Point Configurations Research Articles

A hybrid registration method using the mandibular bone surface for electromagnetic navigation in mandibular surgery.

To utilize navigated mandibular (reconstructive) surgery, accurate registration of the preoperative CT scan with the actual patient in the operating room (OR) is required. In this phantom study, the feasibility of a noninvasive hybrid registration method is assessed. This method consists of a point registration with anatomic landmarks for initialization and a surface registration using the bare mandibular bone surface for optimization. Three mandible phantoms with reference notches on two osteotomy planes were 3D printed. An electromagnetic tracking system in combination with 3D Slicer software was used for navigation. Different configurations, i.e., different surface point areas and number and configuration of surface points, were tested with a dentate phantom (A) in a metal-free environment. To simulate the intraoperative environment and different anatomies, the registration procedure was also performed with an OR bed using the dentate phantom and two (partially) edentulous phantoms with atypical anatomy (B and C). The accuracy of the registration was calculated using the notches on the osteotomy planes and was expressed as the target registration error (TRE). TRE values of less than 2.0mm were considered as clinically acceptable. In all experiments, the mean TRE was less than 2.0mm. No differences were found using different surface point areas or number or configurations of surface points. Registration accuracy in the simulated intraoperative setting was-mean (SD)-0.96 (0.22), 0.93 (0.26), and 1.50 (0.28) mm for phantom A, phantom B, and phantom C. Hybrid registration is a noninvasive method that requires only a small area of the bare mandibular bone surface to obtain high accuracy in phantom setting. Future studies should test this method in clinical setting during actual surgery.

On the minimax spherical designs

AbstractDistributing points on a (possibly high‐dimensional) sphere with minimal energy is a long‐standing problem in and outside the field of mathematics. This paper considers a novel energy function that arises naturally from statistics and combinatorial optimization, and studies its theoretical properties. Our result solves both the exact optimal spherical point configurations in certain cases and the minimal energy asymptotics under general assumptions. Connections between our results and the L1‐principal component analysis and quasi‐Monte Carlo methods are also discussed.

Minkowski summands of cubes

In pioneering works of Meyer and of McMullen in the early 1970s, the set of Minkowski summands of a polytope was shown to be a polyhedral cone called the type cone. Explicit computations of type cones are in general intractable. Nevertheless, we show that the type cone of the product of simplices is simplicial. This remarkably simple result derives from insights about rainbow point configurations and the work of McMullen.

Points and Lines Configurations for Perpendicular Bisectors of Convex Cyclic Polygons

We characterize the topological configurations of points and lines that may arise when placing $n$ points on a circle and drawing the $n$ perpendicular bisectors of the sides of the corresponding convex cyclic $n$-gon. We also provide exact and asymptotic formulas describing a random realizable configuration, obtained either by sampling the points uniformly at random on the circle or by sampling a realizable configuration uniformly at random.

Tolerance for colorful Tverberg partitions

Tverberg’s theorem bounds the number of points Rd needed for the existence of a partition into r parts whose convex hulls intersect. If the points are colored with N colors, we seek partitions where each part has at most one point of each color. In this manuscript, we bound the number of color classes needed for the existence of partitions where the convex hulls of the parts intersect even after any set of t colors is removed. We prove asymptotically optimal bounds for t when r≤d+1, improve known bounds when r>d+1, and give a geometric characterization for the configurations of points for which t=N−o(N).

An intrinsic parameter calibration method for R-LAT system based on CMM

Rotary-laser automatic theodolite (R-LAT) system is a distributed large-scale metrology system, which provides parallel measurement in scalable measurement room without obvious precision loss. Each of R-LAT emits two nonparallel laser planes to scan the measurement space via evenly rotations, while the photoelectric sensors receive these laser plane signals and perform the coordinate calculation based on triangulation. The accurate geometric parameters of the two laser planes play a crucial role in maintaining the measurement precision of R-LAT system. In practice, the geometry of the two laser planes, which is termed as intrinsic parameters, is usually unknown after assembled. Therefore, how to figure out the accurate intrinsic parameter of each R-LAT is a fundamental question for the application of R-LAT system. This paper proposed an easily operated intrinsic parameter calibration method for R-LAT system by adopting coordinate measurement machine. The mathematical model of laser planes and the observing equation group of R-LAT are established. Then, the intrinsic calibration is formulated as a nonlinear least-square problem that minimizes the sum of deviations of target points and laser planes, and the ascertainment of its initial guess is introduced. At last, experience is performed to verify the effectiveness of this method, and simulations are carried out to investigate the influence of the target point configuration in the accuracy of intrinsic parameters.

