  Confinement of filamentary objects is ubiquitous in numerous biological,
medical, and engineering scenarios. Quantitatively determining the mechanical
interaction between flexible filaments and surface confinement is particularly
challenging due to the unknown contact force induced by elasticity interacting
with geometric constraints. Here, we consider a simplified model of confined
filamentary object: an elastic curve under surface confinement. Local force and
moment balance equation incorporating the role of contact is utilized to derive
the contact force exerted by surface on the confined elastic curve. It reveals
the relation between contact force and local geometry at balanced state and
provides a route to obtain the contact force from local confined geometry
directly. Examples are provided to illustrate how to calculate contact force
from obtained geometries. We believe that our results contribute to future
efforts in the mechanics of filamentary objects under surface confinement.
