# Search SciRate

• Can folding a piece of paper flat make it larger? We explore whether a shape $S$ must be scaled to cover a flat-folded copy of itself. We consider both single folds and arbitrary folds (continuous piecewise isometries $S\rightarrow R^2$). The underlying problem is motivated by computational origami, and is related to other covering and fixturing problems, such as Lebesgue's universal cover problem and force closure grasps. In addition to considering special shapes (squares, equilateral triangles, polygons and disks), we give upper and lower bounds on scale factors for single folds of convex objects and arbitrary folds of simply connected objects.
• We discuss in this paper a method of finding skyline or non-dominated points in a set $P$ of $n_P$ points with respect to a set $S$ of $n_S$ sites. A point $p_i \in P$ is non-dominated if and only if for each $p_j \in P$, $j \not= i$, there exists at least one point $s \in S$ that is closer to $p_i$ than $p_j$. We reduce this problem of determining non-dominated points to the problem of finding sites that have non-empty cells in an additive Voronoi diagram with a convex distance function. The weights of the additive Voronoi diagram are derived from the co-ordinates of the points of $P$ and the convex distance function is derived from $S$. In the 2-dimensional plane, this reduction gives a $O((n_S + n_P)\log n_S + n_P \log n_P)$-time randomized incremental algorithm to find the non-dominated points.
• Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that open'' polyhedra with triangular faces may not be unfoldable no matter how they are cut.