index.md

Elastic Surface Embedding

TL;DR

You can make a holdable smooth surface model with this repository.

The main part of this project is how to determine a planer shape from a strip on curved surface. In mathematics, this mapping is called "embedding". We determined the embedding by minimizing its elastic strain energy. This is the meaning of "Elastic Surface Embedding".

Overview: How to make a surface model

step 1 : Define a shape of surface (and split into strips)

The definition must consists of parametric mapping and its domain. For example, a paraboloid can be parametrized as below.

$$\begin{aligned} \bm{p}_{[0]}(u^1, u^2) &= \begin{pmatrix} u^1 \\\ u^2 \\\ (u^1)^2 + (u^2)^2 \end{pmatrix} \\\ D &= [-1,1]\times[-1,1] \end{aligned}$$

The domain D will be split into D_i.

$$\begin{aligned} D_i &= [-1,1]\times\left[\frac{i-1}{10},\frac{i}{10}\right] & (i=1,\dots,10) \end{aligned}$$

step 2 : Numerical analysis

This is the main part. Split the surface into pieces, and compute the Eucledian embedding. For more information, read [numerical computation section](@ref numerical_computation). The image below is a result for the domain D_1.

step 3 : Edit on vector graphics editor

The output files are SVG format. After editing the svg files, you can print the graphics or cut papers by laser cutting machine.

step 4 : Craft a paper model

This is the final step. Cut papers into strips, and weave them into surface.

Directions: If you like..

..making crafts ✂️

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        Download and print a paraboloid example or a hyperbolic paraboloid example from <a href="https://arxiv.org/abs/2211.06372">my paper on arXiv</a>, and <a href="../craft">make your own surface model</a>.
        Laser cutting machine is useful, but it's not necessary.
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..computing 💻

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        Clone this repository, and run <a href="../run-julia">the Julia script</a> or <a href="../run-wolfram">the Wolfram script</a>!
        Any issues and pull requests are welcomed.
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..mathematics or physics 🌐

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        Read <a href="https://arxiv.org/abs/2211.06372">my paper on arXiv</a>. Here's our theoretical framework:
        <ul>
            <li>Mathematical model: <a href="https://www.sciencedirect.com/topics/engineering/geometric-nonlinearity">Nonlinear elasticity</a> on <a href="https://en.m.wikipedia.org/wiki/Riemannian_manifold">Riemannian manifold</a></li>
            <li>Geometric representation: <a href="https://en.wikipedia.org/wiki/Non-uniform_rational_B-spline">B-spline manifold</a></li>
            <li>Numerical analysis: <a href="https://en.wikipedia.org/wiki/Galerkin_method">Galerkin method</a>, <a href="https://en.wikipedia.org/wiki/Newton%27s_method">Newton-Raphson method</a></li>
        </ul>
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..me! 🐢

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        <ul>
            <li>Follow <a href="https://twitter.com/Hyrodium">my twitter account</a>!</li>
            <li>Visit <a href="https://hyrodium.github.io/">my website</a>!</li>
            <li>Read <a href="https://arxiv.org/abs/2211.06372">my paper on arXiv</a>!</li>
            <li>Give star to <a href="https://github.com/hyrodium/ElasticSurfaceEmbedding.jl">this repository</a>!</li>
        </ul>
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