In mathematics, a metric circle is the
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
of
arc length
Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...
on a
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
, or equivalently on any
rectifiable simple closed curve of bounded length. The metric spaces that can be embedded into metric circles can be characterized by a four-point
triangle equality.
Some authors have called metric circles Riemannian circles, especially in connection with the
filling area conjecture in
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
, but this term has also been used for other concepts. A metric circle, defined in this way, is unrelated to and should be distinguished from a
metric ball, the subset of a metric space within a given radius from a central point.
Characterization of subspaces
A metric space is a subspace of a metric circle (or of an equivalently defined metric line, interpreted as a
degenerate case of a metric circle) if every four of its points can be permuted and labeled as
so that they obey the equalities of distances
and
. A space with this property has been called a ''circular metric space''.
Filling
The Riemannian
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
of length 2 can be embedded, without any change of distance, into the metric of
geodesics on a
unit sphere
In mathematics, a unit sphere is a sphere of unit radius: the locus (mathematics), set of points at Euclidean distance 1 from some center (geometry), center point in three-dimensional space. More generally, the ''unit -sphere'' is an n-sphere, -s ...
, by mapping the circle to a
great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Discussion
Any arc of a great circle is a geodesic of the sphere, so that great circles in spher ...
and its metric to
great-circle distance
The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them. This arc is the shortest path between the two points on the surface of the ...
. The same metric space would also be obtained from distances on a
hemisphere. This differs from the boundary of a
unit disk
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose d ...
, for which opposite points on the unit disk would have distance 2, instead of their distance on the Riemannian circle. This difference in internal metrics between the hemisphere and the disk led
Mikhael Gromov to pose his
filling area conjecture, according to which the unit hemisphere is the minimum-area surface having the Riemannian circle as its boundary.
References
{{Bernhard Riemann , state=collapsed
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
Circles
Metric geometry
Bernhard Riemann