Dynamical Moduli Spaces and Polynomial Endomorphisms of Configurations

A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there exist polynomials realizing the dynamics of a given portrait. We present results and pose questions inspired by a large-scale computational survey of intersections of portrait moduli spaces for polynomials in low degree.

Standard Complexes of Matroids and Lattice Paths

Motivated by Gröbner basis theory for finite point configurations, we define and study the class of standard complexes associated to a matroid. Standard complexes are certain subcomplexes of the independence complex that are invariant under matroid duality. For the lexicographic term order, the standard complexes satisfy a deletion-contraction-type recurrence. We explicitly determine the lexicographic standard complexes for lattice path matroids using classical bijective combinatorics.

Arrangements of Approaching Pseudo-Lines

We consider arrangements of n pseudo-lines in the Euclidean plane where each pseudo-line ell _i is represented by a bi-infinite connected x-monotone curve f_i(x), x in mathbb {R}, such that for any two pseudo-lines ell _i and ell _j with i !<! j, the function x !mapsto ! f_j(x) !-! f_i(x) is monotonically decreasing and surjective (i.e., the pseudo-lines approach each other until they cross, and then move away from each other). We show that such arrangements of approaching pseudo-lines, under some aspects, behave similar to arrangements of lines, while for other aspects, they share the freedom of general pseudo-line arrangements. For the former, we prove:There are arrangements of pseudo-lines that are not realizable with approaching pseudo-lines.Every arrangement of approaching pseudo-lines has a dual generalized configuration of points with an underlying arrangement of approaching pseudo-lines. For the latter, we show:There are 2^{Theta (n^2)} isomorphism classes of arrangements of approaching pseudo-lines (while there are only 2^{Theta (n log n)} isomorphism classes of line arrangements).It can be decided in polynomial time whether an allowable sequence is realizable by an arrangement of approaching pseudo-lines. Furthermore, arrangements of approaching pseudo-lines can be transformed into each other by flipping triangular cells, i.e., they have a connected flip graph, and every bichromatic arrangement of this type contains a bichromatic triangular cell.

Strain Response in the Facet Joint Capsule During Physiological Joint Rotation and Translation Following a Simulated Impact Exposure: An In Vitro Porcine Model.

Low back pain (LBP) is frequently reported following rear impact collisions. Knowledge of how the facet joint capsule (FJC) mechanically behaves before and after rear impact collisions may help explain LBP development despite negative radiographic evidence of gross tissue failure. This study quantified the Green strain tensor in the facet joint capsule during rotation and translation range-of-motion tests completed before and following an in vitro simulation of a rear impact collision. Eight FSUs (4 C3-C4, 4 C5-C6) were tested. Following a preload test, FSUs were flexed and extended at 0.5 deg/s until an ±8 N·m moment was achieved. Anterior and posterior joint translation was then applied at 0.2 mm/s until a target ±400 N shear load was imposed. Markers were drawn on the facet capsule surface and their coordinates were tracked during pre- and postimpact range-of-motion tests. Strain was defined as the change in point configuration relative to the determined neutral joint posture. There were no significant differences (p > 0.05) observed in all calculated FJC strain components in rotation and translation before and after the simulated impact. Our results suggest that LBP development resulting from the initiation of strain-induced mechanoreceptors and nociceptors with the facet joint capsule is unlikely following a severe rear impact collision within the boundaries of physiological joint motion.

Accuracy Assessment of a UAV Direct Georeferencing Method and Impact of the Configuration of Ground Control Points

Unmanned aerial vehicles (UAVs) can obtain high-resolution topography data flexibly and efficiently at low cost. However, the georeferencing process involves the use of ground control points (GCPs), which limits time and cost effectiveness. Direct georeferencing, using onboard positioning sensors, can significantly improve work efficiency. The purpose of this study was to evaluate the accuracy of the Global Navigation Satellite System (GNSS)-assisted UAV direct georeferencing method and the influence of the number and distribution of GCPs. A FEIMA D2000 UAV was used to collect data, and several photogrammetric projects were established. Among them, the number and distribution of GCPs used in the bundle adjustment (BA) process were varied. Two parameters were considered when evaluating the different projects: the ground-measured checkpoints (CPs) root mean square error (RMSE) and the Multiscale Model to Model Cloud Comparison (M3C2) distance. The results show that the vertical and horizontal RMSE of the direct georeferencing were 0.087 and 0.041 m, respectively. As the number of GCPs increased, the RMSE gradually decreased until a specific GCP density was reached. GCPs should be uniformly distributed in the study area and contain at least one GCP near the center of the domain. Additionally, as the distance to the nearest GCP increased, the local accuracy of the DSM decreased. In general, UAV direct georeferencing has an acceptable positional accuracy level.

