# Curve-Crease Folded Panels

**Applying dynamic relaxation techniques to form-find and manufacture curve-crease folded panels**

##### Shajay Bhooshan^{1}, Paul Shepherd^{2}, Paul Richens^{3}

^{1}ETH Zurich

^{1}ETH Zurich^{2}University of Bath

^{2}University of Bath^{3}University of Westminster

^{3}University of Westminster**Image 1: The completed sculpture at the Venice Biennale 2012**

The research incorporated in the paper stems from the design and fabrication of a self-supporting, multi-panel installation for the Venice Biennale 2012 and operates against the backdrop of the exciting potentials that the field of curved-crease folding offers in the development of curved surfaces that can be manufactured from sheet material.

The two main challenges were developing an intuitive design strategy and production of information adhering to manufacturing constraints.

**Figure 1: Showing preffered design method of starting with a coarse mesh, subdividing it and ‘relaxing’ it minimal conditions – either minimal mean curvature or minimal Gaussian. Also indicates indicated manufacture method.**

The essential contribution of the paper is a proposed interactive form-finding method for curve-crease geometries that could negotiate the multiple objectives of ease of use in exploratory design, and manufacturing constraints of their architectural scale assemblies.

**Figure 4 shows the forces and fixing scheme**

The research stems from the design and fabrication of a self-supporting, multi-panel installation for the Venice Biennale 2012 and operates against the backdrop of the exciting potentials that the field of curved-crease folding offers in the development of curved surfaces that can be manufactured from sheet material.

**Figure 3: Shows the use of a chamfer and bevel Conway operators (Conway J., 2008) on a cube, the corresponding subdivision surface and derived topology of rulings of a tri-fold panel.**

The two main challenges were developing an intuitive design strategy and production of information adhering to manufacturing constraints.

The essential contribution of the paper is a proposed form-finding method for curve-crease geometries that could negotiate the multiple objectives of ease of use in exploratory design, and manufacturing constraints of their architectural-scale assemblies.

**Figure 7 for boundary vertex and associated gradient.**

There are several seminal design and art precedents within this field – Richard Sweeney, David Huffman, Erik Demaine etc. Most of the precedents projects and available literature on design methods highlight the difficulty in developing an intuitive, exploratory digital-design method to generate feasible 3D geometries.

**Figure 8 differences caused by various force combinations**

Our initial survey of methods included both the simple and common method – the method of reflection – and the involved Planar-Quadmeshes and optimization-based method.

**Figure 9 shows three variants of adding extra faces – in plane, extrude normally out and normally in and the resulting equilibrium meshes.**

Most methods, including the two above, presented difficulties towards incorporation within an intuitive, edit-and-observe method of design; The first one proving difficult to explore variety of generalized solutions free of prior assumptions and the second one being elaborate involving scanning of physical paper models, proprietary optimization algorithms etc.

**Figure 10 showing planar developments. (L to R) conformal, authalic, isomertric.**

For an extensive overview on the precedents, and computational methods related to curved crease folding, we refer the reader to a survey. Further, we were particularly interested in the recent developments of physicallybased, interactive tools that operate on user-specified coarse linear piecewise complexes that are iteratively subdivided and perturbed to produce feasible solutions via energy minimization methods.

**Figure 12 shows various input meshes and resulting output meshes from our algorithm.**

This is in alignment with established benefits of subdivision surface based modeling paradigm in architectural form-finding, and the application of dynamic relaxation techniques on subdivision surfaces to design and fabricate minimal mean curvature surfaces – so called minimal surfaces.

**Figure 14 showing procedural generation of input mesh and out-ofplane measure of the generated mesh.**

The method proposed in this paper follows from these observations, and an explicit intention to perturb input 3D geometries to find feasible geometry as opposed to finding the folded state of a 2D input mesh.

**Figure 15 showing Panel singularities**

It may be noted that the optimization-based method proposed by does in fact solve this problem, albeit it is more difficult to implement. Our method is easier to implement and extend. However, unlike their method relies on the designer to provide an initial mesh with appropriate topology.

**Figure 16 showing panel unfold visualizations**

We show simple procedural methods involving known meshoperations that can be used to produce the initial mesh and the subsequent use of dynamic relaxation (DR) techniques to iteratively perturb the surface towards minimal Gaussian curvature and local planarity.

**Figure 17 showing rearrangement of panels to form adjacency bands**

The paper will proceed by describing key discoveries made in applying DR to design individual panels with a few crease folds , and the subsequent incorporation of those discoveries in the design and manufacture of selfsupporting, multiple-panel configurations.

**Figure 18 showing panel boundary edges before and after the merging process (above and below)**

The various caveats and limitations of the proposed method are noted in the previous section. As a proof-of concept, the proposed method was successfully employed to design and fabricate a self-supporting structure composed on folded panels, in relatively short span of time – design to completion in 3 months.

**Figure 19 showing plotting unfold error per panel edge and per band provided a holistic overview of its relationship to surface topology**

Further the proposed method was found to be designer-friendly in that it utilisizes popular mesh modeling procedures, thereby easing the assimilation into established contemporary design workflows.

The method also allows for multiple possibilities of feedback and iteration within the various steps of the process, thus allowing for multiple collaborative inputs to be assimilated during the process of design. In short, the method allows for integration into a general interactive and iterative form-finding framework that can be employed for finding minimal surfaces – both Gaussian-curvature minimal and mean-curvature minimal – along with their planar developments, allowing for their manufacture from sheet material – paper / metal and stretched fabric.

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