ruumiline

Ruumiline is a geometric concept describing a class of curves in a metric space that balance short length with staying near a prescribed reference set. The term appears in recent differential geometry literature and is used to study curves that are not only efficient in length but also spatially constrained, connecting ideas from geodesics and variational problems.

A ruumiline is defined as a smooth curve γ: I → M in a Riemannian manifold M that minimizes a Ruumi functional R(γ) = ∫0^L [κ(s)^2 + α d(γ(s), S)^2] ds, where κ is curvature, d(·, S) is distance to a reference subset S ⊂ M, α > 0 is a weight, and L is arc-length. When α = 0, minimizers reduce to geodesics; for α > 0, the curve trades length against proximity to S.

Properties include invariance under isometries of M and a higher-order Euler–Lagrange equation. Ruumiline curves depend continuously on α and on S, allowing numerical approximation by standard variational methods.

Examples: in Euclidean space with S a line, ruumiline curves interpolate between straight lines and curves hugging the line depending on α. With S a circle, they form near-circular arcs kept close to the circumference while remaining short.

Applications appear in robotics path planning, where short routes must stay within a corridor, and in data visualization, where ruumiline paths highlight proximity to features. See also geodesics, variational calculus, curvature.