Geometric Construction Method Research Articles

Geometric space construction method combined of a spline-skinning based geometric variation method and PCA dimensionality reduction for ship hull form optimization

In the construction stage of ship hull geometric space, the representation and variation of hull forms pose challenges due to the complex nature of the geometry and the necessity for enhanced flexibility in space. In this study, we utilize T-spline technology for the representation of ship hulls, enabling a unified parameter domain with a reduced number of control points to effectively depict the entire hull surface. Furthermore, we introduce a method for geometric variation based on the T-spline parameter domain, which permits both local and global alterations, thus offering greater flexibility in modifying the geometry. By leveraging the T-spline representation and the geometric variation method, we developed a hybrid geometric approach for constructing the geometric space, which significantly increased the geometric flexibility in ship hull optimization. A geometric space characterized by a high degree of geometric freedom typically requires the use of numerous optimization variables and incurs considerable computational resource consumption. In this research, we apply a linear dimensionality reduction method, principal component analysis, to decrease the dimensionality of the geometric space. This method effectively preserves the original geometric design space, thereby eliminating the need for performance analyses. The numerical optimization results clearly indicate that the proposed method significantly enhances optimization efficiency.

Morphometry of sigmoid notch: A novel method of shape assessment for clinical practice

PurposeUsing a novel technique, we assess and describe the distal radioulnar joint (DRUJ) anatomy with reference to sigmoid notch shapes. The purpose of this study was to develop a non-invasive, fast, reliable and reproducible technique for analysing the shapes of sigmoid notch on CT scans. Methods40 cadaveric wrist specimens exposing the sigmoid notch and 100 normal adult wrist CT scans with end on axial sections showing the sigmoid notch outline were obtained. The shape of the sigmoid notch was assessed on both using the geometric construction method. ResultsOf the 40 cadaveric specimens, 42 % were C-shaped, 33 % were flat,17 % were Ski sloped, and 8 % were S-shaped.Of the 100 wrist CT scans, 48 % were C-shaped sigmoid notch, 17 % were flat, 15 % were ski-sloped, and 20 % were S-shaped shapes. ConclusionsOur sample population had a higher prevalence of C-shaped sigmoids in both cadaveric and CT-based studies. It is considered the most stable shape because of its increased depth. The DRUJ morphology in our sample population can be inferred to have a stable configuration. Clinical relevanceThe relationships and measurements demonstrated in this study can be a guide when considering ligament reconstruction for DRUJ instability, managing complex fractures involving the DRUJ and in decision-making for notchplasty, osteotomy and arthroplasty of DRUJ.

Numerical simulation of the effective thermal conductivity of cement-based composites: Microscopic insights into diatomite/silica aerogel

Silica aerogel used in cement composites to prepare novel thermal superinsulation material has been gradually applied in buildings and can effectively reduce building energy consumption. In this paper, the geometric voxel construction method is used to heterogeneity model the thermal conductivity of a cement-based diatomite/silica aerogel composite (DSAC), and thermal conductivity simulations are carried out using a heat conduction model framework analogous to an electrical resistance model framework. The simulation results show that the simulated thermal conductivities of cementitious DSAC materials with different DSAC composite contents agree well with the corresponding experimental values. The saturated thermal conductivity of the cementitious DSAC material is 2.5 times the dry thermal conductivity at 28 d. This shows that the thermal conductivity model proposed in this paper can effectively predict the effective thermal conductivity of cementitious DSAC materials and can provide strong support for the material design and performance prediction of cementitious DSAC materials.

