Bézier curves are parametric curves that are prominent in computer graphics. They are an important element in computer aided design and engineering. They are used in the design of turbine blades and other smooth geometrical surfaces. Used as functions they currently provide solutions to problems in a variety of disciplines including inverse design, digital imaging, and boundary value problems.
My reason for using the curves:
A Bézier curve is a continuous, smooth, bounded polynomial that can simulate any continuous function over a reasonable range. Bézier curves named after Pierre Bézier [1, 1962] who is principally responsible for letting the world know that it can be used to create aesthetic curves for automotive shapes. He worked for the Reynolds’s Car Company. Paul de Casteljau, a French physicist and mathematician working at Citroën is credited to have developed the algorithm for creating these curves in 1959. His work was unpublished. The curve currently is formally defined by invoking the Bernstein polynomial. Sergei Natanovich Bernstein was a Russian mathematician, who defined the family of mathematical functions in 1912. Through the application of Bernstein polynomials he was able to prove the Weierstrass approximation theorem.
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform. (Wikipedia)
Classical analytical solutions in math are usually a combination of infinite series comprising a combination of polynomials, exponentials, and trigonometric functions. A Bézier curve is a parametric curve that is completely determined by a set of vertices or control points. In my view what this does is to replace the infinite series by a finite set of control points.
My Bezier math is all about the determination of these control points for all kinds of mathematical problems. Essentially a numerical determination (using optimization) of an analytical expression. The examples are in the drop down links for this page.
There are an incredible amount of references for the use of these curves. I have found the following very useful in my work. My approach is original and unique.
References
1. Bézier P., 1968. How Renault Uses Numerical Control for Car Body Design and Tooling, SAE Paper 680010, Society of Automotive Engineers’ Congress, Detroit, MI.
2. Rida, T. F., 2012. The Bernstein polynomial basis: a centennial retrospective, Dept. Of Mechanical and Aerospace Engineering, University of California, Davis.
URL: http://mae.engr.ucdavis.edu/~farouki/bernstein.pdf
3. Rogers, D. F., and Adams, J.A., 1990. Mathematical Elements for Computer Graphics, Second Edition, McGraw-Hill Book Company.
4. Gordon, W.J., Riesenfeld, R.F., 1974. Bernstein-Bézier Methods for the Computer-Aided Design of Free Form Curves and Surfaces, Journal of the Association of Computing Machinery, Vol. 21, No. 2, pp 293 – 310.
5. Mike Kamermans “Pomax”, 2016. A Primer on Bézier Curves.
URL: https://pomax.github.io/Bézierinfo/
6. MATLAB, A Software for Technical Computing”, Ver. 7.0 and +. The MathWorks, Inc., MA.
7. Venkataraman, P., 2009. Applied Optimization with MATLAB Programming, Second Edition” Wiley.