  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.
