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.

Application of Rational B-Splines to the Synthesis of Cam-Follower Motion Programs

1993 ◽  
Vol 115 (3) ◽  
pp. 621-626 ◽  
Author(s):  
D. M. Tsay ◽  
C. O. Huey
Keyword(s):  
Basis Functions ◽  
Spline Functions ◽  
B Spline ◽  
B Splines ◽  
Motion Constraints ◽  
Cam Follower ◽  
Spline Basis

A procedure employing rational B-spline functions for the synthesis of cam-follower motion programs is presented. It differs from earlier techniques that employ spline functions by using rational B-spline basis functions to interpolate motion constraints. These rational B-splines permit greater flexibility in refining motion programs. Examples are provided to illustrate application of the approach.

scholarly journals On the Use of B-Splines as Ritz Variational Basis Functions to Solve the Schrodinger Equation (TISE) for a constrained free Quantum Particle

2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Anandaram Mandyam N
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Quantum Particle ◽  
Bernstein Polynomials ◽  
Energy Levels ◽  
Basis Functions ◽  
Spline Collocation ◽  
Stationary Schrödinger Equation ◽  
B Splines ◽  
Domain Length

B-Splines as piecewise adaptation of Bernstein polynomials (aka, B-polys) are widely used as Ritz variational basis functions in solving many problems in the fields of quantum mechanics and atomic physics. In this paper they are used to solve the 1-D stationary Schrodinger equation (TISE) for a free quantum particle subject to a fixed domain length by using the Python software SPLIPY with different sets of computation parameters. In every case it was found that over 60 percent of energy levels had excellent accuracy thereby proving that the use of B-spline collocation is a preferred method.

On Generalized Jacobi–Bernstein Basis Transformation: Application of Multidegree Reduction of Bézier Curves and Surfaces

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
E. H. Doha ◽  
A. H. Bhrawy ◽  
M. A. Saker
Keyword(s):  
Jacobi Polynomials ◽  
Basis Functions ◽  
Least Square ◽  
Natural Basis ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Computer Aided Geometric Design ◽  
Curves And Surfaces ◽  
Endpoint Constraints

This paper formulates a new explicit expression for the generalized Jacobi polynomials (GJPs) in terms of Bernstein basis. We also establish and prove the basis transformation between the GJPs basis and Bernstein basis and vice versa. This transformation embeds the perfect least-square performance of the GJPs with the geometrical insight of the Bernstein form. Moreover, the GJPs with indexes corresponding to the number of endpoint constraints are the natural basis functions for least-square approximation of Bézier curves and surfaces. Application to multidegree reduction (MDR) of Bézier curves and surfaces in computer aided geometric design (CAGD) is given.

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