 Geodesic circle

For $r$ smaller than the normal-coordinate radius, the geodesic circle on a Riemannian surface consists of points at distance $r$ from $p$. The exponential map restricted to the radius-$r$ circle in $T_{p}S$ gives its smooth embedded parameterization. Outside the injectivity range a distance sphere need not be smooth.

A geodesic circle is a concept in differential geometry, particularly in the study of Riemannian manifolds. It refers to the set of points that are a fixed distance (radius) from a given point on the manifold, along the shortest path, or geodesics, which are the generalization of straight lines in curved spaces.