A framework to describe equilibrium configurations of semiflexible polymers confined by, or bound to a membrane is presented. Such configurations minimize the bending energy of the polymers-- quadratic in their curvature--under the confinement constraint. This constriction is imposed in the variational principle by introducing a local Lagrange multiplier, which is identified as the confining force and is directed along the vector normal to the membrane. Both, the Euler-Lagrange equation--describing equilibrium states of the polymer--and the confining force, are expressed in terms of the local geometry of the curve. This framework is applied to describe the conformation of semiflexible polymer loops confined to membranes with spherical and cylindrical geometries. The residual Euclidean symmetries associated with these geometries allow for the integration of the Euler-Lagrange equation, reducing it to a quadrature and facilitating the explicit reconstruction of the polymer equilibrium configurations. The effect of the geometry and topology of such membranes on the morphology of the polymer is discussed. Also, the non-trivial dependance of the bending energy and the confining normal force on the polymer loop's length is examined.