We present an analytical framework for exploring the design space of hinged kirigami structures that deploy rigidly and uniformly. A hinged kirigami structure consists of rigid planar faces connected by hinges, and is deployable if the faces can rotate about the hinges without deformation. Such deployability depends on both the topology and the geometry, i.e., the combinatorial connectivity of the faces and their spatial embedding. Prior work studies geometric constraints for uniform deployability in restricted settings, most notably quadrilateral kirigami patterns and structures derived from 2-colorable planar graphs. However, existing analysis does not readily generalize to arbitrary kirigami structures, nor does it provide a systematic approach for constructing deployable kirigami from non-2-colorable planar graphs (e.g., graphs with non-manifold embeddings). Moreover, while geometric conditions for deployability have been partially investigated, the structure of the full deployable design space and its associated degrees of freedom remain largely unexplored. In this work, we propose a new framework that enables the derivation of hinged kirigami structures from arbitrary planar graphs, with the key feature that multiple distinct kirigami structures can be generated from the same graph. We derive geometric constraints that ensure uniform rigid deployability and analytically characterize the full design space of deployable embeddings, including its associated degrees of freedom. This characterization allows continuous navigation of the design space and provides a systematic foundation for the design of kirigami-based mechanical metamaterials.

We present a computational framework for designing geometric metamaterials capable of approximating freeform 3D surfaces via rotationally deployable kirigami patterns. While prior inverse design methods typically rely on standard, well-studied patterns, such as equilateral triangles or quadrilaterals, we step back to examine the broader design space of the patterns themselves. Specifically, we derive principled rules to determine whether a given planar tiling can be cut into a rotationally deployable hinged kirigami structure with possible curvature adaptation. These insights allow us to generate and validate a broad family of novel tiling patterns beyond traditional examples. We further analyze two key deployment states of a general pattern: the commonly used maximal area expansion, and the maximal rotation angle reached just before face collisions occur, which we adopt as the default for inverse design as it allows for simple deployment in practice, i.e., rotating the faces to their natural limit. Finally, we solve the inverse problem: given a target 3D surface, we compute a planar tiling that, when cut and deployed to its maximal rotation angle, approximates the input geometry. We show that for a subset of patterns, the deployed configurations are hole-free, demonstrating that curvature can be achieved from planar sheets through local combinatorial changes. Our experiments, including physical fabrications, demonstrate the effectiveness of our approach and validate a wide range of previously unexplored patterns that are both physically realizable and geometrically expressive.

Distortion-minimizing surface parameterization is an essential step for computing 2D pieces necessary to fabricate a target 3D shape from flat material. Garment design and textile fabrication are a prominent application example. Common distortion measures quantify length, angle or area preservation in an isotropic manner, so that when applied to woven textile fabrication, they implicitly assume fabric behaves like paper, which is inextensible in all directions and does not permit shearing. However, woven fabric differs significantly from paper: it exhibits anisotropy along the yarn directions and allows for some degree of shearing. We propose a novel distortion energy based on Chebyshev nets that anisotropically penalizes shearing and stretching. Our energy formulation can be used as an optimization objective for surface parameterization and is simple to minimize via a local-global algorithm. We demonstrate its advantages in modeling nets or woven fabric behavior over the commonly used isotropic distortion energies.

We develop an optimization-based method to model smocking , a surface embroidery technique that provides decorative geometric texturing while maintaining stretch properties of the fabric. During smocking, multiple pairs of points on the fabric are stitched together, creating non-manifold geometric features and visually pleasing textures. Designing smocking patterns is challenging, because the outcome of stitching is unpredictable: The final texture is often revealed only when the whole smocking process is completed, necessitating painstaking physical fabrication and time consuming trial-and-error experimentation. This motivates us to seek a digital smocking design method. Straightforward attempts to compute smocked fabric geometry using surface deformation or cloth simulation methods fail to produce realistic results, likely due to the intricate structure of the designs, the large number of contacts and high-curvature folds. We instead formulate smocking as a graph embedding and shape deformation problem. We extract a coarse graph representing the fabric and the stitching constraints and then derive the graph structure of the smocked result. We solve for the three-dimensional embedding of this graph, which in turn reliably guides the deformation of the high-resolution fabric mesh. Our optimization based method is simple, efficient, and flexible, which allows us to build an interactive system for smocking pattern exploration. To demonstrate the accuracy of our method, we compare our results to real fabrications on a large set of smocking patterns.

We present a novel method for realizing freeform surfaces with pieces of flat fabric, where curvature is created by stitching together points on the fabric using a technique known as smocking. Smocking is renowned for producing intricate geometric textures with voluminous pleats. However, it has been mostly used to realize flat shapes or manually designed, limited classes of curved surfaces. Our method combines the computation of directional fields with continuous optimization of a Tangram graph in the plane, which together allow us to realize surfaces of arbitrary topology and curvature with smocking patterns of diverse symmetries. Given a target surface and the desired smocking pattern, our method outputs a corresponding 2D smocking pattern that can be fabricated by sewing specified points together. The resulting textile fabrication approximates the target shape and exhibits visually pleasing pleats. We validate our method through physical fabrication of various smocked examples.

Abstract Conformal maps between planar domains are an important tool in geometry processing, used for shape deformation and image warping. The Riemann mapping theorem guarantees that there exists a conformal map between any two simply connected planar domains, yet computing this map efficiently remains challenging. In practice, one of the main algorithmic questions is the correspondence between the boundaries of the domains. On the one hand, there exist a number of conformal maps between any two domains, thus many potential boundary correspondences, yet on the other, given full boundary prescription a conformal map might not exist. Furthermore, an approximate boundary fitting can be enough for many applications. We therefore propose an alternating minimization algorithm for finding a boundary‐approximating conformal map given only an initial global alignment of the two input domains. We utilize the Cauchy‐Green complex barycentric coordinates to parameterize the space of conformal maps from the source domain, and thus compute a continuous map without requiring the discretization of the domain, and without mapping to intermediate domains. This yields a very efficient method which allows to interactively modify additional user‐provided constraints, such as point‐to‐point and stroke‐to‐stroke correspondences. Furthermore, we show how to easily generalize this setup to quasi‐conformal maps, thus enriching the space of mappings and reducing the area distortion. We compare our algorithm to state‐of‐the‐art methods for mapping between planar domains, and demonstrate that we achieve less distorted maps on the same inputs. Finally, we show applications of our approach to stroke based deformation and constrained texture mapping.