Offsets Of Curves Research Articles

Automated Exploration of Envelopes and Offsets with Networking of Technologies

We explore envelopes and offsets of plane curves using automated methods, based on the usage of Computer Algebra Systems and Dynamical Geometry Systems. Work is performed using both parametric presentations and implicit equations. Implicitization involves well-known algorithms from the theory of Gröbner bases. In particular, irreducibility of the involved polynomials, whence irreducibility of the obtained curves is checked using the latest developments of the software tool GeoGebra Discovery. We conclude with comments on the 4C’s of twenty-first century Education, applied to mathematics and show how networking between technologies provides a suitable environment to apply and develop a 5th C, namely curiosity.

A Study of Digital Measurement and Analysis Technology for Transformer Excitation Magnetizing Curve

Transformer excitation magnetizing curves (TEMC) reflect the dynamic operation characteristics of iron-core materials. Using numerical analysis and the waveform recording function of digital oscilloscopes, we developed a cost-effective method for determining the TEMC. This approach eliminates the need for conventional analog integrator circuits. To address the potential obstacles to the digital generation of TEMC—namely, curve offsets and curve transients—we proposed solutions involving Fourier filtering and determining the initial point of integration. The results indicate that the proposed approach yields results consistent with those of conventional analog integrator circuits and highlight its promise for applications in data processing.

Offset Approximation of Rational Trigonometric Bezier ´ Curves

Offset curves are one of the crucial curves, but the presence of square root function in the representation is main hindrance towards their applications in CAD/CAM. The presented technique is based on offset approximation using rational trigonometric Bezier curves. The idea is ´ to construct a new control polygon parallel to original one. The two end points of the offset control polygon have been taken as exact offset end points, while the middle control points and weights have been computed using definition of parallel curves. As a result, offsets of rational and nonrational trigonometric Bezier curves have been approximated by rational ´ cubic trigonometric Bezier curve. An error between exact and approxi- ´ mated offset curves have also been computed to show the efficacy of the method.

Machine Learning Models of Vibrating H2CO: Comparing Reproducing Kernels, FCHL, and PhysNet.

Machine learning (ML) has become a promising tool for improving the quality of atomistic simulations. Using formaldehyde as a benchmark system for intramolecular interactions, a comparative assessment of ML models based on state-of-the-art variants of deep neural networks (NNs), reproducing kernel Hilbert space (RKHS+F), and kernel ridge regression (KRR) is presented. Learning curves for energies and atomic forces indicate rapid convergence toward excellent predictions for B3LYP, MP2, and CCSD(T)-F12 reference results for modestly sized (in the hundreds) training sets. Typically, learning curve offsets decay as one goes from NN (PhysNet) to RKHS+F to KRR (FCHL). Conversely, the predictive power for extrapolation of energies toward new geometries increases in the same order with RKHS+F and FCHL performing almost equally. For harmonic vibrational frequencies, the picture is less clear, with PhysNet and FCHL yielding accuracies of ∼1 and ∼0.2 cm-1, respectively, no matter which reference method, while RKHS+F models level off for B3LYP and exhibit continued improvements for MP2 and CCSD(T)-F12. Finite-temperature molecular dynamics (MD) simulations using the PESs from the three ML methods with identical initial conditions yield indistinguishable infrared spectra with good performance compared with experiment except for the high-frequency modes involving hydrogen stretch motion which is a known limitation of MD for vibrational spectroscopy. For sufficiently large training set sizes, all three models can detect insufficient convergence ("noise") of the reference electronic structure calculations in that the learning curves level off. Transfer learning (TL) from B3LYP to CCSD(T)-F12 with PhysNet indicates that additional improvements in data efficiency can be achieved.

Euclidean offset and bisector approximations of curves over freeform surfaces

The computation of offset and bisector curves/surfaces has always been considered a challenging problem in geometric modeling and processing. In this work, we investigate a related problem of approximating offsets of curves on surfaces (OCS) and bisectors of curves on surfaces (BCS). While at times the precise geodesic distance over the surface between the curve and its offset might be desired, herein we approximate the Euclidean distance between the two. The Euclidean distance OCS problem is reduced to a set of under-determined non-linear constraints, and solved to yield a univariate approximated offset curve on the surface. For the sake of thoroughness, we also establish a bound on the difference between the Euclidean offset and the geodesic offset on the surface and show that for a C2 surface with bounded curvature, this difference vanishes as the offset distance is diminished. In a similar way, the Euclidean distance BCS problem is also solved to generate an approximated bisector curve on the surface. We complete this work with a set of examples that demonstrates the effectiveness of our approach to the Euclidean offset and bisector operations.

