# Are all infinite lines just the perimeters of circles from a birdseye view?

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If you'd said, "Are what appear to be infinite lines just the perimeters of circles from a birdseye view?" then I'd have said, "Possibly". Otherwise, no. If it's a line, it's a line. If it's an arc, it's an arc. Just because we don't have enough info to make a proper determination doesn't change that fact.

All the answers given so far are good, and it's hard for me to add anything to this. But since your question starts with "Are" instead of "Could", the answer is "not necessarily". Otherwise, you'd be dismissing the whole of Euclidean geometry, which posits such infinite straight lines. Nobody has ever been able to prove that Euclidean geometry is self-contradicting, so it remains a viable geometry with lots of infinite lines which aren't necessarily circles or arcs.

If you examine the line in the complex plane, then the answer is yes, in a very precise sense. Let z be a complex variable, and consider the mapping that takes z to (az + b)/(cz + d). Assume that c and d are not both zero. This is called a linear-fractional transformation, or a Moebius transformation, after the same mathematician who defined the Moebius strip. This is a conformal, or angle-preserving, mapping of the complex plane onto itself. It maps circles onto either circles or lines, and it maps lines onto either circles or lines. Thus, a line can be viewed as a circle of infinite radius, or as the boundary of a disc of infinite radius. For further applications which lend added insight into this answer, please see the references below.

Source(s): (1) John B. Conway, Functions of One Complex Variable I, Graduate Textx in Mathematics no. 15, Springer-Verlag (2) Tom M. Apostol , Introduction to Analytic Number Theory, Volume 2, Graduate Textx in Mathematics, Springer-Verlag

It depends on which geometry you are using. In the normal, Euclidean system, no, lines and curves are completly different entities. But in many non-euclidean geometries, it is just as you say, an infinite line is the perimeter of a huge circle.