-0)^2} = sqr{5}. How can we extend this to finding the distance between a finite point, such as (3,2), and a point at infinity?

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    Why did you decide to ask this here as opposed to Math SE? Obviously, Euclidean distance can not be extended to points at infinity and still produce finite values, you can e.g. embed the plane into the Riemann sphere via the stereographic projection, then distances to the point at infinity will be finite, but that would alter distances between finite points as well. – Conifold Mar 27 '18 at 19:59
  • Where is "infinity"? Considered differently - what is the distance between infinity and infinity + 3? – Bob Jarvis - Reinstate Monica Mar 27 '18 at 23:54
  • This should be moved to the Math SE. Its phrasing is of a question which can be purely decided by mathematics alone, and I'm not seeing any particular philosophical conundrums being explored via that math. – Cort Ammon - Reinstate Monica Mar 28 '18 at 5:58
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In plane Euclidean geometry the point "infinity" is a limit point. It does not belong to the 2-dimensional Euclidean plane. Therefore the distance from a finite point to the limit point "infinity" is not defined.

To encorporate the point "infinity" into geometry one has to embed Euclidean geometry into projective geometry. Here the point "infinity" becomes a point like all other points. But projective geometry does not have an Euclidean metric to measure distances.

Embedding 2-dimensional plane geometry into projective geometry is similar to considering the surface of a sphere with the north pole as the point "infinity". All other points correspond to the finite points of Euclidean geometry. Distances between two points on the sphere can be measured along great circles. Then each two points have a finite distance, but the geometry is only locally Euclidean.

See "Stereographic projection" concerning the relation between the Euclidean plane and the sphere.

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    You are confusing the Riemann sphere with the projective plane. Geometry on the Riemann sphere is not projective but spherical, and it does have a natural metric, conformally equivalent to the Euclidean one. The projective plane also has a natural metric which is locally spherical. – Conifold Mar 27 '18 at 20:08
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    @Conifold Of course your distinction is correct. But in a philosophical forum I do not want to go into the details concerning the sphere S^2 and the complex projective plane P^1(C). Therefore I wrote "is like"; but I will change it to "is similar to" :-) – Jo Wehler Mar 27 '18 at 20:13
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There is no point at infinity.

The only available points are represented as real numbers. The infinity symbol, that figure-8 on its side, represents unboundedness, not a particular point. One can talk about the limit being unbounded or infinite, but the limit never reaches that particular unbounded value. That is why limits are used rather than specifying a point to represent infinity.

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The projective plane is a common way to extend the Euclidean plane to have points (in fact, an entire line) "at infinity".

In the projective plane, one generally doesn't consider distances between pairs of points. It's more useful to select numerical invariants that better relate to the structure of the projective plane.

For example, given any four points on a projective line, one can obtain their cross ratio. One can see this is related to distance because if O,I,P are three points on a line and ∞ is that line's point at infinity1, then the cross ratio of the quadruple (∞, O, I, P) is precisely the ratio of the (signed) lengths OP / OI. (think of ∞ as fixing an affine perspective, O as an "origin" and I as marking off a "unit distance", and then the cross ratio is computing the length of OP)

1: Note that the projective plane has no intrinsic notion of "at infinity"; every point is just like every other. It's only when we pick a Euclidean plane contained within the projective plane that we get a notion of "at infinity", which is the points of the projective plane that aren't contained in the Euclidean one

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