The polytope of non-crossing graphs on a planar point set

For any finite set \A of $n$ points in $\R^2$, we define a $(3n-3)$-dimensional simple polyhedron whose face poset is isomorphic to the poset of ``non-crossing marked graphs'' with vertex set \A, where a marked graph is defined as a geometric graph together with a subset of its vertices. The poset of non-crossing graphs on \A appears as the complement of the star of a face in that polyhedron. The polyhedron has a unique maximal bounded face, of dimension $2n_i +n -3$ where $n_i$ is the number of points of \A in the interior of \conv(\A). The vertices of this polytope are all the pseudo-triangulations of \A, and the edges are flips of two types: the traditional diagonal flips (in pseudo-triangulations) and the removal or insertion of a single edge. As a by-product of our construction we prove that all pseudo-triangulations are infinitesimally rigid graphs.Comment: 28 pages, 16 figures. Main change from v1 and v2: Introduction has been reshape

Semi-automated geomorphological mapping applied to landslide hazard analysis

Computer-assisted three-dimensional (3D) mapping using stereo and multi-image (“softcopy”) photogrammetry is shown to enhance the visual interpretation of geomorphology in steep terrain with the direct benefit of greater locational accuracy than traditional manual mapping. This would benefit multi-parameter correlations between terrain attributes and landslide distribution in both direct and indirect forms of landslide hazard assessment. Case studies involve synthetic models of a landslide, and field studies of a rock slope and steep undeveloped hillsides with both recently formed and partly degraded, old landslide scars. Diagnostic 3D morphology was generated semi-automatically both using a terrain-following cursor under stereo-viewing and from high resolution digital elevation models created using area-based image correlation, further processed with curvature algorithms. Laboratory-based studies quantify limitations of area-based image correlation for measurement of 3D points on planar surfaces with varying camera orientations. The accuracy of point measurement is shown to be non-linear with limiting conditions created by both narrow and wide camera angles and moderate obliquity of the target plane. Analysis of the results with the planar surface highlighted problems with the controlling parameters of the area-based image correlation process when used for generating DEMs from images obtained with a low-cost digital camera. Although the specific cause of the phase-wrapped image artefacts identified was not found, the procedure would form a suitable method for testing image correlation software, as these artefacts may not be obvious in DEMs of non-planar surfaces.Modelling of synthetic landslides shows that Fast Fourier Transforms are an efficient method for removing noise, as produced by errors in measurement of individual DEM points, enabling diagnostic morphological terrain elements to be extracted. Component landforms within landslides are complex entities and conversion of the automatically-defined morphology into geomorphology was only achieved with manual interpretation; however, this interpretation was facilitated by softcopy-driven stereo viewing of the morphological entities across the hillsides.In the final case study of a large landslide within a man-made slope, landslide displacements were measured using a photogrammetric model consisting of 79 images captured with a helicopter-borne, hand-held, small format digital camera. Displacement vectors and a thematic geomorphological map were superimposed over an animated, 3D photo-textured model to aid non-stereo visualisation and communication of results

The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices

We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of d/2 in dimension d, achieved by the “standard terminal simplices” and direct sums of them. We prove this conjecture up to dimension three and show it to be equivalent to the conjecture of González-Merino and Schymura (Discrete Comput. Geom. 58(3), 663–685 (2017)) that the d-th covering minimum of the standard terminal n-simplex equals d/2, for every n≥d . We also show that these two conjectures would follow from a discrete analog for lattice simplices of Hadwiger’s formula bounding the covering radius of a convex body in terms of the ratio of surface area versus volume. To this end, we introduce a new notion of discrete surface area of non-hollow simplices. We prove our discrete analog in dimension two and give strong evidence for its validity in arbitrary dimension.G. Codenotti and F. Santos were supported by the Einstein Foundation Berlin under grant EVF-2015-230. F. Santos was also supported by grants MTM2017-83750-P/AEI/10.13039/501100011033 and PID2019-106188GB-I00/AEI/10.13039/501100011033 of the Spanish State Research Agency. M. Schymura was supported by the Swiss National Science Foundation (SNSF) within the Project Convexity, geometry of numbers, and the complexity of integer programming (Nr. 163071)

Comparing Features of Three-Dimensional Object Models Using Registration Based on Surface Curvature Signatures

This dissertation presents a technique for comparing local shape properties for similar three-dimensional objects represented by meshes. Our novel shape representation, the curvature map, describes shape as a function of surface curvature in the region around a point. A multi-pass approach is applied to the curvature map to detect features at different scales. The feature detection step does not require user input or parameter tuning. We use features ordered by strength, the similarity of pairs of features, and pruning based on geometric consistency to efficiently determine key corresponding locations on the objects. For genus zero objects, the corresponding locations are used to generate a consistent spherical parameterization that defines the point-to-point correspondence used for the final shape comparison

Fast, accurate, and transferable many-body interatomic potentials by symbolic regression

The length and time scales of atomistic simulations are limited by the computational cost of the methods used to predict material properties. In recent years there has been great progress in the use of machine learning algorithms to develop fast and accurate interatomic potential models, but it remains a challenge to develop models that generalize well and are fast enough to be used at extreme time and length scales. To address this challenge, we have developed a machine learning algorithm based on symbolic regression in the form of genetic programming that is capable of discovering accurate, computationally efficient manybody potential models. The key to our approach is to explore a hypothesis space of models based on fundamental physical principles and select models within this hypothesis space based on their accuracy, speed, and simplicity. The focus on simplicity reduces the risk of overfitting the training data and increases the chances of discovering a model that generalizes well. Our algorithm was validated by rediscovering an exact Lennard-Jones potential and a Sutton Chen embedded atom method potential from training data generated using these models. By using training data generated from density functional theory calculations, we found potential models for elemental copper that are simple, as fast as embedded atom models, and capable of accurately predicting properties outside of their training set. Our approach requires relatively small sets of training data, making it possible to generate training data using highly accurate methods at a reasonable computational cost. We present our approach, the forms of the discovered models, and assessments of their transferability, accuracy and speed

Knot Tightening By Constrained Gradient Descent

We present new computations of approximately length-minimizing polygons with fixed thickness. These curves model the centerlines of "tight" knotted tubes with minimal length and fixed circular cross-section. Our curves approximately minimize the ropelength (or quotient of length and thickness) for polygons in their knot types. While previous authors have minimized ropelength for polygons using simulated annealing, the new idea in our code is to minimize length over the set of polygons of thickness at least one using a version of constrained gradient descent. We rewrite the problem in terms of minimizing the length of the polygon subject to an infinite family of differentiable constraint functions. We prove that the polyhedral cone of variations of a polygon of thickness one which do not decrease thickness to first order is finitely generated, and give an explicit set of generators. Using this cone we give a first-order minimization procedure and a Karush-Kuhn-Tucker criterion for polygonal ropelength criticality. Our main numerical contribution is a set of 379 almost-critical prime knots and links, covering all prime knots with no more than 10 crossings and all prime links with no more than 9 crossings. For links, these are the first published ropelength figures, and for knots they improve on existing figures. We give new maps of the self-contacts of these knots and links, and discover some highly symmetric tight knots with particularly simple looking self-contact maps.Comment: 45 pages, 16 figures, includes table of data with upper bounds on ropelength for all prime knots with no more than 10 crossings and all prime links with no more than 9 crossing