Bernstein Polynomials

Properties and Applications to Bezier Curves, B-Splines and Solution of Boundary Value Problems


  • Anandaram Mandyam Professor of Physics (Retired), Bangalore University, Bangalore, Karnataka, India



Bernstein polynomials (B-polys), B-Splines, Bezier Curves, BVP solution


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.


J. S. Racine,”A Primer on Regression Splines”,(Chapter article), 2019. Pdf (downloadable from web/ packages/ crs/vignettes/spline_primer.pdf )'s_algorithm

M. I. Bhatti and P. Bracken,” Solutions of differential equations in a Bernstein polynomial basis”, Journal of Computational and Applied Mathematics, Vol. 205(1), pp.272-280, 2007.

J. Magoon, "Application of the b-spline collocation method to a geometrically non-linear beam problem" , MS Thesis, 2010, RIT, 2010. (Access from: )

R. Jhaveri, "Design of passive suspension system with non-linear springs using b-spline collocation method", MS Thesis, RIT, 2011. (Accessfrom:

H.Bachau etal., “Applications of B-splines in atomic and molecular physics”, Rep. Prog. Phys. 64, 1815–1942, 2001.

Johnson, “Lectures in Atomic Physics”, University of Notre Dame, USA, 2006. (access this book from) https:// ~johnson/Publications/book.pdf