Tangent Plane Research Articles

Parabolic rectifiability, tangent planes and tangent measures

We define rectifiability in \(\mathbb{R}^{n}\times\mathbb{R}\) with a parabolic metric in terms of \(C^1\) graphs and Lipschitz graphs with small Lipschitz constants and we characterize it in terms of approximate tangent planes and tangent measures. We also discuss relations between the parabolic rectifiability and other notions of rectifiability.

A convergence framework for optimal transport on the sphere

We consider a PDE approach to numerically solving the optimal transportation problem on the sphere. We focus on both the traditional squared geodesic cost and a logarithmic cost, which arises in the reflector antenna design problem. At each point on the sphere, we replace the surface PDE with a generalized Monge–Ampère type equation posed on the tangent plane using normal coordinates. The resulting nonlinear PDE can then be approximated by any consistent, monotone scheme for generalized Monge–Ampère type equations on the plane. Existing techniques for proving convergence do not immediately apply because the PDE lacks both a comparison principle and a unique solution, which makes it difficult to produce a stable, well-posed scheme. By augmenting the discretization with an additional term that constrains the solution gradient, we obtain a strong form of stability. A modification of the Barles–Souganidis convergence framework then establishes convergence to the mean-zero solution of the original PDE.

Thermodynamics analysis of multi-stage temperature swing adsorption cycle for dilute CO2 capture, enrichment and purification

Direct air capture is a carbon-negative technology that absorbs dilute carbon dioxide from the air directly, which requires a significant amount of energy and the construction of large-scale equipment. The main challenge for direct air carbon dioxide capture is the development of non-fossil energy sources for carbon dioxide compression and purification. In this work, a multi-stage adsorption system based on temperature swing adsorption with adsorption materials that can be driven by thermal energy is designed for direct air capture. The model of adsorption process is built on thermodynamic principles and conservation of mass, and the common tangent plane approach of adsorption is adopted at the end of each step calculation to validate the selected solution. Analysis of the process shows that, the CO2 purity in the storage bed can exceed 90% and the pressure can exceed 14 bar with zeolite 13X at a regeneration temperature of 368 K. Exploration of thermal performance has been investigated regarding specific energy and exergy efficiency of various regeneration temperatures. The influence of mass of adsorbent on the purity and specific energy has also been evaluated. Results show that the specific energy declines from 11.92 MJth/molCO2 to 8.73 MJth/molCO2, and the optimal exergy efficiency increases from 0.205 % to 0.238 % as the regeneration temperature grows from 368 K to 373 K. The values of adsorbent mass in the second and third beds for optimal purity and specific energy are calculated. It means that a thermally-powered carbon separation device can not only meet the requirements of direct air capture but also get the optimal purity of carbon dioxide or specific energy by altering the operation parameters.

An Additive Manufacturing Direct Slicing Algorithm Based on a STEP Model

The Standard Template Library (STL) file is the most common data format for the description of an additive manufacturing (AM) geometric model, but it has some disadvantages, such as large errors of the geometric model description, the easy loss of topology information, data duplication, large file sizes, and so on. Aiming at these problems, a direct slicing algorithm based on a Standard for the Exchange of Product Model Data (STEP) model was proposed. For the parts composed of basic types of surfaces such as boundary curves, spherical surfaces and cylindrical surfaces, the traditional geometric method was used to calculate the intersection. For the parts with complex surfaces, the three-dimensional models were described based on Non-Uniform Rational B-Spline (NURBS) surfaces. The NURBS surfaces were layered using a discrete tracking algorithm, the tracking starting point was determined, the intersection line between the tangent plane and each NURBS sub-surface was obtained, and the closed layer contour was formed. Finally, the slicing simulations and printing experiments of solid parts were carried out using the direct slicing algorithm based on the STEP model. It was shown that the dimensional accuracy and surface quality of the printed parts from this algorithm had been significantly improved.

