Elastic Surface Embedding; Weaving Parer Strips

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 the target 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 a surface (and split it into strips)

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

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

The domain will be split into $D^{(i)}$.

$$ \begin{aligned} D^{(i)} = [-1,1] \times \left[\frac{i-1}{10}, \frac{i}{10}\right] \qquad (i = 1,...,10) \end{aligned} $$

step 2: Numerical analysis

This is the main part. Split the surface into pieces, and compute the Euclidean embedding. For more information, read this document. The image below is a result for the domain $D^{(1)}$.