Intrinsic metric
In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second along a path whose arclength is equal to (or very close to) that distance. The distance between two points of a metric space relative to the intrinsic metric is defined as the infimum of the lengths of all paths from the first point to the second. A metric space is a  length metric space if the intrinsic metric agrees with the original metric of the space. If the space has the stronger property that there always exists a path that achieves the infimum of length (a geodesic) then it may be called a geodesic metric space or geodesic space. For instance, the Euclidean plane is a geodesic space, with line segments as its geodesics. The Euclidean plane with the origin removed is not geodesic, but is still a length metric space. Definitions. Let formula_1 be a metric space, i.e., formula_2 is a collection of points (such as all of the points  in the plane, or all points on the circle) and formula_3 is a function that provides us with the "distance" between points formula_4. We define a new metric formula_5 on formula_2, known as the induced intrinsic metric, as follows: formula_7 is the infimum of the lengths of all paths from formula_8 to formula_9. Here, a "path" from formula_8 to formula_9 is a continuous map with formula_13 and formula_14. The "length" of such a path is defined as explained for rectifiable curves. We set formula_15 if there is no path of finite length from formula_8 to formula_9. If for all points  formula_8 and formula_9 in formula_2, we say that formula_1 is a length space or a path metric space and the metric formula_23 is intrinsic. We say that the metric formula_23 has approximate midpoints if for any formula_25 and any pair of points formula_8 and formula_9 in formula_2 there exists formula_29 in formula_2 such that formula_31 and formula_32 are both smaller than