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Volumetric T-Spline Modeling
Published in Yongjie Jessica Zhang, Geometric Modeling and Mesh Generation from Scanned Images, 2018
To represent free-form surfaces precisely for design and manufacturing, NURBS was developed starting from the 1950s [133, 297]. Currently the NURBS surfaces are widely used in aircraft, shipbuilding and automobile industry. For example, in the design of aircraft wings, ship hulls, propellers, turbines and car bodies. Based on the Bernstein polynomials, a French engineer named Pierre Bézier invented the Bézier method [100, 297]. Roughly at the same time Paul de Casteljau proposed a numerical method to evaluate Bézier curves [133], which was named the “de Casteljau algorithm.” This is really a milestone for surface description, translating the representation of mechanical parts from blueprints to computers. To overcome the shortcomings of Bézier, Carl R. de Boor replaced a single segment with piecewise polynomials and generalized the Bézier method to B-spline [133]. In addition, several knot insertion and degree elevation schemes were developed for B-splines, including Boehms knot insertion algorithm [55], the Oslo algorithm [259], the degree elevation algorithm [297] and the blossoming algorithm [308]. As a generalization of B-spline, NURBS was invented in the 1960s to unify the geometry representation of standard analytical shapes such as conic curves and free-form shapes. Currently NURBS has become the standard form for CAD, CAM and CAE.
Multi-Sided Patches via Barycentric Coordinates
Published in Kai Hormann, N. Sukumar, Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics, 2017
While we can evaluate Bézier curves by evaluating the polynomial expressions in the Bernstein basis functions, a more elegant solution exists via de Casteljau’s algorithm. To do so, we introduce a graphical notation for a linear combination of two control points. Figure 8.2 shows a simple pyramid diagram. The arrows denote taking the product of the value at the base of the arrow with the scalar value listed along the arrow. The result of this product is then added to the sum at the end of the arrow. Hence, this figure denotes taking f0 $ f_0 $ , f1 $ f_1 $ and multiplying these values by a, b, respectively, to form the result af0+bf1 $ a f_0 + b f_1 $ . Using this notation, de Casteljau’s algorithm can be written in a very elegant fashion as a pyramid diagram shown in Figure 8.3 [167]. Note that each level of the pyramid produces lower order Bézier functions with the value of the curve appearing at the apex of the pyramid.
Kinematical Lie Algebras and Invariant Functions of Algebras
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2019
J. M. Escobar, J. Núñez, P. Pérez-Fernández
Other example of the application of Lie algebras and groups in Kinematics is found later, in reference [4], where Park and Ravani generalize the concept of Bézier curves to curved spaces, and illustrate this generalization with an application in Kinematics. Indeed, they show how De Casteljau’s algorithm for constructing Bézier curves can be extended in a natural way to a special class of Riemannian manifold, the Lie groups, since that these groups, due to their group structure, admit an elegant and efficient recursive algorithm for constructing those curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Bézier sense) using the recursive algorithm which they construct and apply to the kinematic problem of trajectory generation or motion interpolation for a moving rigid body. The orientation trajectory of motions generated in this way have the important property of being invariant with respect to choices of inertial and body-fixed reference frames.
Controllable three-dimension auxetic structure design strategies based on triply periodic minimal surfaces and the application in hip implant
Published in Virtual and Physical Prototyping, 2023
Bo Liu, Jiawei Feng, Zhiwei Lin, Yong He, Jianzhong Fu
In the Bezier curve fitting process, after the control points are determined, the discrete points on the curve can be calculated with De Casteljau’s algorithm. Then, the periodic curves can be formed through the array of Bezier fitting curves, as shown in Figure 3(c). The expression of the dual-period deformation function based on the Bezier fitting period curve is. where is the Bezier fitting period curve similar to the cosine function and is the Bezier fitting period curve similar to the sine function.