Implications of the connectivity rule in a Gabriel graph. (A) Closing triangles from edges: three points will be mutual neighbors if and only if they form an acute triangle (left). If the angle between $xi$ and $xj$ at $xk$ is at least $π/2$⁠, all three points will lie in $Bij$⁠, so no edge is created (right). (B) The maximum principal curvature in $M$ (shown in blue) that can be reasonably approximated by the resulting graph geodesic (path) is constrained by the sampling interval. The limiting case occurs when three points form a right triangle (top, see equation 3.4). When sampling is too sparse (bottom left), a triangle may be formed, in this case preventing the graph from adequately capturing the manifold's geometry. As sampling frequency increases (bottom right), higher curvatures can be better approximated.