Developable surfaces have a vital part in geometric modeling, architectural design, and material manufacturing. Developable Bézier surfaces are the important tools in the construction of developable ...surfaces, but due to polynomial depiction and having no shape parameter, they cannot describe conics exactly and can only handle a few shapes. To tackle these issues, two straightforward techniques are proposed to the computer-aided design of developable generalized blended trigonometric Bézier surfaces (for short, developable GBT-Bézier surfaces) with shape parameters. A developable GBT-Bézier surface is established by making a collection of control planes with generalized blended trigonometric Bernstein-like (for short, GBTB) basis functions on duality principle among points and planes in 4D projective space. By changing the values of shape parameters, a group of developable GBT-Bézier surfaces that preserves the features of the developable GBT-Bézier surfaces can be generated. Furthermore, for a continuous connection among these developable GBT-Bézier surfaces, the necessary and sufficient
G
1
and
G
2
(Farin–Boehm and beta) continuity conditions are also defined. Some geometric designs of developable GBT-Bézier surfaces are illustrated to show that the suggested scheme can settle the shape and position adjustment problem of developable Bézier surfaces in a better way than other existing schemes. Hence, the suggested scheme has not only all geometric features of current curve design schemes but surpasses their imperfections which are usually faced in engineering.
Full text
Available for:
IZUM, KILJ, NUK, PILJ, PNG, SAZU, UL, UM, UPUK
Aiming at the problem of approximate degree reduction of SG‐Bézier surfaces, a method is proposed to achieve the degree reduction from (n × n) to (m × m) (m < n). Starting from the idea of grey wolf ...optimizer (GWO) algorithm and combining the geometric properties of SG‐Bézier surfaces, this method transforms the degree reduction problem of SG‐Bézier surfaces into an optimization problem. By choosing the fitness function, the degree reduction approximation of shape‐adjustable SG‐Bézier surfaces under unconstrained and angular interpolation constraints is realized. At the same time, some concrete examples of degree reduction and its errors are given. The results show that this method not only achieves good degree reduction effect but also is easy to implement and has high precision.
Full text
Available for:
FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SBCE, SBMB, UL, UM, UPUK
The construction of the generalized Bézier model with shape parameters is one of the research hotspots in geometric modeling and CAGD. In this paper, a novel shape-adjustable generalized Bézier (or ...SG-Bézier, for short) surface of order (m, n) is introduced for the purpose to construct local and global shape controllable free-form complex surfaces. Meanwhile, some properties of SG-Bézier surfaces and the influence rules of shape parameters, as well as the constructions of special triangular and biangular SG-Bézier surfaces, are investigated. Furthermore, based on the terminal properties and linear independence of SG-Bernstein basis functions, the conditions for G1 and G2 continuity between two adjacent SG-Bézier surfaces are derived, and then simplified them by choosing appropriate shape parameters. Finally, the specific steps and applications of the smooth continuity for SG-Bézier surfaces are discussed. Modeling examples show that our methods in this paper are not only effective and can be performed easily, but also provide an alternative strategy for the construction of complex surfaces in engineering design.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
The paper presents the approach for solving 2D elastoplastic problems with singular stress/strain fields by the parametric integral equation system (PIES) method. PIES is applied since it does not ...require the discretization of the plastic zone into elements, and instead, it uses globally declared parametric surface patches. The properties of such surfaces (e.g. the unit square is their domain) are used for automatically concentrating the interpolation nodes around the regions with singular stress/strain fields. This, in turn, limits the global number of nodes in favor of their accumulation in specific places. The proposed approach of node concentration requires only a few parameters to be defined. In order to take into account the complex nature of the plastic strain fields, a special method of their interpolation is selected. The Kriging approach effectively involves an interactive investigation of the spatial behavior of the strains to choose the best estimate. It is also the method that bases on irregularly spaced interpolation nodes. Some examples are solved, and obtained results are compared with the finite element method (FEM).
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
A new algorithm for computing a point on a polynomial or rational curve in Bézier form is proposed. The method has a geometric interpretation and uses only convex combinations of control points. The ...new algorithm’s computational complexity is linear with respect to the number of control points and its memory complexity is O(1). Some remarks on similar methods for surfaces in rectangular and triangular Bézier form are also given.
