An elastic ring interacting with rigid surfaces is a fundamental engineering challenge with vast practical implications in various disciplines. However, the exploration of closed-form solutions for this issue has been limited, and existing studies often present complex, numerically unstable solution methods influenced by specific boundary conditions. Furthermore, the lack of scalable design principles for the ring-in-contact scenario has hindered its broader application across different geometries and materials. This study introduces a streamlined analytical and numerical approach to predict the contact behavior of orthotropic rings against centrosymmetric rigid surfaces, encompassing both flat and curved surfaces. Our approach, which simplifies the closed-form solution for extensible Timoshenko curved beams coupled with a contact algorithm that prevents penetration, yields robust and accurate predictions of nonlinear contact behaviors in elastic rings, including deformation patterns, contact angles, stresses, and stiffness. Additionally, we present a design map that serves as a scalable guideline for engineering elastic rings in contact, facilitating the choice of geometry and materials, such as ring radius, thickness, and elastic moduli. This research enhances the theoretical underpinnings of elasticity concerning ring contact and expands the engineering viewpoint on designing elastic rings in various contact scenarios.