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Straightest Geodesics on Polyhedral Surfaces

ACM SIGGRAPH 2006 Courses on - SIGGRAPH '06, 1998
Geodesic curves are the fundamental concept in geometry to generalize the idea of straight lines to curved surfaces and arbitrary manifolds. On polyhedral surfaces we introduce the notion of discrete geodesic curvature of curves and define straightest geodesics.
Konrad Polthier, Markus Schmies
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Origamizing Polyhedral Surfaces

IEEE Transactions on Visualization and Computer Graphics, 2010
This paper presents the first practical method for "origamizing" or obtaining the folding pattern that folds a single sheet of material into a given polyhedral surface without any cut. The basic idea is to tuck fold a planar paper to form a three-dimensional shape.
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Approximating Surfaces With Polyhedral Ones

The Annals of Mathematics, 1957
If J is a simple closed curve in the plane E2, the Schoenflies Theorem says that there is a homeomorphism of E2 onto itself that takes J onto a circle. The theorem does not generalize directly to E3 because there is a simple surface S in E3 such that there is no homeomorphism of E3 onto itself that takes S onto the surface of a sphere. However, if S is
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Shortest paths on polyhedral surfaces

2005
An algorithm is presented that finds the shortest path between two points on a polyhedral surface in O(n5) time, where n is the number of vertices on the surface, thereby establishing that the problem can be solved in polynomial time. The path is confined to the surface, and shortest is defined in terms of Euclidean distance.
Joseph O'Rourke, Subhash Suri, Heather Booth   +2 more
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Computing minimal distances on polyhedral surfaces

IEEE Transactions on Pattern Analysis and Machine Intelligence, 1989
The authors implement an algorithm that finds minimal (geodesic) distances on a three-dimensional polyhedral surface. The algorithm is intrinsically parallel, in as much as it deals with all nodes simultaneously, and is simple to implement. Although exponential in complexity, it can be used with a companion gradient-descent surface-flattening algorithm
Estarose Wolfson, Eric L. Schwartz
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Generalized shape operators on polyhedral surfaces

Computer Aided Geometric Design, 2011
This paper presents the theory about the approximation of the shape operator of smooth surfaces in \(\mathbb{R}^{3}\). The authors give two shape operators, which are defined as (vector-valued) linear functionals on a Sobolev space of (weakly differentiable) vector fields, for smooth surfaces and polyhedral surfaces, respectively.
Klaus Hildebrandt, Konrad Polthier
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Piecewise Linear Approximation and Polyhedral Surfaces

Journal of Mathematical Sciences, 2014
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Pugach, P. A., Shlyk, V. A.
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Metamorphosis of Polyhedral Surfaces using Decomposition

Computer Graphics Forum, 2002
This paper describes an algorithm for morphing polyhedral surfaces based on their decompositions into patches. The given surfaces need neither be genus-zero nor two-manifolds. We present a new algorithm for decomposing surfaces into patches. We also present a new projection scheme that handles topologically cylinder-like polyhedral surfaces.
Shymon Shlafman, Ayellet Tal, Sagi Katz
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Polyhedral Surface Modeling with a Diffusion System

Computer Graphics Forum, 1997
This paper presents a method of generating polyhedral surfaces by using a diffusion system that calculates the positional and normal vectors on their vertices. The system generates smooth shapes that satisfy the minimum norm property, and can be extended to imitate the shape controls of curvature continuous surfaces with bias and tension parameters ...
Shigeru Kuriyama, K. Tachibana
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On flexible polyhedral surfaces

Proceedings of the Steklov Institute of Mathematics, 2015
A polyhedral 2-dimensional surface in Euclidean 3-space is said to be flexible if its spatial shape can be changed continuously due to changes of its dihedral angles only, i.\,e., if every face remains self-congruent during this deformation. The deformation is called a flex.
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