# Programming Curvature using Origami Tessellations

## TL;DR

This paper introduces a geometric and optimization-based method to design origami patterns that approximate complex curved surfaces, enabling scalable and energy-aware surface fitting.

## Contribution

It presents a novel approach combining geometric constructions and constrained optimization to tailor origami patterns for specific curved surfaces, independent of scale.

## Key findings

- Patterns can approximate target surfaces with controllable accuracy.
- The energetic barrier quantifies the difficulty of folding into the desired shape.
- Trade-offs between pattern accuracy and folding effort are characterized.

## Abstract

Origami describes rules for creating folded structures from patterns on a flat sheet, but does not prescribe how patterns can be designed to fit target shapes. Here, starting from the simplest periodic origami pattern that yields one degree-of-freedom collapsible structures, we show that scale-independent elementary geometric constructions and constrained optimization algorithms can be used to determine spatially modulated patterns that yield approximations to given surfaces of constant or varying curvature. Paper models confirm the feasibility of our calculations. We also assess the difficulty of realizing these geometric structures by quantifying the energetic barrier that separates the metastable flat and folded states. Moreover, we characterize the trade-off between the accuracy to which the pattern conforms to the target surface, and the effort associated with creating finer folds. Our approach enables the tailoring of origami patterns to drape complex surfaces independent of absolute scale, and quantify the energetic and material cost of doing so.