1. Modelling the folding of paper into three dimensions using affine transformations
- Author
-
Thomas C. Hull and sarah-marie belcastro
- Subjects
Paper ,Transformations ,Numerical Analysis ,Algebra and Number Theory ,Non-flat folding ,Affine ,Identity matrix ,Folding ,Folding (DSP implementation) ,Computer Science::Computational Geometry ,Vertex (geometry) ,Combinatorics ,Homogeneous ,Product (mathematics) ,Piecewise ,Discrete Mathematics and Combinatorics ,Order (group theory) ,Geometry and Topology ,Affine transformation ,Mathematics - Abstract
We model the folding of ordinary paper via piecewise isometries R 2 → R 3 . The collection of crease lines and vertices in the unfolded paper is called the crease pattern. Our results generalize the previously known necessity conditions from the more restrictive case of folding paper flat (into R 2 ); if the crease pattern is foldable, then the product (in a non-intuitive order) of the associated rotational matrices is the identity matrix. This condition holds locally in a multiple vertex crease pattern and can be adapted to a global condition. Sufficiency conditions are significantly harder, and are not known except in the two-dimensional single-vertex case.
- Published
- 2002