  Triaxial weaving is a handicraft technique that has long been used to create
curved structures using initially straight and flat ribbons. Weavers typically
introduce discrete topological defects to produce nonzero Gaussian curvature,
albeit with faceted surfaces. We demonstrate that, by tuning the in-plane
curvature of the ribbons, the integrated Gaussian curvature of the weave can be
varied continuously, which is not feasible using traditional techniques.
Further, we reveal that the shape of the physical unit cells is dictated solely
by the in-plane geometry of the ribbons, not elasticity. Finally, we leverage
the geometry-driven nature of triaxial weaving to design a set of ribbon
profiles to weave smooth spherical, ellipsoidal, and toroidal structures.
