Straight line as circle in Euclidean geometry

Question

In Euclidean geometry ,is it possible to have two concentric circles of infinite radius?

Answer 1

The answer to the exact question you're asking is "no" but there are contexts where you want to think of a straight line as a "circle of infinite radius".

One is when you're looking at a pencil of circles. In this picture from wikipedia you can imagine a vertical blue line that is the limit of the blue circles of increasing radius:

The idea is useful when you think about what happens to circles on a sphere when you represent the surface by stereographic projection onto a plane. Some circles become lines:

Answer 2

A circle of infinite radius doesn't exist.

If, instead, you meant generalised circle, then two concentric "generalised-circles-that-turn-out-to-be-lines" are parallel.

Answer 3

This is a model of geometry, which yields all three (spheric, euclidean, hyperbolic), from the notion of homogenious isotropic Gauss-Riemann curvature. It is the closest model to the one you use when you don't know if the universe is closed or not. But i use it to find tilings in hyperbolic geometry.

There is a general catergory of 'isocurves', or curves of constant isotropic curvature. Curves, horocycles, and bollocycles (pseudocycles) are the main examples.

In a given geometry, a curve is 'straight' if it has the same curvature as the space it is in. So great circles are straight lines in the sphere they fall on, but not in lesser ones.

Curvature roughly corresponds to $\frac 1{r^2}$.

Parallel lines are a sub-class of isocurves sharing two parameters. Lines crossing at a point, or circles passing through two points, are other examples. Parallels are of three types of crossing, positive, negative, and zero. They create orthogonals that are of the opposite type.

You can have parallel horocycles both in euclidean and hyperbolic geometries, these are orthogonals to rays emerging from the same point or direction in infinity.

It is of course, possible to have two circles of infinite radius, they can be concentric, or cross each other. In hyperbolic geometry, it is possible that they cross in a circle.