A mixed hyperbolic/trigonometric non‐stationary subdivision scheme for arbitrary topology meshes

In this paper, a new non‐stationary mixed scheme constructed from hyperbolic and trigonometric schemes is presented. It is characterized by a shape parameter that allows the user to modify the shape of the surface obtained from a given initial mesh. Such a non‐stationary subdivision scheme is constructed by taking the tensor product of the univariate mixed hyperbolic/trigonometric schemes by giving appropriate rules in the neighborhoods of the extraordinary points. This new scheme can generate tangent plane continuous surfaces. Some examples are given to show the performance of these new schemes, such as the reconstruction of 3D models extracted from medical images.

Intrusive growth of initials does not affect cambial circumference in Robinia pseudoacacia.

This study aimed to test the hypothesis whether intrusive growth of initial cells is related to the increase in circumference of Robinia pseudoacacia vascular cambium—both qualitatively and quantitatively. The mode of intrusive growth of cambial initial cells was also studied. Samples collected from tree trunks were examined using series of semi-thin transverse sections. Anatomical reconstructions of radial and tangential planes of analysed fragments of cambial tissue were made. Observations and measurements have shown that the intrusive growth of R. pseudoacacia initial cells does not contribute to an increase in tangential dimension of observed tissue fragments where cell rearrangement occurs. Moreover, initially separated tangential walls of cells (between which cambial initial cell elongates intrusively) are transformed into obliquely oriented walls. These results stand in accordance with a statement that only symplastic growth of initials, not intrusive growth, is responsible for the increase in circumference in all woody plants with the continuous cambial cylinder. Moreover, we managed to capture the moment of transition of initial status from one cell to another for the first time. This phenomenon may be explained on the basis of the system of mechanical stresses operating not only in the secondary meristematic tissue but also in a whole plant organism.

Vector graphics on surfaces using straightedge and compass constructions

We port classical straightedge and compass constructions to manifold surfaces under the geodesic metric. We propose two complementary approaches: one working on the tangent plane; and another working directly on the surface. In both cases, many constructions lack some of the geometric properties they have in the Euclidean case. We devise alternative constructions that guarantee at least a subset of such properties. We integrate our constructions in the context of a prototype system supporting the interactive drawing of primitives of vector graphics.

The Bible as a human and fallible book? Contrasting Karl Barth and classical Pentecostal hermeneutical perspectives

Many Pentecostals share an affinity with Karl Barth’s theology of the Word along with a high view of the Bible that Protestants share. Like some Reformed theologians, they interpret his theology as follows: the Bible is not in any objective sense the word of God but becomes the word when it means something to us through the work of the Spirit. The objective content of biblical words meant little for Barth; only the encounter with God in the text is essential. The purpose of this study is to consider the appropriateness of such affinity. The study limits the discussion to Barth’s view about the Bible’s humanness and fallibility and compares it to the Pentecostal perspective, that the Bible requires the Spirit’s inner working in the reader to become God’s revelation, implying that biblical authors point to the witness of the Spirit that entails the revelation; readers should strive to find that witness prayerfully. As a result, the related issues of the Bible’s authority and the task of exegesis are also discussed. The research uses a comparative literature study of some Barthian and Pentecostal resources. As a result, this article submits that some Pentecostals’ perception of Barth’s views is incorrect. However, some tangent planes exist between Barth’s and Pentecostal hermeneutics. The article concludes that by recognising their significant influence, Pentecostals would establish room to further the nuances their hermeneutics represent. It contributes to considering one proponent of Reformed hermeneutics, Barth, and Pentecostal hermeneutics. It expounds one aspect of Pentecostal hermeneutics about the Bible’s humanness and fallibility, that has received scant attention in the available literature.Contribution: The article contributes to the discourse about Pentecostal hermeneutics by investigating links between Karl Barth’s and classical Pentecostals’ theology of the word. It concludes that although some tangent planes exist between Barth’s and Pentecostal hermeneutics, Pentecostals regard the authority of the Bible as derived from the Spirit’s preference to use the Bible to speak to people.

