scholarly journals The Generalized H-Bézier Model: Geometric Continuity Conditions and Applications to Curve and Surface Modeling

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 924 ◽  
Author(s):  
Fenhong Li ◽  
Gang Hu ◽  
Muhammad Abbas ◽  
Kenjiro T. Miura
Keyword(s):  
Linear Independence ◽  
Geometric Continuity ◽  
Shape Parameters ◽  
Computer Design ◽  
First Order ◽  
Complex Curves ◽  
Surface Models ◽  
G2 Continuity ◽  
Geometric Representations ◽  
Shape Controlled

The local controlled generalized H-Bézier model is one of the most useful tools for shape designs and geometric representations in computer-aided geometric design (CAGD), which is owed to its good geometric properties, e.g., symmetry and shape adjustable property. In this paper, some geometric continuity conditions for the generalized cubic H-Bézier model are studied for the purpose of constructing shape-controlled complex curves and surfaces in engineering. Firstly, based on the linear independence of generalized H-Bézier basis functions (GHBF), the conditions of first-order and second-order geometric continuity (namely, G1 and G2 continuity) between two adjacent generalized cubic H-Bézier curves are proposed. Furthermore, following analysis of the terminal properties of GHBF, the conditions of G1 geometric continuity between two adjacent generalized H-Bézier surfaces are derived and then simplified by choosing appropriate shape parameters. Finally, two operable procedures of smooth continuity for the generalized H-Bézier model are devised. Modeling examples show that the smooth continuity technology of the generalized H-Bézier model can improve the efficiency of computer design for complex curve and surface models.

scholarly journals Explicit Continuity Conditions for G1 Connection of S-λ Curves and Surfaces

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1359 ◽  
Author(s):  
Gang Hu ◽  
Huinan Li ◽  
Muhammad Abbas ◽  
Kenjiro T. Miura ◽  
Guoling Wei
Keyword(s):  
Computer Aided Design ◽  
Linear Independence ◽  
Basis Functions ◽  
Geometric Continuity ◽  
Complex Curves ◽  
Curves And Surfaces ◽  
Computer Aided ◽  
Cad Cam ◽  
Geometric Representations ◽  
Aided Design

The S-λ model is one of the most useful tools for shape designs and geometric representations in computer-aided geometric design (CAGD), which is due to its good geometric properties such as symmetry, shape adjustable property. With the aim to solve the problem that complex S-λ curves and surfaces cannot be constructed by a single curve and surface, the explicit continuity conditions for G1 connection of S-λ curves and surfaces are investigated in this paper. On the basis of linear independence and terminal properties of S-λ basis functions, the conditions of G1 geometric continuity between two adjacent S-λ curves and surfaces are proposed, respectively. Modeling examples imply that the continuity conditions proposed in this paper are easy and effective, which indicate that the S-λ curves and surfaces can be used as a powerful supplement of complex curves and surfaces design in computer aided design/computer aided manufacturing (CAD/CAM) system.

scholarly journals Generalized Fractional Bézier Curve with Shape Parameters

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2141
Author(s):  
Syed Ahmad Aidil Adha Said Mad Said Mad Zain ◽  
Md Yushalify Misro ◽  
Kenjiro T. Miura
Keyword(s):  
Basis Functions ◽  
Bézier Curve ◽  
Free Form ◽  
Bezier Curve ◽  
Geometric Continuity ◽  
Shape Parameters ◽  
Complex Curves ◽  
Curves And Surfaces ◽  
New Type ◽  
Fractional Parameter

The construction of new basis functions for the Bézier or B-spline curve has been one of the most popular themes in recent studies in Computer Aided Geometric Design (CAGD). Implementing the new basis functions with shape parameters provides a different viewpoint on how new types of basis functions can develop complex curves and surfaces beyond restricted formulation. The wide selection of shape parameters allows more control over the shape of the curves and surfaces without altering their control points. However, interpolated parametric curves with higher degrees tend to overshoot in the process of curve fitting, making it difficult to control the optimal length of the curved trajectory. Thus, a new parameter needs to be created to overcome this constraint to produce free-form shapes of curves and surfaces while still preserving the basic properties of the Bézier curve. In this work, a general fractional Bézier curve with shape parameters and a fractional parameter is presented. Furthermore, parametric and geometric continuity between two generalized fractional Bézier curves is discussed in this paper, as well as demonstrating the effect of the fractional parameter of curves and surfaces. However, the conventional parametric and geometric continuity can only be applied to connect curves at the endpoints. Hence, a new type of continuity called fractional continuity is proposed to overcome this limitation. Thus, with the curve flexibility and adjustability provided by the generalized fractional Bézier curve, the construction of complex engineering curves and surfaces will be more efficient.

