35 research outputs found
When Can You Fold a Map?
We explore the following problem: given a collection of creases on a piece of
paper, each assigned a folding direction of mountain or valley, is there a flat
folding by a sequence of simple folds? There are several models of simple
folds; the simplest one-layer simple fold rotates a portion of paper about a
crease in the paper by +-180 degrees. We first consider the analogous questions
in one dimension lower -- bending a segment into a flat object -- which lead to
interesting problems on strings. We develop efficient algorithms for the
recognition of simply foldable 1D crease patterns, and reconstruction of a
sequence of simple folds. Indeed, we prove that a 1D crease pattern is
flat-foldable by any means precisely if it is by a sequence of one-layer simple
folds.
Next we explore simple foldability in two dimensions, and find a surprising
contrast: ``map'' folding and variants are polynomial, but slight
generalizations are NP-complete. Specifically, we develop a linear-time
algorithm for deciding foldability of an orthogonal crease pattern on a
rectangular piece of paper, and prove that it is (weakly) NP-complete to decide
foldability of (1) an orthogonal crease pattern on a orthogonal piece of paper,
(2) a crease pattern of axis-parallel and diagonal (45-degree) creases on a
square piece of paper, and (3) crease patterns without a mountain/valley
assignment.Comment: 24 pages, 19 figures. Version 3 includes several improvements thanks
to referees, including formal definitions of simple folds, more figures,
table summarizing results, new open problems, and additional reference
Minkowski operators for voxel based sculpting
A sculpting package provides the user with a set of primitive shapes and a set of operators to operate on them. Voxel based representations are attractive for sculpting due to their ability to work with arbitrary topology with uniform ease. Minkowski operators have been found useful in providing some of the common tools for Interactive Sculpting. This paper gives algorithms for implementing these tools for voxel grids and octrees
Data Structures for Maintaining Set Partitions
Eciently maintaining the partition induced by a set of features is an important problem in building decision-tree classi ers. In order to identify a small set of discriminating features, we need the capability of eciently adding and removing speci c features and determining the eect of these changes on the induced classi cation or partition