# Straight line as circle in Euclidean geometry

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

• What is a circle of infinite radius? – Levent Dec 29 '16 at 14:30
• There's no such thing as "circle of infinite radius" in Euclidean geometry. – Henning Makholm Dec 29 '16 at 14:30

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:

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.

• Thanks.Please suggest some reference so that I can develop a better understanding of the concept of concentric circles becoming parallel lines. – user402705 Dec 29 '16 at 14:44
• Take two concentric circles of radius $1$ and $2$, centered at the origin. Then two concentric circles of radius $2$ and $3$ centered at $(1,0)$. Then two concentric circles of radius $11$ and $12$ centered at $(10,0)$. Keep going like this, having circles centered at $(r,0)$ with raduis $r+1$ and $r+2$, and see that the two generalised circles you eventually get are the two lines $x = -1$ and $x = -2$. – Arthur Dec 29 '16 at 14:47
• Thanks. Please consider my doubt given below. In the limit as radius tends to infinity,considering the fact that (infinity minus infinity = infinity),will it be contradictory that the difference in radii in this case is one unit(a finite number). I don't know clearly about what infinity is,but I have seen that "infinity-infinity=infinity". – user402705 Dec 29 '16 at 15:12
• I never said anything about $\infty-\infty$. I said "large number minus almost the same number", and see what happens as the large number gets larger and larger, beyond all bounds. This is often called "$\infty-\infty$", but it's not what it is. There are many things that is called $\infty-\infty$, some of them finite, some of them infinite, some of them negative infinite, this is just one of them. – Arthur Dec 29 '16 at 15:33
• Thanks for the clarification. – user402705 Dec 29 '16 at 15:39

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.

• Thanks for the detailed explanation. – user402705 Dec 29 '16 at 15:21