We study curves obtained by tracing triangle centers within special families of triangles, focusing on centers and families that yield (semi-)invariant triangle curves, meaning that varying the initial triangle changes the loci only by an affine transformation. We identify four two-parameter families of triangle centers that are semi-invariant and determine which are invariant, in the sense that the resulting curves for different initial triangles are related by a similarity transformation. We further observe that these centers, when combined with the aliquot triangle family, yield sheared Maclaurin trisectrices, whereas the nedian triangle family yields Limaçon trisectrices.

We present a computational framework for numerical homogenization of origami materials—thin sheets structured with periodic crease patterns that, once folded, exhibit diverse and often unusual mechanical properties. Whereas the in-plane stiffness of conventional sheet materials is typically orders of magnitude larger than their resistance to bending, origami-based folding introduces geometric structure that can drastically reshape both bending and stretching behavior. However, predicting how a particular crease pattern gives rise to effective macroscopic properties remains challenging due to the complex coupling of crease geometry, folding kinematics, and surface deformations. In this work, we introduce a computational framework that integrates simulation-based folding and numerical homogenization to explore the relationship between crease pattern and effective material behavior. To describe the macromechanical response of origami materials, we employ a quadratic energy model based on Classical Laminate Theory, together with a simplified treatment of crease plasticity. Our unified representation accommodates both straight- and curved-crease designs, revealing a rich space of origami materials with diverse behavior. In particular, we examine how pattern symmetry governs material symmetries, demonstrating examples that span the full spectrum from perfectly isotropic to highly anisotropic membrane and bending responses. Our framework further enables controlled exploration of parameter variations, illustrating how geometric features such as crease curvature shape macroscopic mechanical behavior. We showcase the potential of this approach through a broad set of examples, ranging from canonical straight-crease patterns such as Miura-ori to complex curved-crease tessellations. While a quantitative analysis is left for future work, we validate our homogenized descriptions against native-scale simulations and qualitatively compare deformation behaviors with real-world prototypes.

We introduce C-tubes , 3D tubular structures composed of developable surface strips. C-tubes can be understood as a generalization of Monge surfaces—a special class of sweep surfaces—towards the recently introduced conenets. This observation allows formulating a constructive algorithm to create tubular structures that ensures developability of the constituent surfaces, while significantly broadening the design space. Our novel form-finding tool enables design exploration by solving for the input variables of the constructive algorithm so that the C-tube best conforms to user-specified objectives. We discuss several case studies that illustrate the versatility of our approach for the design and fabrication of complex structures, with applications in architecture, furniture, and lighting design.

In this paper we study pleated structures generated by folding paper along curved creases. We discuss their properties and the special case of principal pleated structures. A discrete version of pleated structures is particularly interesting because of the rich geometric properties of the principal case, where we are able to establish a series of analogies between the smooth and discrete situations, as well as several equivalent characterizations of the principal property. These include being a conical mesh, and being flat-foldable. This structure-preserving discretization is the basis of computation and design. We propose a new method for designing pleated structures and reconstructing reference shapes as pleated structures: we first gain an overview of possible crease patterns by establishing a connection to pseudogeodesics, and then initialize and optimize a quad mesh so as to become a discrete pleated structure. We conclude by showing applications in design and reconstruction, including cases with combinatorial singularities. Our work is relevant to fabrication in so far as the offset properties of principal pleated structures allow us to construct curved sculptures of finite thickness.