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The construction of parametric curve and surface plays an important role in computer aided geometric design (CAGD), computer aided design (CAD), and geometric modeling. In this paper, we define a new kind of blending functions associated with a real points set, called generalized toric-Bernstein (GT-Bernstein) basis functions. Then, the generalized toric-Bézier (GT-Bézier) curves and surfaces are constructed based on the GT-Bernstein basis functions, which are the projections of the (irrational) toric varieties in fact and the generalizations of the classical rational Bézier curves/surfaces and toric surface patches. Furthermore, we also study the properties of the presented curves and surfaces, including the limiting properties of weights and knots. Some representative examples verify the properties and results.

The spatial expression and ordering of social processes is one of the primary aims of architecture. Such is the view of Zaha Hadid Architects (ZHA), where Shajay Bhooshan heads the computation and design group (CoDe). Here he explains how the practice has followed in the footsteps of the automotive, aircraft and shipbuilding industries in adopting a hybrid approach to design development. As demonstrated by a mathematics-themed gallery conceived by ZHA for London's Science Museum, it assimilates historical knowledge while facilitating fabrication and allowing for future flexibility. [ABSTRACT FROM AUTHOR]

The Bernstein–Fourier algorithm for the evaluation of polynomial curves is extended for the evaluation of polynomial tensor product surfaces. Under a natural hypothesis, accurate evaluation of Bézier curves and surfaces through several algorithms is discussed. Numerical experiments comparing the accuracy of the corresponding Horner, de Casteljau, VS and Bernstein–Fourier algorithms are presented. [ABSTRACT FROM AUTHOR]