Augmented reality system based on full-color holographic optical elements lens

Holographic optical element (HOE) lens is an imaging element fabricated through recording wavefront by interference. Because of its advantages of small form factor and wavelength, angle selectivity and arbitrary wavefront formation, it has a good application prospect in augmented reality display. To make the system more compact, the HOE lens is adopted as an off-axis optical element. At the same time, according to diffraction principle, its wavelength response is more sensitive than those of traditional refractive and reflective optical elements. Thus the fabrication and design of a full-color HOE lens is a challenge to optimizing the free-space head-up display system. To systematically analyze the HOE imaging system, the conjugate relation between the object and image is derived by scalar diffraction theory. Then the Gaussian conjugate imaging equation is obtained and the off-axis aberration of distortion and astigmatism in the HOE imaging system are analyzed. In addition, A head-up display with field of view (FOV) of 18° and eyebox of 10 mm is simulated and its imaging process is visualized through the geometric optics method of <i> <b>k</b> </i>-vector diagram and ray-tracing. A full-color HOE lens with high diffraction efficiency is fabricated by interference. Its average peak diffraction efficiency is 56.7%, reaching a high level in the world. A prototype of augmented reality system is established by integrating laser pico-projectior with HOE lens. The experimental results of distortion effect and astigmatism effect of the system are obtained, which are consistent with the simulation results. The modulation transfer function (MTF) parameter of the system is measured, and its definition basically meets the requirements of the human eyes for resolution. The aberration of the system is analyzed and the optimization method is proposed. To optimize the monochromatic image quality, an extra cylindrical lens is added to ensure the same optical power of meridian and sagittal plane to eliminate the astigmatism. Besides, a freeform wavefront is designed by the geometric construction method and forms a freeform HOE to deal with the distortion problem. The local recording freeform wavefront can be calculated by the imaging equation. When full-color HOE is applied to the display system, the images of three channels may separate in the space because of their different reconstruction wavelengths and angles. We propose a pre-compensation method of recording process to solve this problem. If these above-mentioned problems can be solved, due to its good image uniformity, sufficient field angle and eyebox area, the head-up display based on HOE lens with extra optical power will have a better application in augmented reality technology.

Trough-Type Free-Form Secondary Solar Concentrator for CPV/T Application

Imaging concentrators like the parabolic trough solar concentrators have been widely employed for energy production in solar power plants. The conventional imaging solar concentrators form a non-uniform Gaussian distribution on receiving absorbers yielding the highest temperatures. The traditional CSP system normally truncated a peripheral region of heat flux to better use the central part. CPV/T systems using the waste heat recovery method can largely improve the total efficiency. However, for the CPV module, the coolant temperature was usually below 80 °C, which limited the applications of the thermal cycle such as the ORC system. In this article, a novel trough-type free-form secondary solar concentrator (TFSC) for PV/Thermal hybrid application has been proposed. Different from other CPV/T concepts using a combined PV panel and cooling tunnel/tube, the current concept separates the receiver in two parts. The secondary free-form reflector is generated by the geometric construction method, resulting in uniform heat flux in the edge region and high concentration in the central region. Through the ray tracing method, the optical properties have been verified. Sensitivity analysis of the concentrating structure is also conducted. The results provide supports for the design and applications of novel CPV/T systems.

Comparison of negative blended lenticular lens design methods for high myopic spectacles

The power distribution, high-order polynomial, bicubic spline interpolation, and geometric construction methods were used to reduce the edge thickness and weight of negative blended lenticular lenses. Four lenses with the same optical parameters were designed, simulated, machined, and evaluated using the four methods. Spherical and cylindrical power maps were generated and compared. The methods reduced the edge thickness by 78.73% for power distribution method, 77.40% for high-order polynomial method, 78.12% for bicubic spline interpolation method and 79.86% for geometric construction method for a 35 mm radius compared to that with the spherical surface. Geometric construction, power distribution, high-order polynomial, and bicubic spline interpolation reduced the lens weight by 43.86%, 35.98%, 33.06%, and 29.76%, respectively. The center optical area for the power distribution and geometric construction methods were larger than those with the other methods.

Stitching and Geometric Modeling Approach Based on Multi-Slice Satellite Images

Time delay and integration (TDI) charge-coupled device (CCD) is an image sensor for capturing images of moving objects at low light levels. This study examines the model construction of stitched TDI CCD original multi-slice images. The traditional approaches, for example, include the image-space-oriented algorithm and the object-space-oriented algorithm. The former indicates concise principles and high efficiency, whereas the panoramic stitching images lack the clear geometric relationships generated from the image-space-oriented algorithm. Similarly, even though the object-space-oriented algorithm generates an image with a clear geometric relationship, it is time-consuming due to the complicated and intensive computational demands. In this study, we developed a multi-slice satellite images stitching and geometric model construction method. The method consists of three major steps. First, the high-precision reference data assist in block adjustment and obtain the original slice image bias-corrected RFM to perform multi-slice image block adjustment. The second process generates the panoramic stitching image by establishing the image coordinate conversion relationship from the panoramic stitching image to the original multi-slice images. The final step is dividing the panoramic stitching image uniformly into image grids and employing the established image coordinate conversion relationship and the original multi-slice image bias-corrected RFM to generate a virtual control grid to construct the panoramic stitching image RFM. To evaluate the performance, we conducted experiments using the Tianhui-1(TH-1) high-resolution image and the Ziyuan-3(ZY-3) triple liner-array image data. The experimental results show that, compared with the object-space-oriented algorithm, the stitching accuracy loss of the generated panoramic stitching image was only 0.2 pixels and that the mean value was 0.799798 pixels, achieving the sub-pixel stitching requirements. Compared with the object-space-oriented algorithm, the RFM positioning difference of the panoramic stitching image was within 0.3 m, which achieves equal positioning accuracy.

