Reconstruction approximating method by biquadratic splines of offset surfaces holes

Standard Offset surfaces are defined as locus of the points which are at constant distance along the unit normal direction from the generator surfaces. Offset are widely used in various practical applications, such as tolerance analysis, geometric optics and robot path-planning. In some of the engineering applications, we need to extend the concept of standard offset to the generalized offset where distance offset is not necessarily constant and offset direction are not necessarily along the normal direction. Normally, a generalized offset is functionally more complex than its progenitor because of the square root appears in the expression of the unit normal vector. For this, an approximation method of its construction is necessary. In many situation it is necessary to fill or reconstruct certain function defined in a domain in which there is a lack of information inside one or several sub-domains (holes). In some practical cases, we may have some specific geometrical constrains, of industrial or design type, for example, the case of a specified volume inside each one of these holes. The problem of filling holes or completing a 3D surface arises in all sorts of computational graphics areas, like CAGD, CAD-CAM, Earth Sciences, computer vision in robotics, image reconstruction from satellite and radar information, etc. In this work we present an approximation method of filling holes of the generalized offset of a surface when there is a lack information in a sub-domain of the function that define it. We prove the existence and uniqueness of solution of this problem, we show how to compute it and we establish a convergence result of this approximation method. Finally, we give some graphical and numerical examples.

SUSY in the sky with gravitons

Picture yourself in the wave zone of a gravitational scattering event of two massive, spinning compact bodies (black holes, neutron stars or stars). We show that this system of genuine astrophysical interest enjoys a hidden $$ \mathcal{N} $$ N = 2 supersymmetry, at least to the order of spin-squared (quadrupole) interactions in arbitrary D spacetime dimensions. Using the $$ \mathcal{N} $$ N = 2 supersymmetric worldline action, augmented by finite-size corrections for the non-Kerr black hole case, we build a quadratic-in-spin extension to the worldline quantum field theory (WQFT) formalism introduced in our previous work, and calculate the two bodies’ deflection and spin kick to sub-leading order in the post-Minkowskian expansion in Newton’s constant G. For spins aligned to the normal vector of the scattering plane we also obtain the scattering angle. All D-dimensional observables are derived from an eikonal phase given as the free energy of the WQFT that is invariant under the $$ \mathcal{N} $$ N = 2 supersymmetry transformations.

Intelligent Point Cloud Edge Detection Method Based on Projection Transformation

An edge detection method based on projection transformation is proposed. First, the vertical projection transformation is carried out on the target point cloud. Data X and data Y are normalized to the width and height of the image, respectively. Data Z is normalized to the range of 0-255, and the depth represents the gray level of the image. Then, the Canny algorithm is used to detect the edge of the projection transformed image, and the detected edge data is back projected to extract the edge point cloud in the point cloud. Evaluate the performance by calculating the normal vector of the edge point cloud. Compared with the normal vector of the whole data point cloud of the target, the normal vector of the edge point cloud can well express the characteristics of the target, and the calculation time is reduced to 10% of the original.

Effective action of string theory at order $$\alpha '$$ in the presence of boundary

Recently, using the assumption that the string theory effective action at the critical dimension is background independent, the classical on-shell effective action of the bosonic string theory at order $$\alpha '$$ α ′ in a spacetime manifold without boundary has been reproduced, up to an overall parameter, by imposing the O(1, 1) symmetry when the background has a circle. In the presence of the boundary, we consider a background which has boundary and a circle such that the unit normal vector of the boundary is independent of the circle. Then the O(1, 1) symmetry can fix the bulk action without using the lowest order equation of motion. Moreover, the above constraints and the constraint from the principle of the least action in the presence of boundary can fix the boundary action, up to five boundary parameters. In the least action principle, we assume that not only the values of the massless fields but also the values of their first derivatives are arbitrary on the boundary. We have also observed that the cosmological reduction of the leading order action in the presence of the Hawking–Gibbons boundary term, produces zero cosmological boundary action. Imposing this as another constraint on the boundary couplings at order $$\alpha '$$ α ′ , we find the boundary action up to two parameters. For a specific value for these two parameters, the gravity couplings in the boundary become the Chern–Simons gravity plus another term which has the Laplacian of the extrinsic curvature.

Harmonic Evolute Surface of Tubular Surfaces via B -Darboux Frame in Euclidean 3-Space

In this article, we look at a surface associated with real-valued functions. The surface is known as a harmonic surface, and its unit normal vector and mean curvature have been used to characterize it. We use the Bishop-Darboux frame ( B -Darboux frame) in Euclidean 3-space E 3 to study and explain the geometric characteristics of the harmonic evolute surfaces of tubular surfaces. The characterizations of the harmonic evolute surface’s ϱ and ς parameter curves are evaluated, and then, they are compared. Finally, an example of a tubular surface’s harmonic evolute surface is presented, along with visuals of these surfaces.