scholarly journals Geometric Modeling of Novel Generalized Hybrid Trigonometric Bézier-Like Curve with Shape Parameters and Its Applications

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 967 ◽  
Author(s):  
Samia BiBi ◽  
Muhammad Abbas ◽  
Kenjiro T. Miura ◽  
Md Yushalify Misro
Keyword(s):  
Geometric Modeling ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Geometric Continuity ◽  
Shape Parameters ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Curvature Continuity ◽  
Bernstein Basis Functions

The main objective of this paper is to construct the various shapes and font designing of curves and to describe the curvature by using parametric and geometric continuity constraints of generalized hybrid trigonometric Bézier (GHT-Bézier) curves. The GHT-Bernstein basis functions and Bézier curve with shape parameters are presented. The parametric and geometric continuity constraints for GHT-Bézier curves are constructed. The curvature continuity provides a guarantee of smoothness geometrically between curve segments. Furthermore, we present the curvature junction of complex figures and also compare it with the curvature of the classical Bézier curve and some other applications by using the proposed GHT-Bézier curves. This approach is one of the pivotal parts of construction, which is basically due to the existence of continuity conditions and different shape parameters that permit the curve to change easily and be more flexible without altering its control points. Therefore, by adjusting the values of shape parameters, the curve still preserve its characteristics and geometrical configuration. These modeling examples illustrate that our method can be easily performed, and it can also provide us an alternative strong strategy for the modeling of complex figures.

scholarly journals Normalized potentials of minimal surfaces in spheres

1999 ◽  
Vol 156 ◽  
pp. 187-214
Author(s):  
Quo-Shin Chi ◽  
Luis Fernández ◽  
Hongyou Wu
Keyword(s):  
Twistor Space ◽  
Dimensional Sphere ◽  
Large Area ◽  
Type Representation ◽  
First Order ◽  
Birational Map ◽  
Complex Curves ◽  
The North ◽  
Almost Complex ◽  
Superconformal Surface

We determine explicitly the normalized potential, a Weierstrass-type representation, of a superconformal surface in an even-dimensional sphere S2n in terms of certain normal curvatures of the surface. When the Hopf differential is zero the potential embodies a system of first order equations governing the directrix curve of a superminimal surface in the twistor space of the sphere. We construct a birational map from the twistor space of S2n into ℂPn(n+1)/2. In general, birational geometry does not preserve the degree of an algebraic curve. However, we prove that the birational map preserves the degree, up to a factor 2, of the twistor lift of a superminimal surface in S6 as long as the surface does not pass through the north pole. Our approach, which is algebro-geometric in nature, accounts in a rather simple way for the aforementioned first order equations, and as a consequence for the particularly interesting class of superminimal almost complex curves in S6. It also yields, in a constructive way, that a generic superminimal surface in S6 is not almost complex and can achieve, by the above degree property, arbitrarily large area.

scholarly journals Optimization of Tooth Root Profile Using Bezier Curve with G2 Continuity to Reduce Bending Stress of Asymmetric Spur Gear Tooth

2018 ◽  
Vol 237 ◽  
pp. 03010 ◽  
Author(s):  
Priyakant Vaghela ◽  
Jagdish Prajapati
Keyword(s):  
Stress Concentration ◽  
Gear Tooth ◽  
Spur Gear ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Tooth Root ◽  
Geometric Continuity ◽  
Bending Stress ◽  
Gear Design ◽  
G2 Continuity

This research describes simple and innovative approach to reduce bending stress at tooth root of asymmetric spur gear tooth which is desire for improve high load carrying capacity. In gear design at root of tooth circular-filleted is widely used. Blending of the involute profile of tooth and circular fillet creates discontinuity at root of tooth causes stress concentration occurs. In order to minimize stress concentration, geometric continuity of order 2 at the blending of gear tooth plays very important role. Bezier curve is used with geometric continuity of order 2 at tooth root of asymmetric spur gear to reduce bending stress.

scholarly journals Modeling of Top Scroll Profile Using Equidistant-Curve Approach for a Scroll Compressor

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Tao Liu ◽  
Zaixin Wu
Keyword(s):  
Dynamic Models ◽  
Rotation Angle ◽  
Volume Ratio ◽  
Geometric Continuity ◽  
Leakage Loss ◽  
Scroll Compressor ◽  
Order Continuity ◽  
Circular Arc ◽  
First Order ◽  
Generation Line

Scroll profile plays a key role in determining the performance of a scroll compressor. In this study geometric and dynamic characteristics of the scroll profile are analyzed to investigate the influence of its geometric continuity on property of a scroll compressor. Firstly, scroll profiles are created to redesign the geometry of scroll wrap by using the equidistant-curve approach on the basis of a generation line consisting of involute of circle and circular arc. Subsequently, the geometric and dynamic models of the scroll compressor are established. These models are related to parameters of the generation line of scroll profile and rotation angle of a moving scroll. Lastly, some simulation examples of second-order continuity (SOC) scroll profile are compared with first-order continuity (FOC) scroll profiles and some important conclusions are obtained. Results show that SOC scroll profile is superior to FOC profile in terms of volume ratio, stability of gas force, and possible leakage loss in a scroll compressor.

