Möbius strips are prototypical examples of ribbon-like structures. Inspecting their shapes and features provides useful insights into the rich mechanics of elastic ribbons. Despite their ubiquity and ease of construction, quantitative experimental measurements of the three-dimensional shapes of Möbius strips are surprisingly non-existent in the literature. We propose two novel stereo vision-based techniques to this end—a marker-based technique that determines a Lagrangian description for the construction of a Möbius strip, and a structured light illumination technique that furnishes an Eulerian description of its shape. Our measurements enable a critical evaluation of the predictive capabilities of mechanical theories proposed to model Möbius strips. We experimentally validate, seemingly for the first time, the developable strip and the Cosserat plate theories for predicting shapes of Möbius strips. Equally significantly, we confirm unambiguous deficiencies in modelling Möbius strips as Kirchhoff rods with slender cross-sections. The experimental techniques proposed and the Cosserat plate model promise to be useful tools for investigating a general class of problems in ribbon mechanics.

莫比乌斯带是带状结构的典型例子。检查它们的形状和特征可以提供对弹性带丰富力学的有用见解。尽管它们无处不在且易于构建，但令人惊讶的是，文献中并不存在莫比乌斯带三维形状的定量实验测量。为此，我们提出了两种新颖的基于立体视觉的技术——一种基于标记的技术，用于确定莫比乌斯带构造的拉格朗日描述，以及一种结构光照明技术，用于提供其形状的欧拉描述。我们的测量结果能够对建议用于模拟莫比乌斯带的机械理论的预测能力进行批判性评估。我们通过实验验证，似乎是第一次，用于预测莫比乌斯带形状的可展带和 Cosserat 板理论。同样重要的是，我们确认了将莫比乌斯带建模为具有细长横截面的基尔霍夫棒的明确缺陷。提出的实验技术和 Cosserat 板模型有望成为研究带状力学中一类一般问题的有用工具。