72 research outputs found
Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths
When can a plane graph with prescribed edge lengths and prescribed angles
(from among \}) be folded flat to lie in an
infinitesimally thin line, without crossings? This problem generalizes the
classic theory of single-vertex flat origami with prescribed mountain-valley
assignment, which corresponds to the case of a cycle graph. We characterize
such flat-foldable plane graphs by two obviously necessary but also sufficient
conditions, proving a conjecture made in 2001: the angles at each vertex should
sum to , and every face of the graph must itself be flat foldable.
This characterization leads to a linear-time algorithm for testing flat
foldability of plane graphs with prescribed edge lengths and angles, and a
polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure
Flat Folding an Unassigned Single-Vertex Complex (Combinatorially Embedded Planar Graph with Specified Edge Lengths) without Flat Angles
A foundational result in origami mathematics is Kawasaki and Justin's simple,
efficient characterization of flat foldability for unassigned single-vertex
crease patterns (where each crease can fold mountain or valley) on flat
material. This result was later generalized to cones of material, where the
angles glued at the single vertex may not sum to . Here we
generalize these results to when the material forms a complex (instead of a
manifold), and thus the angles are glued at the single vertex in the structure
of an arbitrary planar graph (instead of a cycle). Like the earlier
characterizations, we require all creases to fold mountain or valley, not
remain unfolded flat; otherwise, the problem is known to be NP-complete (weakly
for flat material and strongly for complexes). Equivalently, we efficiently
characterize which combinatorially embedded planar graphs with prescribed edge
lengths can fold flat, when all angles must be mountain or valley (not unfolded
flat). Our algorithm runs in time, improving on the previous
best algorithm of .Comment: 17 pages, 8 figures, to appear in Proceedings of the 38th
International Symposium on Computational Geometr
A Pseudopolynomial Algorithm for Alexandrov's Theorem
Alexandrov's Theorem states that every metric with the global topology and
local geometry required of a convex polyhedron is in fact the intrinsic metric
of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a
differential equation whose solution leads to the polyhedron corresponding to a
given metric. We describe an algorithm based on this differential equation to
compute the polyhedron to arbitrary precision given the metric, and prove a
pseudopolynomial bound on its running time. Along the way, we develop
pseudopolynomial algorithms for computing shortest paths and weighted Delaunay
triangulations on a polyhedral surface, even when the surface edges are not
shortest paths.Comment: 25 pages; new Delaunay triangulation algorithm, minor other changes;
an abbreviated v2 was at WADS 200
Folding Every Point on a Polygon Boundary to a Point
We consider a problem in computational origami. Given a piece of paper as a
convex polygon and a point located within, fold every point on a
boundary of to and compute a region that is safe from folding, i.e.,
the region with no creases. This problem is an extended version of a problem by
Akitaya, Ballinger, Demaine, Hull, and Schmidt~[CCCG'21] that only folds
corners of the polygon. To find the region, we prove structural properties of
intersections of parabola-bounded regions and use them to devise a linear-time
algorithm. We also prove a structural result regarding the complexity of the
safe region as a variable of the location of point , i.e., the number of
arcs of the safe region can be determined using the straight skeleton of the
polygon .Comment: Preliminary results appeared in JCDCGGG'2
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by Erd˝os
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n ≥ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version
Origami as a Tool for Mathematical Investigation and Error Modelling in Origami Construction
Origami is the ancient Japanese art of paper folding. It has inspired applications in industries ranging from Bio-Medical Engineering to Architecture. This thesis reviews ways in which Origami is used in a number of fields and investigates unexplored areas providing insight and new results which may lead to better understanding and new uses.
The OSME conference series arguably covers most of the research activities in the field of Origami and its links to Science and Mathematics. The thesis provides a comprehensive review of the work that has been presented at these conferences and published in their proceedings.