On k ‐point configuration sets with nonempty interior

We give conditions for $k$-point configuration sets of thin sets to have nonempty interior, applicable to a wide variety of configurations. This is a continuation of our earlier work \cite{GIT19} on 2-point configurations, extending a theorem of Mattila and Sj\"olin \cite{MS99} for distance sets in Euclidean spaces. We show that for a general class of $k$-point configurations, the configuration set of a $k$-tuple of sets, $E_1,\,\dots,\, E_k$, has nonempty interior provided that the sum of their Hausdorff dimensions satisfies a lower bound, dictated by optimizing $L^2$-Sobolev estimates of associated generalized Radon transforms over all nontrivial partitions of the $k$ points into two subsets. We illustrate the general theorems with numerous specific examples. Applications to 3-point configurations include areas of triangles in $\mathbb R^2$ or the radii of their circumscribing circles; volumes of pinned parallelepipeds in $\mathbb R^3$; and ratios of pinned distances in $\mathbb R^2$ and $\mathbb R^3$. Results for 4-point configurations include cross-ratios on $\mathbb R$, triangle area pairs determined by quadrilaterals in $\mathbb R^2$, and dot products of differences in $\mathbb R^d$.

The Effect of Fresnel Lens Focal Point Location on Heat Transfer in Phase Change Material (PCM) Enhanced Dynamic Solar Facade

Abstract In recent years, the demand for energy-efficient technological solutions in the building sector has risen significantly worldwide. The exploitation of phase change material as a medium for thermal energy storage in building envelopes has increased due to its superior properties. There is still a knowledge gap to cover in the way to the effective solar thermal energy storage in the building envelope – to enhance the heat transfer, to reduce the heat loss, etc. This paper deals with the optimisation of heat transfer using a solar concentrator (Fresnel lens). This study examines the effect of Fresnel lens focal point location on heat transfer in a dynamic solar facade prototype that stores thermal energy in phase change material. Nine different setups (solar façade compositions) were tested in the laboratory – two parameters with three alternatives each. Testing conditions simulate the relevant Northern Europe climate. By changing the air gap configuration and location of the Fresnel lens focal point, the heat transfer to phase change material was observed by measuring temperatures in the phase change material container using five thermocouples. The results show the improved thermal performance in test modules with larger cone diameter by 7.2 % and Fresnel lens focal point positioning closer to the back of the phase change material container by 5.4 %.

UAV-based landfill operation monitoring: A year of volume and topographic measurements.

Unmanned Aerial Vehicles (UAVs) for photogrammetry operations configures a technology capable of extracting quantitative information from land surface in a fast, accurate and safe way, reproducing it in high-resolution Digital Elevation Models (DEMs) and orthomosaics. Due to the operational efficiency of this technique, there is an interest in evaluating its quality compared to other methodologies traditionally used for monitoring procedures in infrastructure earthwork. In sanitary landfills, operational monitoring is directly linked to topographic services, as these are the main source of data for the geometric assessment of the work. In this context, the aim of the study was to verify accuracy and application range of UAV photogrammetry for geometrical and volumetric measurements, when compared to usual conventional survey procedures using total station, and how it can aggregate reliable data to landfills monitoring activities. UAV flights were carried on monthly basis, over a year. For accuracy analysis, the maximum RMSE error observed was 7.1cm for horizontal axis and 0.37cm for vertical axis for the monitoring period. Volumetric measurements were tested using Ground Control Point (GCPs) configurations distributed first at the landfill perimeter, which resulted in an average difference of 9% from that calculated by conventional topography, and measurements where GCPs were placed also in the landfill operation fronts, when a 4% average difference diverging from conventional topography was obtained. The conclusion shows that such monitoring routines, when performed periodically, provides a robust database with a high level of operational performance, covering effective information for preventive and corrective monitoring in landfill projects.

Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams

We introduce a differential refinement of Cohomotopy cohomology theory, defined on Penrose diagram spacetimes, whose cocycle spaces are unordered configuration spaces of points. First we prove that brane charge quantization in this differential 4-Cohomotopy theory implies intersecting p/(p+2)-brane moduli given by ordered configurations of points in the transversal 3-space. Then we show that the higher (co-)observables on these brane moduli, conceived as the (co-)homology of the Cohomotopy cocycle space, are given by weight systems on horizontal chord diagrams and reflect a multitude of effects expected in the microscopic quantum theory of Dp/D(p+2)-brane intersections: condensation to stacks of coincident branes and their Chan-Paton factors, BMN matrix model and fuzzy funnel states, M2-brane 3-algebras, the Hanany-Witten rules, AdS3-gravity observables, supersymmetric indices of Coulomb branches as well as gauge/gravity duality between all these. We discuss this in the context of the hypothesis that the M-theory C-field is charge-quantized in Cohomotopy theory.

