The Bernstein-Bezier representation was founded by mathematician Sergei Bernstein and perfected and popularized by mathematician Pierre Bezier. It utilizes polynomials to create curves on graphs. The polynomials are used to chart multiple points on a graph, and then connected to form lines and curves to create objects. The polynomials can be used to create circles on a graph instead of curves that only move from left to right. This is useful for adding curves to planes and cars during their design process, for plotting missile trajectory and for use in 3D graphics. A sample formula is: f(t)=a0+a1t+a2t(2)+a3t(3)+a4t(4)+... +adt(d).
Splines are used in the creation and design of jets and airplanes. They use quadratic equations to determine boundaries, shapes and curves. In his book, "Spline Functions," Larry Schumaker explains that basic polynomials are good for small projects but splines are more efficient for larger projects. Splines utilize approximation theory, piecewise polynomials and numerical analysis in their calculations. Piecewise polynomials use low degree polynomials and intervals divided into smaller pieces to calculate the whole. Approximation theory is the basis of determining relationships with simple and complex functions.
Non-Uniform Rational Basis Spline (NURBS) modeling is a mathematical method used to design sports cars, organic shapes and other mechanical items. NURBS curves and surfaces allow mathematicians to align surfaces and curves together as if there is fluid movement from one object to the next. In her book, "Maya 8 Windows and Macintosh," Morgan Robinson writes that "NURBS models usually consist of many separate NURBS surfaces, all working together to form the semblance of a continuous surface." NURBS utilizes multiple control points on a surface to create curves.
When two separate Bernstein-Bezier curve or surface equations are joined together to create continuity it is called a B-Spline curve. Zhejian University professors Hongxin Zhang and Jieqing Feng write that B-Spline curves and surfaces are defined by P(t) = n/{E}(P(i)N(I,k)(t). The surface equation may be represented by B and the curve equation may be represented by "C." The equation that combines the segments that join the curves or surfaces is represented by "G."