The mathematics of Origami has been explored before and much of the fundamental work in this field is presented in chapter 3. Here an attempt is made to push the bounds of this field by suggesting ways in which Origami can be used as a mathematical tool for in-depth exploration of non trivial problems. A particular problem we consider is the 4-colour theorem and its proof. Looking at some well known methods for producing angles and lengths mathematically the thesis also explores how accurate these might be. This leads to the surprisingly unstudied field of error modelling in Origami. Errors in folding processes have not previously been looked at from a mathematical point of view. The thesis develops a model for error estimation in crease patterns and a framework for error modelling in Origami applications. By introducing a standardised error into alignments, uniform error bounds for each of the one-fold constructions are generated. This defines a region in which a crease could lie in order to satisfy the alignments of a given fold within a specified tolerance. Analysis of this method on some examples provides insight into how this might be used in multi-fold constructions. An algorithm to that effect is introduced
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Skeleton Structures and Origami Design
In this dissertation we study problems related to polygonal skeleton structures that have applications to computational origami. The two main structures studied are the straight skeleton of a simple polygon (and its generalizations to planar straight line graphs) and the universal molecule of a Lang polygon. This work builds on results completed jointly with my advisor Ileana Streinu.
Skeleton structures are used in many computational geometry algorithms. Examples include the medial axis, which has applications including shape analysis, optical character recognition, and surface reconstruction; and the Voronoi diagram, which has a wide array of applications including geographic information systems (GIS), point location data structures, motion planning, etc.
The straight skeleton, studied in this work, has applications in origami design, polygon interpolation, biomedical imaging, and terrain modeling, to name just a few. Though the straight skeleton has been well studied in the computational geometry literature for over 20 years, there still exists a significant gap between the fastest algorithms for constructing it and the known lower bounds.
One contribution of this thesis is an efficient algorithm for computing the straight skeleton of a polygon, polygon with holes, or a planar straight-line graph given a secondary structure called the induced motorcycle graph.
The universal molecule is a generalization of the straight skeleton to certain convex polygons that have a particular relationship to a metric tree. It is used in Robert Lang\u27s seminal TreeMaker method for origami design. Informally, the universal molecule is a subdivision of a polygon (or polygonal sheet of paper) that allows the polygon to be ``folded\u27\u27 into a particular 3D shape with certain tree-like properties. One open problem is whether the universal molecule can be rigidly folded: given the initial flat state and a particular desired final ``folded\u27\u27 state, is there a continuous motion between the two states that maintains the faces of the subdivision as rigid panels? A partial characterization is known: for a certain measure zero class of universal molecules there always exists such a folding motion. Another open problem is to remove the restriction of the universal molecule to convex polygons. This is of practical importance since the TreeMaker method sometimes fails to produce an output on valid input due the convexity restriction and extending the universal molecule to non-convex polygons would allow TreeMaker to work on all valid inputs. One further interesting problem is the development of faster algorithms for computing the universal molecule. In this thesis we make the following contributions to the study of the universal molecule. We first characterize the tree-like family of surfaces that are foldable from universal molecules. In order to do this we define a new family of surfaces we call Lang surfaces and prove that a restricted class of these surfaces are equivalent to the universal molecules. Next, we develop and compare efficient implementations for computing the universal molecule. Then, by investigating properties of broader classes of Lang surfaces, we arrive at a generalization of the universal molecule from convex polygons in the plane to non-convex polygons in arbitrary flat surfaces. This is of both practical and theoretical interest. The practical interest is that this work removes the case from Lang\u27s TreeMaker method that causes TreeMaker to fail to produce output in the presence of non-convex polygons. The theoretical interest comes from the fact that our generalization encompasses more than just those surfaces that can be cut out of a sheet of paper, and pertains to polygons that cannot be lied flat in the plane without self-intersections. Finally, we identify a large class of universal molecules that are not foldable by rigid folding motions. This makes progress towards a complete characterization of the foldability of the universal molecule
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