•A new algorithm for computing a point on a polynomial or rational curve in Bézier form is proposed.•The method has a geometric interpretation and uses only convex combinations of control points.•The new algorithm’s computational complexity is linear with respect to the number of control points and its memory complexity is O(1).•Some remarks on similar methods for surfaces in rectangular and triangular Bézier are also given.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
•The shape of generalized developable H-Bézier surfaces (GDHBS) can be adjusted by changing parameters.•Three shape optimization models of GDHBS are presented.•An adaptive CS optimization algorithm ...is applied to the solutions of GDHBS shape optimization models.•Some representative and convictive examples are performed.
The shape optimization design of developable surface is a critical and intractable technique in CAD/CAM and used in many manufacturing planning operations. In this paper, we study shape optimization by using a new cuckoo search (CS) algorithm to promote the development of generalized developable H-Bézier surfaces (or GDHBS, for short). According to the duality between points and planes in 3D projective space, the GDHBS shape optimization problem is formulated as an arc length, energy and curvature variation minimization problem of dual curve, respectively. Then, an adaptive CS optimization algorithm is applied to the solutions of those GDHBS shape optimization models. Finally, we illustrate the effectiveness and performance of the proposed methods by some representative and convictive examples.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK, ZRSKP
7.
On α-Bézier curves and surfaces Kaur, Jaspreet; Goyal, Meenu
Bollettino della Unione matematica italiana (2008),
09/2023, Volume:
16, Issue:
3
Journal Article
In the given note, we present the generalization of Bézier curves depending upon the parameter
α
, named as
α
-
Bézier curves. Also, we introduce tensor product
α
-
Bézier surfaces. We study some ...properties and degree elevation of these curves and surfaces. In the end, we show that the parameter provides us the flexibility to modify the curves as well as surfaces. To present it, we give some numerical examples with the help of Matlab.
Full text
Available for:
EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
•A class of novel H-Bézier basis functions is presented.•Two methods of design for developable H-Bézier surfaces are proposed.•The shape of developable H-Bézier surfaces can be adjusted by changing ...parameters.•The continuity conditions of developable H-Bézier surfaces are obtained.•The numerical results demonstrate that the proposed methods are effective.
To solve the problem of shape adjustment for developable surfaces, we propose a novel method for constructing local controlled generalized developable H-Bézier surfaces with shape parameters. The generalized developable H-Bézier surfaces are designed by using control planes with generalized H-Bézier basis functions and their shapes can be adjusted by altering the values of shape parameters. When the shape parameters assume different values, a family of developable H-Bézier surfaces can be constructed, which retain the characteristics of the developable Bézier surfaces. Furthermore, we derive the necessary and sufficient conditions for G1 continuity, Farin-Boehm G2 continuity and G2 Beta continuity between two adjacent generalized developable H-Bézier surfaces. Finally, some properties of the new developable surfaces are discussed, and the influence rules of shape parameters on the new developable surfaces are studied. Modeling examples provided show that the proposed methods are effective and hence can greatly improve problem-solving abilities in engineering appearance design by adjusting the position and shape of developable surfaces.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK, ZRSKP
•A new multi-sided, control point-based surface over concave polygonal domains.•Generalized Bézier patches with an extended side-based control structure.•An algorithm to define concave domains from ...3D boundaries.•Parameterization and blending functions for concave domains.•Adding internal control points for shape editing.
Display omitted
A new multi-sided, control point based surface representation is introduced, based on the Generalized Bézier patch 1. While the original surface is based on convex polygonal domains and a specific, uniform arrangement of control points, the new construction permits domains with concave angles and supports a more general control point structure, where independent “half-Bézier” interpolants, or ribbons, are blended together. The ribbons may have arbitrary degrees along the boundary and also in the cross-derivative direction; the related control points ensure tangent- or curvature-continuous connection to adjacent quadrilateral Bézier patches and permit shape editing and optimization, when needed.
The surface comprises four components: (i) a concave domain generated from a 3D loop of boundary edges, (ii) half-Bézier interpolants, (iii) parameterizations that cover the full domain for each interpolant, and (iv) blending functions that guarantee both Bézier-like behavior along the boundaries and a smooth, C∞-continuous composition in the interior of the patch. Editing concave Bézier patches using additional control points is also discussed. A few interesting test examples illustrate the benefits of the method.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK, ZRSKP
PDF
