Folding Polyominoes into (Poly)Cubes

We study the problem of folding a polyomino $P$ into a polycube $Q$, allowing faces of $Q$ to be covered multiple times. First, we define a variety of folding models according to whether the folds (a) must be along grid lines of $P$ or can divide squares in half (diagonally and/or orthogonally), (b) must be mountain or can be both mountain and valley, (c) can remain flat (forming an angle of $180^\circ$), and (d) must lie on just the polycube surface or can have interior faces as well. Second, we give all the inclusion relations among all models that fold on the grid lines of $P$. Third, we characterize all polyominoes that can fold into a unit cube, in some models. Fourth, we give a linear-time dynamic programming algorithm to fold a tree-shaped polyomino into a constant-size polycube, in some models. Finally, we consider the triangular version of the problem, characterizing which polyiamonds fold into a regular tetrahedron.Comment: 30 pages, 19 figures, full version of extended abstract that appeared in CCCG 2015. (Change over previous version: Fixed a missing reference.

ConTesse: Accurate Occluding Contours for Subdivision Surfaces

This paper proposes a method for computing the visible occluding contours of subdivision surfaces. The paper first introduces new theory for contour visibility of smooth surfaces. Necessary and sufficient conditions are introduced for when a sampled occluding contour is valid, that is, when it may be assigned consistent visibility. Previous methods do not guarantee these conditions, which helps explain why smooth contour visibility has been such a challenging problem in the past. The paper then proposes an algorithm that, given a subdivision surface, finds sampled contours satisfying these conditions, and then generates a new triangle mesh matching the given occluding contours. The contours of the output triangle mesh may then be rendered with standard non-photorealistic rendering algorithms, using the mesh for visibility computation. The method can be applied to any triangle mesh, by treating it as the base mesh of a subdivision surface.Comment: Accepted to ACM Transactions on Graphics (TOG

From Curves to Words and Back Again: Geometric Computation of Minimum-Area Homotopy

Let $\gamma$ be a generic closed curve in the plane. Samuel Blank, in his 1967 Ph.D. thesis, determined if $\gamma$ is self-overlapping by geometrically constructing a combinatorial word from $\gamma$. More recently, Zipei Nie, in an unpublished manuscript, computed the minimum homotopy area of $\gamma$ by constructing a combinatorial word algebraically. We provide a unified framework for working with both words and determine the settings under which Blank's word and Nie's word are equivalent. Using this equivalence, we give a new geometric proof for the correctness of Nie's algorithm. Unlike previous work, our proof is constructive which allows us to naturally compute the actual homotopy that realizes the minimum area. Furthermore, we contribute to the theory of self-overlapping curves by providing the first polynomial-time algorithm to compute a self-overlapping decomposition of any closed curve $\gamma$ with minimum area.Comment: 27 pages, 16 figure

ConTesse: Accurate Occluding Contours for Subdivision Surfaces

International audienceThis paper proposes a method for computing the visible occluding contours of subdivision surfaces. The paper first introduces new theory for contour visibility of smooth surfaces. Necessary and sufficient conditions are introduced for when a sampled occluding contour is valid, that is, when it may be assigned consistent visibility. Previous methods do not guarantee these conditions, which helps explain why smooth contour visibility has been such a challenging problem in the past. The paper then proposes an algorithm that, given a subdivision surface, finds sampled contours satisfying these conditions, and then generates a new triangle mesh matching the given occluding contours. The contours of the output triangle mesh may then be rendered with standard non-photorealistic rendering algorithms, using the mesh for visibility computation. The method can be applied to any triangle mesh, by treating it as the base mesh of a subdivision surface

Self-overlapping curves revisited

Let S be a surface embedded in space in such a way that each point has a neighborhood within which the surface is a terrain. Then S projects to an immersed surface in the plane, the boundary of which is a (possibly self-intersecting) curve. Under what circumstances can we reverse these mappings algorithmically? Shor and van Wyk considered one such problem, determining whether a curve is the boundary of an immersed disk; they showed that the self-overlapping curves defined in this way can be recognized in polynomial time. We show that several related problems are more difficult: it is NP-complete to determine whether an immersed disk is the projection of a disk embedded in space, or whether a curve is the boundary of an immersed surface in the plane that is not constrained to be a disk. However, when a casing is supplied with a self-intersecting curve, describing which component of the curve lies above and which below at each crossing, we may determine in time linear in the number of crossings whether the cased curve forms the projected boundary of a surface in space. As a related result, we show that an immersed surface with a single boundary curve that crosses itself n times has at most 2n/2 combinatorially distinct spatial embeddings, and we discuss the existence of fixed-parameter tractable algorithms for related problems