scholarly journals A GC2 Joining Procedure for B-Spline Patches in Reverse Engineering

1997 ◽  
Author(s):  
Tachung Yang ◽  
Cheng-Chung Wang
Keyword(s):  
Reverse Engineering ◽  
Coordinate Measuring Machine ◽  
Surface Patch ◽  
Geometric Continuity ◽  
Huge Amount ◽  
B Spline ◽  
Surface Patches ◽  
Spline Surface ◽  
Surface Models ◽  
Measuring Machine

Reconstruction of surface models is a vital part in reverse engineering. Because of the huge amount of data from Coordinate Measuring Machine (CMM), processes for division of data into groups, surface patch reconstruction, and patch joining are inevitable in the CAD systems tailored for reverse engineering applications. Existing techniques of surface patch joining have the disadvantages, such as computational complication or lack of desired geometric continuity. A GC2 joining technique for B-spline surface patches by utilising a Bezier patch joining technique was proposed in this paper. This method possesses the merits in which only the control vertices near the joining boundaries of patches are modified and no additional blending surfaces at the joints of patches are created.

scholarly journals Geometric continuity, shape parameters, and geometric constructions for Catmull-Rom splines

1988 ◽  
Vol 7 (1) ◽  
pp. 1-41 ◽  
Author(s):  
Tony D. DeRose ◽  
Brian A. Barsky
Keyword(s):  
Geometric Continuity ◽  
Shape Parameters ◽  
Geometric Constructions

scholarly journals “Efficiency Space”: A Framework for Evaluating Joint Evaporation and Runoff Behavior*

2015 ◽  
Vol 96 (3) ◽  
pp. 393-396 ◽  
Author(s):  
Randal Koster
Keyword(s):  
Soil Moisture ◽  
Land Surface ◽  
Optimal Combination ◽  
Balance Model ◽  
Net Radiation ◽  
Analysis Framework ◽  
Land Surface Models ◽  
First Order ◽  
Surface Models ◽  
Functional Forms

Abstract At the land surface, higher soil moisture levels generally lead to both increased evaporation for a given amount of incoming radiation (increased “evaporation efficiency”) and increased runoff for a given amount of precipitation (increased “runoff efficiency”). Evaporation efficiency and runoff efficiency can thus be said to vary with each other, motivating the development of a unique hydroclimatic analysis framework. Using a simple water balance model fitted, in different experiments, with a wide variety of functional forms for evaporation and runoff efficiency, the author transforms net radiation and precipitation fields into fields of streamflow that can be directly evaluated against observations. The optimal combination of the functional forms—the combination that produces the most skillful streamflow simulations—provides an indication for how evaporation and runoff efficiencies vary with each other in nature, a relationship that can be said to define the overall character of land surface hydrological processes, at least to the first order. The inferred optimal relationship is represented herein as a curve in “efficiency space” and should be valuable for the evaluation and development of GCM-based land surface models, which by this measure are often found to be suboptimal.

scholarly journals Parallel Tempering Monte Carlo Studies of Phase Transition of Free Boundary Planar Surfaces

2018 ◽  
Author(s):  
Andrey Shobukhov ◽  
Hiroshi Koibuchi
Keyword(s):  
Monte Carlo ◽  
Free Boundary ◽  
Functional Materials ◽  
Thermal Fluctuations ◽  
Order Transition ◽  
Parallel Tempering ◽  
First Order ◽  
Monte Carlo Studies ◽  
Surface Models ◽  
First Order Transition

We numerically study surface models defined on hexagonal disks with a free boundary. 2D surface models for planer surfaces have recently attracted interest due to the engineering applications of functional materials such as graphene and its composite with polymers. These 2D composite meta-materials are strongly influenced by external stimuli such as thermal fluctuations if they are sufficiently thin. For this reason, it is very interesting to study the shape stability/instability of thin 2D materials against thermal fluctuations. In this paper, we study three types of surface models including Landau-Ginzburg (LG) and Helfirch-Polyakov models defined on triangulated hexagonal disks using the parallel tempering Monte Carlo simulation technique. We find that the planer surfaces undergo a first-order transition between the smooth and crumpled phases in the LG model and continuous transitions in the other two models. The first-order transition is relatively weaker compared to the transition on spherical surfaces already reported. The continuous nature of the transition is consistent with the reported results, although the transitions are stronger than that of the reported ones.

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