Multiple Cylinder Extraction from Organized Point Clouds

Most man-made objects are composed of a few basic geometric primitives (GPs) such as spheres, cylinders, planes, ellipsoids, or cones. Thus, the object recognition problem can be considered as one of geometric primitives extraction. Among the different geometric primitives, cylinders are the most frequently used GPs in real-world scenes. Therefore, cylinder detection and extraction are of great importance in 3D computer vision. Despite the rapid progress of cylinder detection algorithms, there are still two open problems in this area. First, a robust strategy is needed for the initial sample selection component of the cylinder extraction module. Second, detecting multiple cylinders simultaneously has not yet been investigated in depth. In this paper, a robust solution is provided to address these problems. The proposed solution is divided into three sub-modules. The first sub-module is a fast and accurate normal vector estimation algorithm from raw depth images. With the estimation method, a closed-form solution is provided for computing the normal vector at each point. The second sub-module benefits from the maximally stable extremal regions (MSER) feature detector to simultaneously detect cylinders present in the scene. Finally, the detected cylinders are extracted using the proposed cylinder extraction algorithm. Quantitative and qualitative results show that the proposed algorithm outperforms the baseline algorithms in each of the following areas: normal estimation, cylinder detection, and cylinder extraction.

Criterion for the occurrence of the macro destruction and formation of breaking during metal deformation

When generalizing the geometrically nonlinear law of Murnaghan elasticity to plasticity, a formally mathematical criterion was introduced for deformational macrofracture (macrocrack appearance) associated with an increase in elastic and plastic anisotropy as a failure cause. The use of the double potentiality of the governing equations in stresses and their velocities made it possible to obtain the reliable information on the structure of the deviatory section of the yield surface, the existence of which is a classical hypothesis in solid mechanics. The normal vector to the surface of the deviatory section is selected from two mutually orthogonal eigenvectors of the constructed operator. There are two families of regular concave surfaces, and a section surface is formed by joining the parts of two representatives of the families at singular points. To select normal vectors, the obtained ratio for them is used for isotropy. In connection with the considered problem of a double simple shift, it is established that multiple eigenvalues appear for the both normal vectors. To unambiguously determine the normal vector at a regular point, it is necessary to exclude the presence of multiple eigenvalues for the both normal vectors at the same time. At a singular point, the appearance of a multiple eigenvalue of one of the normal vectors is still unacceptable. These two conditions are necessary and sufficient to validate the governing equations of the generalized Murnaghan model. Otherwise, a macrocrack occurs. The theoretical construction is supported by the developed software complexes.

Topographical evaluation of laser generated surfaces using statistical analysis of surface-normal vector distributions

Surfaces generated by Additive Manufacturing or laser texturing can involve the solidification of droplets of liquid, which can give rise to overhanging features on the solidified surface. Overhanging features add a layer of complexity to the surface topography and are undetectable by standard surface roughness measurement techniques such as profilometry. Such features are important because they can have a considerable effect on surface properties such as wettability. New techniques and algorithms are therefore required to analyse and quantify convoluted surfaces with overhanging (re-entrant) features. Earlier work by the authors introduced the concept of using X-ray micro-computed tomography (Micro-CT) to identify the directions of vectors normal to the surface at any point and thus indicate the presence or absence of overhanging features. This paper divides overhanging features into two types; simple and compound, and introduces new, size independent, analysis techniques which measure what proportion of each type is on the surface. Another extension of the analysis is the comparison of surface profiles taken in different directions in order to identify any surface roughness anisotropies.

3D Point Cloud of Single Tree Branches and Leaves Semantic Segmentation Based on Modified PointNet Network

Semantic segmentation of single tree 3D point cloud is one of the key technologies in building tree model. It plays an important role in tree skeleton extraction, tree pruning, tree model reconstruction and other fields. Because the area of a single leaf is smaller than that of the whole tree, the segmentation of branches and leaves is a challenging problem. In view of the above problems, this paper first migrates PointNet to the tree branch and leaf point cloud segmentation, and proposes an automatic segmentation method based on improved PointNet. According to the difference of normal direction between leaves and branches, the point cloud information of three dimensions coordinates, color and normal vector is input into the point feature space. In data processing, increase the number of each block data, so that the network can better learn features. MLP is added to the original PointNet network to improve the ability of extracting and learning local features. In addition, in the process of feature extraction, jump connection is added to realize feature reuse and make full use of different levels of features. The original 1×1 filter of PointNet is replaced by 3×1 filter to improve the segmentation accuracy of tree point cloud. The focus loss function focal loss is introduced into the field of 3D point cloud to reduce the impact of the imbalance of point cloud samples on the results. The results show that the improved method improves the accuracy of tree branch point cloud segmentation compared with the original PointNet for branch and leaf segmentation. The segmentation accuracy of structural elements of branches and leaves is more than 88%, and MIoU is 48%.

Determination of the geometry of laser-cutting fronts with high spatial and temporal resolution

The melt flow velocity and the local surface angles of the cutting front during laser fusion cutting of 10 mm AISI 304 were determined for a laser power of 8 kW and a feed rate of 2 m/min. The cut front was recorded with a polarization goniometer, which uses the polarization of the process emission to determine the local surface angles, allowing to calculate the orientation of the normal vector of the surface. The records in this work were carried out with a frame rate of 75 kHz and a spatial resolution of about 30 µm. This allowed to identify big and small structures moving down the cutting front and to determine their velocities. The approximate velocity of the small structures was 9.1 m/s and for the big structures approx. 2.5 m/s. The information of a usual high-speed video was compared with the additionally obtained geometry information.