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For many shape analysis problems in computer vision and scientific imaging (e.g., computational anatomy, morphological cytometry), the ability to align two closed curves in the plane is crucial. In this paper, we concentrate on rigidly…

This paper presents a method of reconstruction a primary structure of a protein that folds into a given geometrical shape. This method predicts the primary structure of a protein and restores its linear sequence of amino acids in the…

The reassembling of a simple connected graph G = (V,E) is an abstraction of a problem arising in earlier studies of network analysis. The reassembling process has a simple formulation (there are several equivalent formulations) relative to…

Let $\mathcal{T}$ be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles,…

Leaves and flowers frequently have a characteristic rippling pattern at their edges. Recent experiments found similar patterns in torn plastic. These patterns can be reproduced by imposing metrics upon thin sheets. The goal of this paper is…

We study maps from a 2D world-sheet to a 2D target space which include folds. The geometry of folds is discussed and a metric on the space of folded maps is written down. We show that the latter is not invariant under area preserving…

The evaluation of a web page with respect to a query is a vital task in the web information retrieval domain. This paper proposes the evaluation of a web page as a bottom-up process from the segment level to the page level. A model for…

A reduced model, which can fold both helix and sheet structures, is proposed to study the problem of protein folding. The goal of this model is to find an unbiased effective potential that has included the effects of water and at the same…

We consider a disk-shaped thin elastic sheet bonded to a compliant sphere. (Our sheet can slip along the sphere; the bonding controls only its normal displacement.) If the bonding is stiff (but not too stiff), the geometry of the sphere…

The Planar Contraction problem is to test whether a given graph can be made planar by using at most k edge contractions. This problem is known to be NP-complete. We show that it is fixed-parameter tractable when parameterized by k.

A cornerstone of geometric reconstruction, rotation averaging seeks the set of absolute rotations that optimally explains a set of measured relative orientations between them. In spite of being an integral part of bundle adjustment and…

The protein folding problem must ultimately be solved on all length scales from the atomic up through a hierarchy of complicated structures. By analyzing the stability of the folding process using physics and mathematics, this paper shows…

We study flip graphs of triangulations whose maximum vertex degree is bounded by a constant $k$. In particular, we consider triangulations of sets of $n$ points in convex position in the plane and prove that their flip graph is connected if…

We consider a problem in computational origami. Given a piece of paper as a convex polygon $P$ and a point $f$ located within, fold every point on a boundary of $P$ to $f$ and compute a region that is safe from folding, i.e., the region…

For a polygonal linkage, we produce a fast navigation algorithm on its configuration space. The basic idea is to approximate the configuration space by the vertex-edge graph of its cell decomposition discovered by the first author. The…

We study the problem of rotating a simple polygon to contain the maximum number of elements from a given point set in the plane. We consider variations of this problem where the rotation center is a given point or lies on a line segment, a…

We study the spontaneous folding of a 2D template of microscopic panels into a 3D pyramid, driven by thermal fluctuations. Combining numerical simulations and analytical calculations, we find that the total folding time is a non-monotonic…

We study the problem of finding maximum-area triangles that can be inscribed in a polygon in the plane. We consider eight versions of the problem: we use either convex polygons or simple polygons as the container; we require the triangles…

For a drop on an incline with small tilt angle or small Bond number, when the contact line is a circle, we show that the retentive force factor is exactly pi/2 at first order in the tilt angle or the Bond number. The retentive force factor…

A folded disk is bistable, as it can be popped through to an inverted state with elastic energy localized in a small, highly-deformed region on the fold. Cutting out this singularity relaxes the surrounding material and leads to a loss of…