This paper presents a theoretical and mathematical framework for the design of a conformal reconfigurable intelligent surface (RIS) that adapts to non-planar geometries, which is a critical advancement for the deployment of RIS on non-planar and irregular surfaces as envisioned in smart radio environments. Previous research focused mainly on the optimization of RISs assuming a predetermined shape, while neglecting the intricate interplay between shape optimization, phase optimization, and mutual coupling effects. Our contribution, the T3DRIS framework, addresses this fundamental problem by integrating the configuration and shape optimization of RISs into a unified model and design framework, thus facilitating the application of RIS technology to a wider spectrum of environmental objects. The mathematical core of T3DRIS is rooted in optimizing the 3D deployment of the unit cells and tuning circuits, aiming at maximizing the communication performance. Through rigorous full-wave simulations and a comprehensive set of numerical analyses, we validate the proposed approach and demonstrate its superior performance and applicability over contemporary designs. This study-the first of its kind-paves the way for a new direction in RIS research, emphasizing the importance of a theoretical and mathematical perspective in tackling the challenges of conformal RISs.