Finite point configurations and the regular value theorem in a fractal setting

Finite point configurations and the regular value theorem in a fractal setting

Particulate emissions in an engine fueled with biodiesel/Diesel blends at different fuel injection system configurations

ABSTRACT: The present study aimed to evaluate the particulate emissions in a Diesel engine with an electronic injection system using different biodiesel blends and variations in the fuel injection point and volume, comparing it with the MAR-1 standard. The particulate emissions are analyzed using a two-factor (12×8) completely randomized design that resulted from the interaction of three biodiesel/Diesel blends, two injection point configurations, two fuel injection volumes and eight crankshaft speeds, with 30 repetitions. According to the results, the particulate emissions meet the requirements established by the MAR-I standard when the engine operates at the original factory settings, fueled with blends containing up to 30% of biodiesel. With the engine angular speed at maximum torque (1,500 rpm), the injection point advanced and the increase in fuel volume, the particulate emissions increased by 115% and 314% for the 10 and 30% biodiesel blends, respectively, regarding the engine operating with the original settings.

Pinned point configurations and Hausdorff dimension

We prove that if the Hausdorff dimension of a compact subset E of Rd with d≥2 is sufficiently large, and if G is a star-like graph with two parts, and each of its parts is arigid graph, then the Lebesgue measure in the appropriate dimension, of the set of distances in E specified by the graph is positive. We also prove that ifdimH(E)is sufficiently large, then∫νG(r~t)dνG(~t)&gt;0,whereνGis the measure on the space of distances specified by G which is induced by a Frostman measure. In particular, this means that for any r&gt;0 there exist many configurations encoded by ~t with vertices in E such that the vertices of r~t are also in E.

Interpolation of Subdivision Features for Curved Geometry Modeling

We present a nodal interpolation method to approximate a subdivision model. The main application is to model and represent curved geometry without gaps and preserving the required simulation intent. Accordingly, we devise the technique to maintain the necessary sharp features and smooth the indicated ones. This sharp-to-smooth modeling capability handles unstructured configurations of the simulation points, curves, and surfaces. The surfaces correspond to initial linear triangulations that determine the sharp point and curve features. The method automatically suggests a subset of sharp features to smooth which the user modifies to obtain a limit model preserving the initial points. This model reconstructs the curvature by subdivision of the initial mesh, with no need of an underlying curved geometry model. Finally, given a polynomial degree and a nodal distribution, the method generates a piece-wise polynomial representation interpolating the limit model. We show numerical evidence that this approximation, naturally aligned to the subdivision features, converges to the model geometrically with the polynomial degree for nodal distributions with sub-optimal Lebesgue constant. We also apply the method to prescribe the curved boundary of a high-order volume mesh. We conclude that our sharp-to-smooth modeling capability leads to curved geometry representations with enhanced preservation of the simulation intent.

Numerical simulations of hypervelocity collisions scenarios against a large satellite

This paper describes a set of numerical simulations of hypervelocity collisions involving the ESA LOFT spacecraft, with CubeSat impactors ranging from 1 U to 48 U and diverse collision scenarios. The study was done with the purpose to investigate (1) the transition between sub-catastrophic and catastrophic collision with increasing impactor's kinetic energy, and (2) the effect of impact point and encounter configuration on the collision severity.The simulations were performed with the new semi-empirical Collision Simulation Tool developed by CISAS-University of Padova (prime contractor) and etamax GmbH in the framework of an ESA contract recently completed. The CST makes possible to model a large variety of collision scenarios with spacecraft design details included, and is capable to provide statistically accurate impact fragments distributions with low computational expense. It is shown that (1) the ratio between the kinetic energy of the impactor and the mass of the target (Energy-to-Mass Ratio, EMR) is not sufficient to discriminate between sub-catastrophic and catastrophic impacts, and (2) the damage suffered by the target and the number of collision fragments are limited when impacts are not central and the ratio between impactor size and target size is small. In particular, results suggest that the NASA Standard Breakup Model overpredicts the fragments for glancing impacts and/or small impactors, showing that both the impact point and the size of the impactor at a given energy play a significant role to determine the severity of the event.