Slant Helices of (k,m)-Type According to the ED-Frame in Minkowski 4-Space

In differential geometry, relations between curves are a large and important area of study for many researchers. Frame areas are an important tool when studying curves, specially the Frenet–Serret frame along a space curve and the Darboux frame along a surface curve in differential geometry. In this paper, we obtain slant helices of k-type according to the extended Darboux frame (or, for brevity, ED-frame) field by using the ED-frame field of the first kind (or, for brevity, EDFFK), which is formed with an anti-symmetric matrix for ε1=ε2=ε3=ε4∈{−1,1} and the ED-frame field of the second kind (or, for brevity, EDFSK), which is formed with an anti-symmetric matrix for ε1=ε2=ε3=ε4∈{−1,1} in four-dimensional Minkowski space E14. In addition, we present some characterizations of slant helices and determine (k,m)-type slant helices for the EDFFK and EDFSK in Minkowski 4-space.

Mid-Air Drawing of Curves on 3D Surfaces in Virtual Reality

Complex 3D curves can be created by directly drawing mid-air in immersive environments (Augmented and Virtual Realities). Drawing mid-air strokes precisely on the surface of a 3D virtual object, however, is difficult, necessitating a projection of the mid-air stroke onto the user “intended” surface curve. We present the first detailed investigation of the fundamental problem of 3D stroke projection in VR. An assessment of the design requirements of real-time drawing of curves on 3D objects in VR is followed by the definition and classification of multiple techniques for 3D stroke projection. We analyze the advantages and shortcomings of these approaches both theoretically and via practical pilot testing. We then formally evaluate the two most promising techniques spraycan and mimicry with 20 users in VR. The study shows a strong qualitative and quantitative user preference for our novel stroke mimicry projection algorithm. We further illustrate the effectiveness and utility of stroke mimicry to draw complex 3D curves on surfaces for various artistic and functional design applications.

DETERMINATION OF RATIONAL PARAMETERS OF SUPPORTING STRUCTURES MADE OF SOIL-CEMENT PILES ON LANDSLIDE-PRONE SLOPES

Purpose. The article proposes a method for determining the rational parameters of supporting structures made of soil-cement piles on landslide-prone slopes. Methodology. To achieve this purpose, the authors performed comparative calculations of finite-element models of landslide-prone layered slope with the arrangement of supporting structures of soil-cement piles: vertical, located at an angle to the vertical axis, combined with a concrete grillage, without grillage, with different diameters, with different distances between piles, but of the same length, which is justified by the location determined by preliminary calculations of the sliding surface. Findings. According to the results of calculating the contact problem of the interaction of the soil mass and the supporting structure, the regularities of formation of strength and stability of the reinforced landslide-prone slope were obtained, which makes it possible to predict its geomechanical stability depending on the possible location of the sliding surface curve. The method of increasing the stability coefficient of the soil slope depending on the variation of the parameters of the soil-cement supporting pile structure is substantiated. Numerical calculations substantiate the parameters of soil-cement pile supporting structures – the length of the piles, their diameter, the distance between the piles, the angle of their inclination and the feasibility of combining the piles with a concrete grillage. Originality. For the first time, the formation regularities of strength and stability of soil-cement supporting structure for strengthening the landslide-prone slopes are obtained, which makes it possible to predict their geomechanical stability depending on the possible location of the sliding surface curve. For the first time, the dependences of changes in absolute displacements, relative deformations, coefficients of strength, stability, as well as Mises stresses in the elements of supporting structures of soil-cement piles on the angle of their inclination, distance between piles and the presence of a concrete grillage that joins them. Methods for assessing and predicting the landslide danger of soil slopes were further developed, which allowed to substantiate the method of increasing the strength coefficient depending on the variation of the parameters of the soil-cement supporting structure. For the first time, the parameters of the method of increasing the stability of landslide-prone slopes with pile supporting structures based on the complex mathematical and centrifugal modeling using real landslides and similarity criteria are substantiated. Practical value. The authors have developed and patented a method of strengthening landslide slopes with the help of soil-cement piles, combined with a concrete grillage, located perpendicular to the movement vector of the landslide body.

The Algorithm for Crossing the N-dimensional Hyperquadric with N-1-dimensional Hyperspace

In descriptive geometry, the problem of finding a surface curve section with a plane is common. One such surface curve is a quadric. Due to the increased demand for tasks related to quadric, the synthetic modeling method becomes relevant. In recent years, geometric constructions of dimensions of more than three began to be studied more and more often. Multidimensional geometric shapes in multidimensional space are typically constructed using geometric modeling software. However, without additional building automation tools, software does not sufficiently facilitate human labor. The larger the dimension of the constructions, the more cumbersome and time consuming the drawing process becomes. The increasing complexity of constructions requires automation of constructions that can be traditimatized. Geometric constructions made using automation tools make us rethink the process of structural geometric modeling in descriptive geometry. Within the framework of the article, the algorithm for crossing the N-dimensional hyperquadric with N-1-dimensional hyperspace is presented. Special cases of this geometric construction are also considered: intersection of a three- dimensional quadric with a plane and intersection of a four-dimensional hyperquadric with a three-dimensional space. The implementation of the developed algorithm is carried out using the Simplex system and the built-in interpreter of the prolog logical programming language.

A new approach to the bienergy and biangle of a moving particle lying in a surface of lorentzian space

In this research, we study bienergy and biangles of moving particles lying on the surface of Lorentzian 3-space by using their energy and angle values. We present the geometrical characterization of bienergy of the particle in Darboux vector fields depending on surface. We also give the relationship between bienergy of the surface curve and bienergy of the elastic surface curve. We conclude the paper by providing bienergy-curve graphics for different cases.

Ergonomics of a Chinese folk bamboo lounge chair

Bamboo is an environmentally friendly sustainable resource and is commonly used in furniture production in China. Limited research has been conducted in the field of ergonomics of bamboo furniture, which can greatly affect the seating experience. This study provides a qualitative and quantitative analysis of the main effects of seat surface curve, bending deflection of bamboo strips of seat, functions, and usability in a bamboo lounge chair provided in Zhejiang Province. The shape of the seat surface curve of the bamboo lounge chair included the seat-back angle of 134°, and the slope of the bamboo lounge chair seat was 16°. The deflection of bamboo strips of seat under vertical loading force of 336 N was 30.7 mm. In addition, the angles, deflection characteristic, and functions of the bamboo lounge chair were all human-centered design and suitable for human comfort, which can reduce maximum pressure and maintain normal lumbar spine curve in adult.

A Survey and Comparative Study for Connecting 2D points

Reconstruction from noisy point sets has many ap-plications in the areas of science and engineering. Research effort in reconstructing shape from noisy point sets. Reconstruction on planar point including shape, surface, curve and manifold recon-struction. Good algorithms are required to create a good shape from a given point set. Better local and global sampling conditions form the base of these algorithms. Reconstruction from noisy point set is not extensively studied and therefore the researchers do not have a successful algorithm. Reconstruction from the stage is begun before many decades and these activities are now being extended for a few days. Extending any older reconstruction algorithms needs a good understanding and comparison of all previous algorithms. This survey is spamming on different reconstruction algorithms, various local sampling conditions, extension of different works and their working conditions and reconstruction implementation from point sets. Survey begins after 1997 and compares various extension works. The sampling condition for all these algorithms contributes significantly to the construction of algorithms, thus different local sampling conditions are investigated. During this study, all algorithms for reconstruction are tabulated and different parameters for these algorithms are compared. This survey is concluding with several promising directions for the future works on reconstruction.