Optical System Design of a Planar Waveguide Spectrometer

In this paper, an optical design for a hollow planar waveguide spectrometer with Czerny-Turner is proposed. To decrease the propagation loss of the spectrometer, the design strategy of designing the tangential plane and the sagittal plane separately is proposed, corresponding to resolution and energy, respectively. The Czerny–Turner optical path is designed on the tangential plane, and the sagittal design theory and method are analyzed in detail. The ray tracing results show that the resolution of the spectrometer is better than 4 nm on the tangential plane, while on the sagittal plane, the detector receives the highest energy when the detector pixel height matches the distance between the two mirrors.

On rectifiable measures in Carnot groups: Marstrand–Mattila rectifiability criterion

In this paper we continue the study of the notion of P-rectifiability in Carnot groups. We say that a Radon measure is Ph-rectifiable, for h∈N, if it has positive h-lower density and finite h-upper density almost everywhere, and, at almost every point, it admits a unique tangent measure up to multiples.In this paper we prove a Marstrand–Mattila rectifiability criterion in arbitrary Carnot groups for P-rectifiable measures with tangent planes that admit a normal complementary subgroup. Namely, in this co-normal case, even if a priori the tangent planes at a point might not be the same at different scales, a posteriori the measure has a unique tangent almost everywhere.Since every horizontal subgroup of a Carnot group has a normal complement, our criterion applies in the particular case in which the tangents are one-dimensional horizontal subgroups. Hence, as an immediate consequence of our Marstrand–Mattila rectifiability criterion and a result of Chousionis–Magnani–Tyson, we obtain the one-dimensional Preiss's theorem in the first Heisenberg group H1. More precisely, we show that a Radon measure ϕ on H1 with positive and finite one-density with respect to the Koranyi distance is absolutely continuous with respect to the one-dimensional Hausdorff measure H1, and it is supported on a one-rectifiable set in the sense of Federer, i.e., it is supported on the countable union of the images of Lipschitz maps from A⊆R to H1.

Joints of Varieties

We generalize the Guth–Katz joints theorem from lines to varieties. A special case says that N planes (2-flats) in 6 dimensions (over any field) have O(N^{3/2}) joints, where a joint is a point contained in a triple of these planes not all lying in some hyperplane. More generally, we prove the same bound when the set of N planes is replaced by a set of 2-dimensional algebraic varieties of total degree N, and a joint is a point that is regular for three varieties whose tangent planes at that point are not all contained in some hyperplane. Our most general result gives upper bounds, tight up to constant factors, for joints with multiplicities for several sets of varieties of arbitrary dimensions (known as Carbery’s conjecture). Our main innovation is a new way to extend the polynomial method to higher dimensional objects, relating the degree of a polynomial and its orders of vanishing on a given set of points on a variety.

On the stability of tangential discontinuity in the interaction of solar wind and cometary atmospheres

ABSTRACT In 2004, European Space Agency launched the spacecraft Rosetta. After almost 10-yr flight it arrived at the Churyumova–Gerasimenko comet. Rosetta was orbiting about the comet for some time observing its environment. One of the most interesting observations obtained during this mission was multiple disappearance of magnetic field. It was suggested that this phenomenon is related to the instability of cometopause, which is a tangential discontinuity separating the solar wind flow from the plasma component of the flow from a comet. In this article, we aim to study this instability. We numerically calculated the stationary state with the account of charge exchange on water molecules and resonant charge exchange. Then, we solved the non-stationary problem using the stationary state as the initial condition. We also studied the stability of the cometopause analytically in the short-wavelength approximation where we substituted the curved cometopause by a tangent plane. Both the numerical and analytical results show that the velocity jump at the cometopause is too weak to overtake the stabilizing effect of the magnetic field and cause the Kelvin–Helmholtz instability. We conclude that the main course of the cometopause instability is the charge exchange, and the instability is similar to the Rayleigh–Taylor instability that occurs in a gravitational field.