Phase Curves of WASP-33b and HD 149026b and a New Correlation between Phase Curve Offset and Irradiation Temperature

Abstract We present new 3.6 and 4.5 μm Spitzer phase curves for the highly irradiated hot Jupiter WASP-33b and the unusually dense Saturn-mass planet HD 149026b. As part of this analysis, we develop a new variant of pixel-level decorrelation that is effective at removing intrapixel sensitivity variations for long observations (>10 hr) where the position of the star can vary by a significant fraction of a pixel. Using this algorithm, we measure eclipse depths, phase amplitudes, and phase offsets for both planets at 3.6 and 4.5 μm. We use a simple toy model to show that WASP-33b’s phase offset, albedo, and heat recirculation efficiency are largely similar to those of other hot Jupiters despite its very high irradiation. On the other hand, our fits for HD 149026b prefer a very high albedo. We also compare our results to predictions from general circulation models, and we find that while neither planet matches the models well, the discrepancies for HD 149026b are especially large. We speculate that this may be related to its high bulk metallicity, which could lead to enhanced atmospheric opacities and the formation of reflective cloud layers in localized regions of the atmosphere. We then place these two planets in a broader context by exploring relationships between the temperatures, albedos, heat transport efficiencies, and phase offsets of all planets with published thermal phase curves. We find a striking relationship between phase offset and irradiation temperature: the former drops with increasing temperature until around 3400 K and rises thereafter. Although some aspects of this trend are mirrored in the circulation models, there are notable differences that provide important clues for future modeling efforts.

The Small Magellanic Cloud Investigation of Dust and Gas Evolution (SMIDGE): The Dust Extinction Curve from Red Clump Stars

Abstract We use Hubble Space Telescope (HST) observations of red clump stars taken as part of the Small Magellanic Cloud Investigation of Dust and Gas Evolution (SMIDGE) program to measure the average dust extinction curve in a ∼200 pc × 100 pc region in the southwest bar of the Small Magellanic Cloud (SMC). The rich information provided by our eight-band ultraviolet through near-infrared photometry allows us to model the color–magnitude diagram of the red clump accounting for the extinction curve shape, a log-normal distribution of A V , and the depth of the stellar distribution along the line of sight. We measure an extinction curve with . This measurement is significantly larger than the equivalent values of published Milky Way (MW) R V = 3.1 ( ) and SMC Bar R V = 2.74 ( ) extinction curves. Similar extinction curve offsets in the Large Magellanic Cloud (LMC) have been interpreted as the effect of large dust grains. We demonstrate that the line-of-sight depth of the SMC (and LMC) introduces an apparent “gray” contribution to the extinction curve inferred from the morphology of the red clump. We show that no gray dust component is needed to explain extinction curve measurements when FWHM depth of 10 ± 2 kpc in the stellar distribution of the SMC (5 ± 1 kpc for the LMC) is considered, which agrees with recent studies of Magellanic Cloud stellar structure. The results of our work demonstrate the power of broadband HST imaging for simultaneously constraining dust and galactic structure outside the MW.

The Thermal Phase Curve Offset on Tidally and Nontidally Locked Exoplanets: A Shallow Water Model

Abstract Using a shallow water model with time-dependent forcing, we show that the peak of an exoplanet thermal phase curve is, in general, offset from the secondary eclipse when the planet is rotating. That is, the planetary hot spot is offset from the point of maximal heating (the substellar point) and may lead or lag the forcing; the extent and sign of the offset are functions of both the rotation rate and orbital period of the planet. We also find that the system reaches a steady state in the reference frame of the moving forcing. The model is an extension of the well-studied Matsuno–Gill model into a full spherical geometry and with a planetary-scale translating forcing representing the insolation received on an exoplanet from a host star. The speed of the gravity waves in the model is shown to be a key metric in evaluating the phase curve offset. If the velocity of the substellar point (relative to the planet’s surface) exceeds that of the gravity waves, then the hot spot will lag the substellar point, as might be expected by consideration of forced gravity wave dynamics. However, when the substellar point is moving slower than the internal wave speed of the system, the hottest point may lead the passage of the forcing. We provide an interpretation of this result by consideration of the Rossby and Kelvin wave dynamics, as well as, in the very slowly rotating case, a one-dimensional model that yields an analytic solution. Finally, we consider the inverse problem of constraining planetary rotation rate from an observed phase curve.

