4 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
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
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 {0,180°, 360°}) be folded flat to lie in an infinitesimally thick 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 360°, 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