The standard definition of a circle is the locus at a distance r from the centre O. It’s obvious that a line on Euclidean plane is not a circle. But have we really see the centre to check whether a piece of curve is a piece of some circle? We haven’t, because, on Euclidean plane, line of constant non-zero curvature is a criterion of a circle (let us consider only curves with only one sign of curvature, i.e. all chords must lie on one side of the curve). For example, we can check whether Menger curvature for all triples of distinct points is constant. If it is, and is a non-zero quantity, then our curve is a piece of circle. There are different definitions of curvature, such as differentially-geometric one, but it isn’t important which definition we use if it can work for a small piece of curve. Local definition of curvature implies a local criterion for a circle. On Euclidean plane, circles with very small curvature (i.e. great radius) are locally almost indistinguishable from straight lines, where straight lines are lines of zero curvature.
How two definitions, of r,O-circles and of lines of constant curvature, are related? We see, on Euclidean plane, each (unterminated) line of constant curvature is either an r,O-circle or a straight line.
We also have another two homogeneous geometries of a plane: elliptic geometry and Lobachevski’s geometry. They also have distances, and differential geometry works there as well. In any of these geometries both straight lines and r,O-circles have constant curvature, but how they are related? In elliptic geometry there is no r,O-circles with r > πR/2 (where R is constant) and, curiously, circles of radius πR/2 are the same as straight lines (great circles). So, in elliptic geometry, (unterminated) lines of constant curvature and r,O-circles define the same class of curves.
In Lobachevski’s geometry opposite conditions exist: there are r,O-circles for any r > 0, but any r,O-circle has curvature greater than 1/R, where R is constant (sorry, do not remember exact dependence of curvature on r). But there are lines of constant curvature for any curvature. Lines of constant curvature between 0 and 1/R inclusively extend indefinitely (are not closed). The case of curvature 1/R was mentioned in the answer of @Will Jagy. So, in Lobachevski’s geometry, both straight lines and r,O-circles are proper subsets of the set of all (unterminated) lines of constant curvature, and there are also some lines in between.
That’s how relation between straight lines and r,O-circles demonstrates Euclidean geometry’s intermediate position between elliptic geometry and Lobachevski’s geometry. Euclidean straight lines are not r,O-circles, unlike situation in elliptic spaces. But Euclidean straight lines are adjacent to r,O-circles (can be obtained as a limit of r,O-circles, in the sense of pictures presented), unlike situation in hyperbolic spaces.
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