Efficient computation of dual space and directional multiplicity of an isolated point

Isolated singularities typically occur at self-intersection points of planar algebraic curves, curve offsets, intersections between spatial curves and surfaces, and so on. The information characterizing the singularity can be captured in a local dual basis, expressing combinations of vanishing derivatives at the singular point. Macaulay's algorithm is a classic algorithm for computing such a basis, for a point in an algebraic set. The integration method of Mourrain constructs much smaller matrices than Macaulay's approach, by performing integration on previously computed elements.In this work we are interested in the efficiency of dual basis computation, as well as its relation to orthogonal projection. First, we introduce an easy to implement criterion that avoids redundant computations during the computation of the dual basis, by deleting certain columns from the matrices in the integration method. In doing so, we explore general (non-monomial) bases for the associated primal quotient ring. Experiments show the efficient behavior of the improved method. Second, we introduce the notion of directional multiplicity, which expresses the multiplicity structure with respect to an axis, and is useful in understanding the geometry behind projection. We use this notion to shed light on the gap between the degree of the generator of the elimination ideal and the corresponding factor in the resultant.

A gradient-based morphological method to produce planar curve offsets

Two-dimensional curve offsets have a wide application area ranging from manufacturing to medical imaging. To that end, this paper concentrates on two novel techniques to produce planar curve offsets. Both methods, which are based on mathematical morphology, employ the concept that the boundaries formed by a circular structuring element whose center moves across the points on a base curve comprise the entire offsets of the progenitor. The first technique titled IMOBS was introduced in our former paper and was shown to have superior properties in terms of its high accuracy, low computational complexity, and its ability to handle complex curves if compared to the techniques available in the literature. Consequently, an all-purpose algorithm titled AMOBS is introduced to enhance further the performance of the former technique by making good use of gradient information to find globally the most suitable candidate points in the boundary data set via grid search techniques. Thus, the new paradigm is demonstrated to overcome some of the problems (like orphan curve offsets) encountered in extreme cases. Both algorithms, which have similar attributes in terms of run-time complexity and memory cost, are comparatively tested via two experimental cases where most CAD/CAM packages fail to yield acceptable results.

New morphological methods to generate two-dimensional curve offsets

For the purpose of generating 2D curve offsets used in 2.5D machining, four new methods based on morphological operations on different mathematical entities are presented in this paper. All of the methods, which lend themselves for parallel processing, exploit the idea that the boundaries formed by a circular structuring element whose center sweeps across the points on a generator/base curve comprise the entire offsets of the progenitor. The first approach, which is a carry-over from image processing, makes good use of morphological operations on binary images to produce 2D offsets via contour tracing algorithms. The second method, which is to rectify the high memory cost associated with the former technique, utilizes morphological operations on (boundary data) sets. The implementation of this basic technique is illustrated by two Matlab functions given in the paper. Despite its simplicity, the time complexity of this paradigm is found to be high. Consequently, the third method, which is evolved from the preceding one, reduces the time complexity significantly with the utilization of a geometric range search method. This technique, which has a considerable margin for improvement, is found to be suitable to be used as a part of the real-time motion command generator for CNC applications. Unlike the previous schemes, the final approach uses polygon operations to generate such curves. The run-time of this technique is highly governed by the complexity of the polygon overlay algorithm selected. The paper analyzes the complexity of each technique. Finally, the presented methods are evaluated (in terms of run-time and geometric accuracy) via two test cases where most CAD/CAM packages fail to yield acceptable results.

Interpreting Ultimate Capacity of Driven Piles with S-lgt Curve and Davisson Offset Limit

This paper explores the methods used widely in China and western world interpreting the ultimate capacity of driven piles based on axial static load tests. A database consisting of 57 prestressed high- compression-capacity concrete pipe piles, which are widely used in soft soil ground conditions for building and transportation projects in China, is built. Two methods widely used in China and western world to interpret the ultimate capacity of driven piles based on axial static load testing data, S-lg(t) curve and Davisson Offset Limit, are applied to the database. Findings indicate that both methods work well and reach almost the same estimation. Conclusions suggest that both academia and industry of China and western world should take the advantages of the methods and apply them to engineering practice.

Atomic Disorder in Fossil Tooth and Bone Mineral: An FTIR Study Using the Grinding Curve Method

Bone and tooth mineral generally undergo diagenetic changes. These changes in the carbonate hydroxyapatite structure and composition can affect the signals embedded in the mineral phase, such as migration behavior, age of the specimen and the reconstruction of past environments. Mineral preservation state can be assessed using infrared spectroscopy which provides information on crystal disorder at the atomic level and mineral composition. Here we present a new approach to evaluate carbonate hydroxyapatite atomic disorder using infrared spectroscopy and the standard KBr sample mounting method. We show that by repeated grinding of the sample and then plotting the infrared splitting factor against the width of the major phosphate absorption peak after each grinding, grinding curves with well defined trend lines can be obtained. The offsets between curves reflect differences in atomic disorder. We show that grinding curve offsets can be used to evaluate the state of preservation of bone, dentine and enamel mineral.