Contact With Circles and Euclidean Invariants of Smooth Surfaces in ℝ3

Abstract We investigate the vertex curve, that is the set of points in the hyperbolic region of a smooth surface in real 3-space at which there is a circle in the tangent plane having at least 5-point contact with the surface. The vertex curve is related to the differential geometry of planar sections of the surface parallel to and close to the tangent planes, and to the symmetry sets of isophote curves, that is level sets of intensity in a 2-dimensional image. We investigate also the relationship of the vertex curve with the parabolic and flecnodal curves, and the evolution of the vertex curve in a generic 1-parameter family of smooth surfaces.

Dynamics of a gyrostat satellite with the vector of gyrostatic moment tangent to the orbital plane

In this paper, a gyrostat satellite in a circular orbit with its gyrostatic moment tangent to the orbital plane and collinear with the orbital speed is studied regarding its equilibria, bifurcation of equilibria, and asymptotic stability conditions. In the general case, where any gyrostat angular momentum is aligned with any of the orbital coordinate frames, interesting results arose regarding its equilibria bifurcation regarding conditions near to the ones presented in this paper, namely equilibria regions outside their main regions near to the orbital plane tangent. For equilibria and bifurcation of equilibria, a symbolic-numerical method is used to obtain the polynomial equations in function of non-dimensional parameters whose roots are equivalent to the number of equilibria positions. For the asymptotic stability, the results are tested using the Lyapunov stability theory scheme.

Modeling and study of properties of surfaces equidistant to a sphere and a plane

In the present paper geometric locus of points (GLP) equidistant to a sphere and a plane is considered; the properties of the acquired surfaces are studied. Four possible cases of mutual location of a sphere and a plane are considered: the plane passing through the center of the sphere, the plane intersecting the sphere, the plane tangent to the sphere and the plane passing outside the sphere. GLP equidistant to a sphere and a plane constitutes two co-axial co-focused paraboloids of revolution. General properties of the acquired paraboloids were studied: the location of foci, vertices, axis and directing planes, distance between the sphere center and the vertices, the distance between the vertices. GLP for each case of mutual location of a plane and a sphere constitutes: in case one passes through the center of other, two co-axial co-focused oppositely directed paraboloids of revolution symmetrical with respect to the given plane; in case they intersect each other, two co-axial co-focused oppositely directed non-symmetrical paraboloids; in case they are tangent to each other, a paraboloid and a straight line passing through the tangency point; in case they have no common points, a pair of co-axial co-focused mutually directed paraboloids of revolution.

Primary aberration optimization for double-plane symmetric beam shaping systems using a pair of curved reference surfaces.

The geometric aberration of centered refracting double-plane symmetric optical systems (DPSOS) is investigated. For DPSOS with different defocus values in the tangential plane and the sagittal plane (astigmatic wavefront), a pair of curved reference surfaces which vanishes the quadratic terms of the optical path difference (OPD) between a general ray and a reference ray are deduced. With the curved reference surfaces, the primary (fourth-order) wave aberration function for DPSOS is calculated and analyzed, which can be used for beam shaping designs with astigmatic input wavefront, such as slab lasers and semiconductor lasers. Further, the proposed curved reference surfaces can be applied to analyze the aberrations of general DPSOS.

An effective multiple-image encryption algorithm based on 3D cube and hyperchaotic map

In accordance with the principle that the cube graph can be obtained by superimposing planes, a multiple-image encryption scheme based on hyperchaotic map and 3D cube is designed in this paper. Multiple images are segmented by columns, and a regular cube is obtained by stacking multiple fixed-size planes. Each stacked plane is considered as a stacked plane of the cube along the z-axis direction. Firstly, each tangent plane is rotated by different angles in the counterclockwise direction and then DNA encoded, followed by face swap, row swap, and column swap in turn. Then, the DNA-encoded image cube is subjected to DNA addition with the chaotic cube. Finally, two DNA sequences are randomly selected for DNA mutation. The mutated cube is decoded into a digital cube by DNA and the multiple-image encryption algorithm is completed. Theoretically, the cube used in the encryption process can be infinitely large, which means that the number and the size of 2D images that can be encrypted can be unrestricted. Experimental simulations and performance tests prove that the proposed image encryption scheme is effective and secure.

