Vertex-Unfoldings of Simplicial Manifolds

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Title: Vertex-Unfoldings of Simplicial Manifolds

Abstract: This research focuses on the problem of unfolding a triangulated 2-manifold (a type of surface made up of triangles) into a connected, planar layout without overlapping. The main goal is to develop an algorithm that can perform this unfolding in a linear amount of time. The algorithm is designed to work with connected triangulated 2-manifolds, which include simplicial polyhedra of any genus, manifolds with any number of boundary components, and even complex shapes like the Klein bottle.

The algorithm is based on the concept of a facet path, a sequence of triangles that are connected at common vertices but not necessarily along edges. The main idea is to lay out each triangle of the path in an empty vertical strip of the plane. This process is repeated for all facets in the path, resulting in a non-overlapping vertex-unfolding.

The research also extends the algorithm to higher dimensions, providing a solution for simplicial manifolds of arbitrary dimension. The results of this research have implications in various fields, including computer graphics, manufacturing, and topology.

Keywords:

  • Simplicial Manifolds
  • Triangulated 2-Manifolds
  • Vertex-Unfoldings
  • Non-Overlapping Layouts
  • Connected Planar Layouts
  • Linear Time Algorithms
  • Hinged Dissections
  • Ideal Rendering

Link to Article: https://arxiv.org/abs/0110054v1 Authors: arXiv ID: 0110054v1

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