A novel approach for computing C2-continuous offset of NURBS curves

Computing offset curves is an important geometric operation in areas of CAD/CAM, robotics, cam design and many industrial applications. In this paper, an algorithm for computing offsets of NURBS curves using C2-continuous B-spline curves is presented. The progenitor curve in database is initially approximated by a line-fitting curve, and then the exact offset of this line-fitting curve is introduced as an initial offset. Based on the initial offset and a set of selected knots, an intended C2-continuous B-spline curve is subsequently constructed. The method uses a new error-measuring scheme, which is based on the convex hull property of Bezier curves and the idea of cumulative errors, to calculate the global error bound of offset approximation. The method obtains offset curves with C2 continuity and guarantees that the actual error bound is precisely within the prescribed tolerance. In addition, it also allows one to selectively parametrize the offset curve.

Shapes with offsets of nearly constant surface area

This article addresses several issues of offset sizes. A new type of isoperimetric problems is introduced: find the shapes, which have offsets of minimal surface area change. This problem arises in some physical and chemical processes with constant energy release. A computational solution is presented for two cases: convex sets, and a subclass of non-convex sets (star-like sets). Both cases are discussed in the plane and in the space. The solution uses methods from convex theory including the theory of mixed volumes and also some optimization techniques. The formulas developed in the optimization problem can be applied to get analytical formulas of the length of the curve offsets, as well as of the surface area and the volume of the surface offsets. Evaluating properties of offsets without constructing them proves useful for preliminary design in solid modeling, for approximating offsets curves and for planning the velocity of offset paths.

Computing offsets of NURBS curves and surfaces

This paper presents algorithms for computing offsets of NURBS curves and surfaces. The basic approach consists of four steps: (1) recognition of special curves and surfaces; (2) sampling the offset curve or surface based on bounds on second derivatives; (3) interpolating these points; and (4) removing all unwanted knots using the offset tolerance. The method provides a good handle on error control and results in the fewest number of control points compared to all published work. It also allows one to control the degree and the parametrization of the offset approximation.

Band alignments and offsets in In(As,Sb)/InAs superlattices

The band alignments and band offsets were investigated for In(As,Sb)/InAs superlattices of various periods and compositions. Magnetoabsorption experiments allowed identification of subband energies and in-plane reduced masses. Using an 8\ifmmode\times\else\texttimes\fi{}8 k\ensuremath{\cdot}p band-structure calculation which takes account of nonparabolicity, valence subband mixing, and strain effects, we are able to fit calculated absorption curves to transition energies and reduced masses. Comparison of fits on several samples confirm that the electrons lie in the In(As,Sb) layer and the holes in the InAs layer. By using the fitted curve offsets, we are able to calculate the fractional conduction-band-offset parameter, ${\mathrm{Q}}_{\mathrm{c}}$, which for a type-II heterostructure can be greater than 1 and which we found to be 2.06\ifmmode\pm\else\textpm\fi{}0.11.

Computation of singularities and intersections of offsets of planar curves

Abstract The objective of this paper is to compute the singularities of a normal offset of a planar integral polynomial curve and the intersections of two specific normal offsets of two planar integral polynomial curves. Singularities include irregular points (such as isolated points and cusps) and self-intersections. The key element in the above techniques is the computation of all real roots within a finite ☐ of systems of nonlinear equations involving polynomials and square roots of polynomials. The curves that we are investigating are described by polynomial functions, but their offset curve representations involve polynomials and square roots of polynomials. A methodology based on adaptive subdivision techniques to solve the resulting systems of nonlinear equations is investigated. Key components of our methods are the reduction of the problems into solutions of systems of polynomial equations of higher dimensionality through the introduction of auxiliary variables and the use of rounded interval arithmetic in the context of Bernstein subdivision to enhance the robustness of floating point implementation. Examples illustrate our techniques.

Computation of singularities and intersections of offsets of planar curves

Abstract: The objective of this paper is to compute the singularities of a normal offset of a planar integral polynomial curve and the intersections of two specific normal offsets of two planar int...

Offsets of curves on rational B-spline surfaces

The construction of offset curves is an important problem encountered often in design processes and in interrogation of geometric models. In this paper the problem of construction of offsets of curves lying on the same parametric surface is addressed. A novel algorithm is introduced, whose main feature is the use of geodesic paths to determine points of the offset. The offset is then approximated in the underlying surface parameter space by B-splines interpolating data points obtained by traveling a known distance along the geodesics departing from corresponding points of the progenitor in a direction perpendicular to the latter. A comprehensive error checking scheme has been devised allowing adaptive improvement of the approximation of the offset. The applicability of the algorithm is demonstrated by number of numerical examples.