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The subject matter of surfaces in computer aided geometric design (CAGD) is the depiction and design of surfaces in the computer graphics arena. Due to their geometric features, modeling of Bézier curves and surfaces with their shape parameters is the most well-liked topic of research in CAGD/computer-aided manufacturing (CAM). The primary challenges in industries such as automotive, shipbuilding, and aerospace are the design of complex surfaces. In order to address this issue, the continuity constraints between surfaces are utilized to generate complex surfaces. The parametric and geometric continuities are the two metrics commonly used for establishing connections among surfaces. This paper proposes continuity constraints between two generalized Bézier-like surfaces (gBS) with different shape parameters to address the issue of modeling and designing surfaces. Initially, the generalized form of C3 and G3 of generalized Bézier-like curves (gBC) are developed. To check the validity of these constraints, some numerical examples are also analyzed with graphical representations. Furthermore, for a continuous connection among these gBS, the necessary and sufficient G1 and G2 continuity constraints are also developed. It is shown through the use of several geometric designs of gBS that the recommended basis can resolve the shape and position adjustment problems associated with Bézier surfaces more effectively than any other basis. As a result, the proposed scheme not only incorporates all of the geometric features of curve design schemes but also improves upon their faults, which are typically encountered in engineering. Mainly, by changing the values of shape parameters, we can alter the shape of the curve by our choice which is not present in the standard Bézier model. This is the main drawback of traditional Bézier model. [ABSTRACT FROM AUTHOR]

In this paper, we examine q-Bernstein-Bézier surfaces in Minkowski space- R13 with q as the shape parameter. These surfaces, a generalization of Bézier surfaces, have applications in mathematics, computer-aided geometric design, and computer graphics for the surface formation and modeling. We analyze the timelike and spacelike cases of q-Bernstein-Bézier surfaces using known boundary control points. The mean curvature and Gaussian curvature of these q-Bernstein-Bézier surfaces are computed by finding the respective fundamental coefficients. We also investigate the shape operator dependency for timelike and spacelike q-Bernstein-Bézier surfaces in Minkowski space- R13 , and provide biquadratic and bicubic q-Bernstein-Bézier surfaces as illustrative examples for different values of the shape controlling parameter q. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the properties of timelike and spacelike shifted-knots Bézier surfaces in Minkowski space- E13. These surfaces are commonly used in mathematical models for surface formation in computer science for computer-aided geometric design and computer graphics, as well as in other fields of mathematics. Our objective is to analyze the characteristics of timelike and spacelike shifted-knots Bézier surfaces in Minkowski space- E13. To achieve this, we compute the fundamental coefficients of shifted-knots Bézier surfaces, including the Gauss-curvature, mean-curvature, and shape-operator of the surface. Furthermore, we present numerical examples of timelike and spacelike bi-quadratic (m = n = 2) and bi-cubic (m = n = 3) shifted-knots Bézier surfaces in Minkowski space- E13 to demonstrate the applicability of the technique in Minkowski space. [ABSTRACT FROM AUTHOR]

Based on the finite element theory, a joint-plane modeling method is employed to connect the corresponding nodes at the joint surface of the woodworking computer numerical control (CNC) machining center bed with a 2-node 12-degree-of-freedom unit. A spatial element model is established, which can show the state of the nodes between joint surfaces when they are stretched, compressed, or twisted; and it can help build a woodworking CNC machining center on a finite element model of bed with the characteristics of the joint surface. The simulated analysis is performed on the model and is compared with the result of simulated analysis on the bed model that ignores the characteristics of the joint surface and modal experiment. The comparison verifies the effectiveness of the modeling method based on the characteristics of the joint surface. The weak link of the machine bed structure is analyzed and optimized. The natural frequency of the bed is improved by2.55% ~ 11.3%. The displacement is reduced by a maximum of 19.4%, and dynamic performance of the bed is improved. [ABSTRACT FROM AUTHOR]

Purpose: To present a three-dimensional non-parametric method for detecting scleral asymmetry using corneoscleral topography data that are free of edge-effect artefacts. Methods: The study included 88 participants aged 23 to 65 years (37.7±9.7), 47 women and 41 men. The eye topography data were exported from the Eye Surface Profiler software in MATLAB binary data container format then processed by custom built MATLAB codes entirely independent from the profiler software. Scleral asymmetry was determined initially from the unprocessed topography before being determined again after removing the edge-effect noise. Topography data were levelled around the limbus, then edge-effect was eliminated using a robust statistical moving median technique. In addition to comparing raw elevation data, scleral elevation was also compared through fitting a sphere to every single scleral surface and determining the relative elevation from the best-fit sphere reference surface. Results: When considering the averaged raw topography elevation data in the scleral section of the eye at radius 8 mm, the average raw elevations of the right eyes’ sclera were -1.5±1.77, -1.87±2.12, -1.36±1.82 and -1.57±1.87 mm. In the left eyes at the same radius the average raw elevations were -1.62±1.78, -1.82±2.07, -1.28±1.76 and -1.68±1.93 mm. While, when considering the average raw elevation of the sclera after removing the edge effect, the average raw elevations of the right eyes were -3.71±0.25, -4.06±0.23, -3.95±0.19 and -3.95±0.23 mm. In the left eyes at the same radius the average raw elevations were -3.71±0.19, -3.97±0.22, -3.96±0.19 and -3.96±0.18 mm in the nasal, temporal, superior and inferior sides respectively. Maximum raw elevation asymmetry in the averaged scleral raw elevation was 1.6647±0.9015 mm in right eyes and 1.0358±0.6842 mm in left eyes, both detected at -38° to the nasal side. Best-fit sphere-based relative elevation showed that sclera is more elevated in three main meridians at angles -40°, 76°, and 170° in right eyes and -40°, 76°, and 170° in left eyes, all measured from the nasal meridian. Maximum recorded relative elevation asymmetries were 0.0844±0.0355 mm and 0.068±0.0607 mm at angular positions 76° and 63.5° for right and left eyes in turn. Conclusions: It is not possible to use corneoscleral topography data to predict the scleral shape without considering a method of removing the edge-effect from the topography data. The nasal side of the sclera is higher than the temporal side, therefore, rotationally symmetric scleral contact lenses are more likely to be translated towards the temporal side. The scleral shape is best described by levelled raw elevation rather than relative elevation. [ABSTRACT FROM AUTHOR]