On a generalization of Bernstein polynomials and Bézier curves based on umbral calculus (II): de Casteljau algorithm

2015 ◽  
Vol 39 ◽  
pp. 1-16 ◽  
Author(s):  
Rudolf Winkel
Keyword(s):  
Bernstein Polynomials ◽  
Umbral Calculus ◽  
Bezier Curves ◽  
Bézier Curves ◽  
De Casteljau Algorithm

scholarly journals Bernstein Polynomials

2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Anandaram Mandyam N
Keyword(s):  
Fourth Order ◽  
Bernstein Polynomials ◽  
Basis Functions ◽  
Spline Functions ◽  
Bezier Curves ◽  
Bézier Curves ◽  
B Splines

Bernstein polynomials (aka, B-polys) have excellent properties allowing them to be used as basis functions in many applications of physics. In this paper, a brief tutorial description of their properties is given and then their use in obtaining B-polys, B-splines or Basis spline functions, Bezier curves and ODE solution curves, is computationally demonstrated.  An example is also described showing their application to solving a fourth-order BVP relating to the bending at the free end of a cantilever.

Bernstein polynomials and composite Bézier curves

2006 ◽  
Vol 46 (11) ◽  
pp. 1872-1881 ◽  
Author(s):  
M. I. Grigor’ev ◽  
V. N. Malozemov ◽  
A. N. Sergeev
Keyword(s):  
Bernstein Polynomials ◽  
Bezier Curves ◽  
Bézier Curves

scholarly journals Calculation of Retention Indices of Essential Oils with the aid of the Van den Dool and Kratz equation and Bézier Curves

2021 ◽  
Author(s):  
Rifat BATTALOGLU
Keyword(s):  
Gas Chromatography ◽  
Lower Bound ◽  
Essential Oil ◽  
Applied Mathematics ◽  
Bernstein Polynomials ◽  
Gas Chromatography Mass Spectrometry ◽  
Chemical Compositions ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Open Problems

The aim of this article is to study the relationships and models among the Van den Dool and Kratz equation, the gas chromatography, and the Bézier curves constructed by aid of the Bernstein polynomials. Another aim of this article is to introduce open problems that contribute to real-world problems involving mathematics, chemistry, and plant biology, including the Van den Dool and Kratz equation, the gas chromatography, and Bézier curves. Searching for the solutions of these problems may have qualities that will create the potential that can enter the field of study of many researchers. As a result of these goals, the usability of Bézier curves was investigated while determining the chemical composition of essential oil obtained from P. Aladaghensis Leblebici. By applying the retention index from the Van den Dool and Kratz equation and evaluating chemical compositions of the essential oil are characterized by gas chromatography-mass spectrometry (GC-MS). The Van den Dool and Kratz equation results have the potential to be used not only in the chemical compositions of the oils, but also in applied mathematics and other fields. Moreover, we construct a new special finite sum. A lower bound and inequality are also given for the finite special sum involving the dead time associated with the isocratic step. Some applications and criticisms are given that include this lower bound and inequality for these sums and its effects on the chemical compositions of essential oil and the Van den Dool and Kratz equation.

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