GitHub

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. 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 ✂️

Download and print a paraboloid example or a hyperbolic paraboloid example from my paper on arXiv, and make your own surface model. Laser cutting machine is useful, but it's not necessary.

..computing 💻

Clone this repository, and run the Julia script or the Wolfram script! Any issues and pull requests are welcomed.

..mathematics or physics 🌐

Read my paper on arXiv. Here's our theoretical framework:

..me! 🐢