Constructing Continuous Functions

Summary This paper introduces a simple geometric construction method for graphs of continuous functions. Graphs of a nowhere differentiable continuous function, a Lipschitz function that is not differentiable on a perfect set, and a continuous function whose level sets are countable are constructed using this method. We argue that introducing students to this method early on in Real Analysis fosters proper geometric intuition about continuity and provides for simple geometric proofs of non-differentiability properties.

Joint estimation of transmitter and receiver IQ imbalances based on GSOP and GCM for optical coherent systems

In this paper, we propose an in-phase/quadrature (IQ) imbalance estimation scheme for mixed transmitter (Tx) and receiver (Rx) IQ imbalance in the optical coherent transmission system. The proposed scheme is based on traditional IQ imbalance compensation algorithm Gram-Schmidt orthogonalization procedure (GSOP) and the proposed geometric construction method (GCM). In the actual transmission system, frequency offset (FO) and phase noise (PN) are recognized as part of received signal, which has an effect on the estimation of IQ imbalance. At the beginning of the proposed scheme, the GSOP algorithm is used to estimate and compensate Rx IQ imbalance according to the statistical characteristics of the received signal, then standard frequency offset compensation algorithm and k-means clustering aided blind phase search (BPS) algorithm are used to compensate FO and PN respectively. Finally, the proposed method GCM is used to estimate Tx IQ imbalance. 34 Gbaud QPSK and 16QAM signals are experimentally performed to demonstrate the proposed IQ imbalance estimation scheme, the results show the proposed scheme has accurate estimation capability with low estimation algorithm complexity.

Improvement of the performance of parabolic trough solar concentrator using freeform optics and CPV/T design

<abstract> The developmental tendency of parabolic trough collector (PTC) is larger aperture area for energy harvest and novel optical design for higher solar concentration. Larger aperture faces a higher demand in tracking accuracy and lower tolerances with respect to wind loads, quality of mirrors, control, and mounting imprecisions. With the increase of reflection angle, the divergence size of concentrated focal spot gets enlarged on receiving surface, forming a Gaussian distribution. Receiving more energy and making best use of the concentrated solar power have become a key problem. In current study, a novel trough free-form solar concentrator (TFFC) has been developed by extending the aperture size with the aid of freeform optics and combined PV/thermal utilization. The structure model is composed by traditional parabolic trough with thermal tube, and extended freeform reflector with solar panel in slope configuration. The free-form surface is generated by geometric construction method for the sake of uniform heat flux distribution. The optical characteristics are validated by ray tracing method. The advantages will be revealed by compared with traditional system. The sensitivity analysis and error factors would be discussed as well. The initial results are promising and significant for the enhancement of trough type solar concentrator systems. </abstract>

Ashlar Staircases with Warped Vaults in Sixteenth- to Eighteenth-Century Spain

During the sixteenth to eighteenth centuries, three-section cantilevered stone stairways supported by vaults of complex warped shapes were built along Spain’s Mediterranean coast. Presented here are five case studies. The geometric analyses and the comparative study with the relevant treatise illustrate the existence of a geometric construction method. The process involves handling elevation and plan projections simultaneously to obtain the geometric center of the curves that delimit these vaults. The simplicity of the geometric formula applied relies on the use of dimensional references obtained from the stairwell measurements.

