The distance underlying the definition of the projection measures the similarity between designs as the minimum number of topological grammar rules to apply to modify one design into another. This approach is expressed as the projection to the two-colourable subspace of the design space. Based on an algebra for the exploration of the topology of quad meshes, including a grammar and a distance, a topology-finding algorithm is proposed to find the closest two-colour quad-mesh patterns from an input quad-mesh pattern. This paper presents a search strategy to obtain patterns that fulfil this topological requirement, which represent only a fraction of the general design space. Complying with such constraints does not depend on the geometry but the topology of the structure, and, more specifically, on its singularities. Such examples include top versus bottom layer of continuous beams in elastic gridshells, corrugated versus non-corrugated directions in corrugated shells or warp versus weft threads in woven structures. The patterns of many structural systems must fulfil a property of two-colourability to partition their elements into two groups. They are thus easily embedded into a design workflow involving standard operations like re-meshing, trimming, and merging operations. The meshes we employ are combinatorially regular quad meshes with isolated singularities but are otherwise not required to follow any special curves. In combination with standard global optimization procedures, we are able to perform developable lofting, approximation, and design. This criterion is expressed in terms of the canonical checkerboard patterns inscribed in a quad mesh which already was successful in describing discrete-isometric mappings. In this paper, we propose a new and efficient discrete developability criterion that is based on a property well-known from differential geometry, namely a rank-deficient second fundamental form. So far, a local criterion expressing the developability of general quad meshes has been lacking.
Contributions range from special parametrizations to discrete-isometric mappings. There are different ways to capture the property of a surface being developable, i.e., it can be mapped to a planar domain without stretching or tearing. The methods we employ include a differential–geometric interpretation of the morph, besides drawing on recent progress in geometric computing This amounts to a solution of the so-called inverse problem for kirigami cut and fold patterns. The shapes involved can be arbitrary in fact we are able to approximate any mapping between shapes whose principal distortions do not exceed certain bounds. We use optimization to compute kirigami patterns that realize a morph between shapes, in particular between a flat sheet and a surface in space. We here present an affirmative solution to a fundamental geometric question, namely the targeted programming of a shape morph. It is intimately connected with the interpretation of patterned sheets as mechanical metamaterials, typically of negative Poisson ratio.
This manufacturing paradigm has been receiving much attention in recent years and poses challenges in both fabrication and computation. Small-scale cut and fold patterns imposed on sheet material enable its morphing into three-dimensional shapes.
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