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