Development of realistic food microstructure considering the structural heterogeneity of cells and intercellular space

Abstract The geometric characterization of the microstructure is required to model the micro level transport process inside the plant-based food material during thermal processing. However, the existing geometric construction methods are unable to consider the distinctive geometrical characteristics of cells and intercellular spaces during generation of the virtual microstructure. The main aim of this study is to develop a new method to build microstructure of the plant-based food materials with clear identification of cell and intercellular spaces. Granny smith and red delicious apple were taken as samples to implement and validate the proposed method. The virtual microstructure was generated from the scanning electron microscope image (SEM) of the apples. A new algorithm was developed based on the modified ellipse fitting method to generate the microstructure from microscopic images. Linear Discriminant Analysis (LDA) was used to classify the cells and the intercellular spaces. Statistical analysis was performed between the developed geometrical model and microscopic image to determine the accuracy of the proposed method. The area distribution of the microstructural model found in this study is significantly better than that found in the literature.

Establishing Base Elements of Perspective in Order to Reconstruct Architectural Buildings from Photographs

The use of perspective images, especially historical photographs for retrieving information about presented architectural environment is a fast developing field recently. The photography image is a perspective image with secure geometrical connection with reality, therefore it is possible to reverse this process. The aim of the herby study is establishing requirements which a photographic perspective representation should meet for a reconstruction purpose, as well as determination of base elements of perspective such as a horizon line and a circle of depth, which is a key issue in any reconstruction. The starting point in the reconstruction process is geometrical analysis of the photograph, especially determination of the kind of perspective projection applied, which is defined by the building location towards a projection plane. Next, proper constructions can be used. The paper addresses the problem of establishing base elements of perspective on the basis of the photograph image in the case when camera calibration is impossible to establish. It presents different geometric construction methods selected dependently on the starting assumptions. Therefore, the methods described in the paper seem to be universal. Moreover, they can be used even in the case of poor quality photographs with poor perspective geometry. Such constructions can be realized with computer aid when the photographs are in digital form as it is presented in the paper. The accuracy of the applied methods depends on the photography image accuracy, as well as drawing accuracy, however, it is sufficient for further reconstruction. Establishing base elements of perspective presented in the paper is especially useful in difficult cases of reconstruction, when one lacks information about reconstructed architectural form and it is necessary to lean on solid geometry.

Elastica surface generation of curved-crease origami

Curved-crease origami are studied for many novel applications across engineering and architecture, as they are developable but possess a non-zero principal curvature and a corresponding energy storage capability when folded. However, geometric modelling techniques are limited, with most methods requiring numerical discretisation of a target curved surface to allow developability constraints to be enforced at vertices. The discretised surface can approximate a physical surface through relaxation for minimum bending energy, however such methods are cumbersome and their accuracy is largely unknown. This paper presents an analytical geometric construction method for curved-crease origami that avoids the need for surface discretisation. The new method combines a 1D elastica solution for large elastic bending deformation with a straight-crease origami projection and reflection process; it can thus concisely and accurately capture the principal surface curvature and developability characteristics of elastically-bent curved-crease origami. A surface error analysis of 3D scanned physical prototypes is used to validate the model, which is shown to be accurate to within ± 50% of the sheet thickness for a 2 mm thick model for a range of elastica surface profiles. Limitations of the model are also explored including the derivation of a maximum compressibility limit; investigation of accuracy of numerical folding motion simulation; and an investigation of a free edge distortion behaviour which occurs in certain origami forms.

Design of a smooth freeform illumination system for a point light source based on polar-type optimal transport mapping.

A design method is proposed to generate smooth freeform illumination optics for a point light source based on the L2 optimal transport (LOT) mapping. In this method, the LOT mapping between an assumed circular planar source and a prescribed target is first obtained by solving a polar-type LOT problem. Then, the mapping calculated for the circular source is applied for a point light source. Finally, the freeform optical surface is generated by a geometric construction method to realize the ray mapping. As examples, a series of smooth-surface freeform lenses are designed for a point light source to form uniform and complex illumination patterns on rectangular targets. The ray-tracing results show that all the designs achieve excellent performance with the light utilization efficiency η over 0.87 (Fresnel loss considered) and the relative standard deviation (RSD) of the simulated illumination distribution less than 0.051 simultaneously.