Simulation and Analysis of Electromagnetic Scattering from Anisotropic Plasma-Coated Electrically Large and Complex Targets

An efficient physical optics (PO) calculation method is proposed for the electromagnetic (EM) scattering of electrically large targets coated with magnetized plasma characterized by asymmetric tensor dielectric parameters. The outer surface of the arbitrarily shaped target is discretized into triangular elements. According to the principle of tangent plane approximation and by using the plane wave spectrum expansion method, the scattered field from one triangular element is derived as a double integral in the spectral domain. To obtain the solution in the spatial domain, the saddle point method is used to asymptotically calculate the integral. Then, the equivalent surface currents (ESCs) are constructed by calculating the surface field at the outer surface of the planar model, from which the PO solution is derived by using the Stratton–Chu integral. Moreover, to interpret the field propagation process in the plasma layer quantitatively, the total scattered field of the coated planar model is decomposed into the superposition of different mode field components. It is observed that the scattered fields demonstrate an inherent cross-polarization phenomenon due to the nonreciprocal constitutive relation of the plasma, which is a distinct feature and is different from the general anisotropic medium whose dielectric parameters can be diagonalized. The effectiveness of the proposed method is verified by numerical results. Furthermore, the proposed algorithm consumes less calculation time and memory as compared to commercial full solvers.

3D topology optimization of sandwich structures with anisotropic shells

The development of a density-based 3D minimum compliance topology optimization approach for sandwich structures with anisotropic shells is presented. A constant thickness shell is enforced at the interface of a base structure through a two-step filtering approach. Homogenous effective properties of a cellular material are assigned to the base structure while the shell is treated as a constant stiffness laminate with anisotropic properties. Local shell normal vectors are obtained from the density gradient introduced in the second filtering step and local material orientations in the shell’s tangent plane are approximated based on the fabrication process of the structure. Simple approximating strategies using the Gauss map are presented for a fabric draping process and the fused filament fabrication additive manufacturing process. The effect of material anisotropy on the optimal topology is studied on the Messerschmitt Bölkow Blohm beam benchmark problem. An application to composite structures is then presented where a square, simply supported laminate is reinforced on one face by a topology optimized sandwich structure. Though manufacturability cannot be ensured in the current implementation, careful consideration of the optimization parameters allow to constrain the problem towards manufacturable solutions using additively manufactured cores.

Effect of interfacial roughness on thermal boundary conductance: An elastic wave model using the Kirchhoff approximation

Interfacial roughness plays an important role in nanoscale thermal transport, while this issue still lacks a theoretical description due to the difficulty in quantifying the phonon scattering at the disordered interface. We report an elastic wave model (EWM) that describes the phonon–interface interaction by the interface elasticity. The transmissivity calculation based on the interfacial displacement and stress continuity incorporates the mode conversion and nonlinear vibration spectra of phonon waves, and the resulting cut-off frequency. Compared with the acoustic mismatch model that also assumes specular scattering of phonons waves, the applicable range of EWM is extended to room temperature and shows better accuracy. Based on this model, we use a statistical method with the Kirchhoff approximation and introduce the tangent plane slope distribution to determine the phonon scattering at a Gaussian-distributed rough interface. And the effective transmission coefficient is calculated by integrating the contributions of all points to the energy transmission. Calculated thermal boundary conductance (TBC) by a Landauer formula agrees well with the experiments of Al/SiO2/Si and Au/SiO2/Si rough interfaces. And the results show that the TBC of interface with sub-nanometer RMS roughness (10−10 m) is close to that of the smooth interface. After this transition, the increasing σ can significantly impact the interfacial heat transfer. Compared with existing models, the EWM can quantitatively estimate the effect of roughness on TBC and give a transmission image of wave-like phonons at the interface. This work contributes to an in-depth understanding of phonon wave effects in nanostructures.