Generic frame fields are important for many applications in computer graphics such

as texture mapping and quadrangulation, however, unlike their simpler version - cross

fields, they are hard to design. One category of approaches relies on creating an

intermediate metric in which the frame field becomes a cross field. The vast majority

of existing approaches achieve it by deforming the original surface in the Euclidean

space. It’s a known fact that not all Riemannian metrics can be accommodated by

the three-dimensional Euclidean space, thus some authors seek solutions in higher

dimensional Euclidean spaces, thus increasing the complexity of the problem. Intrinsic approaches, designing a suitable Riemannian metric directly have been introduced recently, however, they rely on a complicated formulation and numerical approximations.

Moreover, they require developing new tools for quadrangulating the surface in this

new metric since the surface is curved and not a polytope. We propose an intrinsic

approach that obviates the necessity of an embedding and has a simple formulation. We compute a new Riemannian metric that results from warping a surface by local linear transformations and impose a compatibility constraint to ensure that the resulting space is a metric space and a polytope. This allows for a larger space of solutions than is possible to accommodate in the three- dimensional Euclidean space as well as a simple formulation. We show how to adapt existing methods for designing smooth vector fields for this framework and demonstrate the robustness of the algorithm relatively to extrinsic approaches.