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It is well known that the ratio of the circumference of a circle to its diameter in Euclidean geometry is the constant $\pi$. I also understand that in the case of non-euclidean geometry this ratio is in general not a constant.

What I would like to know, is whether or not $d=2r$ holds in these non-euclidean geometries. For the purpose of this question a circle is defined as the set of points with a constant distance (the radius) to a given point, for any metric, and the diameter as the length of the largest distance between any two points on this circle. In particular, I am interested in geometries with a non-constant curvature.

John Gould
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    This is approximately true for very small radius, and the discrepancy is related to the curvature at the center of the circle. – Qiaochu Yuan Nov 06 '15 at 22:25
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    The triangle inequality is true in non-e geometries so d <= 2r. But is it always true that for colinear points x, y, z that d(x,z) = d(x,y) + d(y, z)? (I'm not sure) If so than d =2r simply mean the center is colinear with two points of a circle. (I think this is true for spherical geometry but I wonder if for hyperbolic geometry the triangle inequality is a strict inequality and if so the diameter is always less than twice the radius.) – fleablood Nov 09 '15 at 00:20
  • @fleablood: it depends on what you mean by collinear. In fact you can take that condition to be the definition of collinearity in a metric space. On a Riemannian manifold you can also try to define collinearity in terms of geodesics. – Qiaochu Yuan Nov 09 '15 at 07:19
  • But are there any metric spaces where for any distinct three point d(x,z) < d(x,y) + d(y,z) is strictly inequal. In such spaces, d < 2r and never equal. Otherwise (as in hyperbolic and spherical) d = 2r. – fleablood Nov 09 '15 at 16:57

1 Answers1

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If circles can be defined in a geometry then the radii of a circle will be of the same length, say, $r$. Assume then that in our geometry there are always two points on any circle whose distance, $d$, is maximal. We can say only that $$2r\ge d.$$

As depicted in the figure below, let the diameter of the circle centered at $C$ be the path connecting $A'$ and $B'$. $A'B'$ is the shortest possible path between $A'$ and $B'$ and at the same time it is the longest possible path between any two points that lie on the circle. This path either goes through the center $C$ (then $d=2r$) or it does not. If it does not then $A'C+CB'=2r>d.$ Note that $AB<AC+CB=2r$ if the shortest path connecting $A$ and $B$ does not go through the center.

enter image description here

In the Euclidean, the hyperbolic, and the elliptic geometries the shortest path is straight and $d=2r$. This is because in these geometries the following theorem holds: If we have two triangles $ACB$ and $A'CB'$ and $AC=CB=A'C=CB'$ and the angle $ACB>A'CB'$ then $AB>A'B'$. Also, the straight line determined by $A$ and $B$ is unique.

zoli
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