Using conics to construct geometric 3-configurations, part II: the generalized Steiner construction

In a number of previous papers, techniques have been developed to construct geometric 3-configurations whose reduced Levi graph has at least one pair of parallel edges, but construction techniques for configurations whose reduced Levi graph is a simple bipartite cubic graph remained elusive. In Using conics to construct geometric 3-configurations, part I: symmetrically generalizing the Pappus configuration (2016), a primarily straightedge-and-compass construction, along with the construction of a certain conic section, was developed to construct configurations whose reduced Levi graph is an edge-labelled digraph with underlying graph \(K_{3,3}\). In this paper, we extend that construction to produce a construction technique for certain classes of configurations whose reduced Levi graphs are bipartite simple cubic graphs that contain at least one quadrilateral; in particular, we produce a geometric construction method that produces configurations whose reduced Levi graphs have as their underlying graphs the cyclic fence graphs, which unify even prism graphs and odd Moebius ladders.

A vector based freeform approach for reflecting concentrator of solar energy

A vector based freeform approach is proposed for the geometrical reverse design of reflecting solar concentrators. The effect of solar cone angle of incident vectors is taken into consideration in the method based on three-dimensional freeform optics. The method developed is a closed-loop program and did not need any other commercial software. According to the transmission pattern of solar concentrators, the source-target mapping should be initially established prior to constructing an initial geometry formed by discrete points using the Geometric Construction Method. The freeform surface is generated by connecting several triangular flat elements defined by these points and then the Monte-Carlo ray tracing was conducted. The freeform surface can then be further optimized through feedback modifications. Finally, the optimal shape is obtained when the design examples are the off-axis freeform reflectors with target circular/rectangular receivers for the use of Concentrated Photovoltaic. This method provides a practical approach for improving the optical performance of solar concentrators.

A novel free-form Cassegrain concentrator for PV/T combining utilization

The conventional imaging solar concentrators normally obtain a Gaussian distribution on the receiving surface. The aperture size of a thermal receiver (TR) is always limited primarily to the high concentration area, discarding the edge part to be lost. The current study proposes a novel free-form Cassegrain concentrator (FFCC) for PV/T combining utilization. The remaining heat flux by the edge of focal spot is directed by the secondary mirror for electric generation. The profile of secondary reflector is constructed by the geometric construction method (GCM), aiming at achieving high concentration for the TR and uniform flux distribution for the PV receiver (PVR). The energy transmission characteristics of FFCC, including the incident vectors and heat flux distribution on the TR/PVR are both investigated. The effects of changing the receiving locations, PV/T dividing ratio, etc. are analyzed. The sun size/solar conic angular is also considered for practical use. The GCM method and simulation results in this article can provide a reference for the PV/T solar energy system and free form surface design.

Polar-grids based source-target mapping construction method for designing freeform illumination system for a lighting target with arbitrary shape.

The freeform optical system plays a key role in illumination engineering, and several methods have been reported to manage the design of such system. In this paper, an approach to generate the polar-grids based flux transportation mapping for an arbitrarily-shaped target is proposed based on the conventional variable separation method. The source emitting grid is divided along the azimuth angle and the zenith angle respectively under the spherical coordinate system. Then, the target grid is achieved by solving the flux integral equations in polar coordinates using separation of variables method. When establishing the target grid along the polar radius, a strategy based on uniformly scaling down the external contour of the target is introduced. According to the mapping, a smooth freeform surface is then generated using the geometric construction method according to Snell's law. Finally, an iterative feedback process is adopted to compensate the deterioration of the target distribution caused by surface construction errors and the extension of a real source. Based on this method, a series of freeform lenses are designed for a 1 × 1 mm(2) LED source to generate uniform, Gaussian and multiple-rings illumination distributions within different target regions. High-performance optical systems with the light utilization efficiency η over 0.8 and the relative standard deviation (RSD) of the simulated illumination distribution less than 0.1 are obtained simultaneously for all the cases.

The Key Technology Research of the Pushing Closure Process about Jiashao Bridge Based on Geometric Control Method

Based on the characteristics of the geometric control method during the construction of cable-stayed bridge, the key goal to complete the bridge is to ensure that the non-stress dimensions and profile of every component is constant. Because one of the key points to complete the cable-stayed bridge is to ensure that the non-stress length of the closure segments is constant, it is necessary to adjust the relative position of the final closure by the geometric closure method of incremental launching construction to satisfy the requirement of the non-stress length of the closure segments. Aimed at the critical technical problems existing in incremental launching construction process, taking Jiashao bridge as the research subject, the analysis of the incremental launching plan and the local stress of pushing bracket and bearing block are carried out. The research will provide scientific basis for the feasibility of geometric control incremental launching construction method and it will guarantee the smoothly incremental launching construction.