In mathematics, the variation diminishing property of certain mathematical objects involves diminishing the number of changes in sign (positive to negative or vice versa).

Variation diminishing property for Bézier curves

The variation diminishing property of

Bézier curve A Bézier curve ( ) is a parametric curve used in computer graphics and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approximate a real-world shape t ...

s is that they are smoother than the polygon formed by their control points. If a line is drawn through the curve, the number of intersections with the curve will be less than or equal to the number of intersections with the control polygon. In other words, for a Bézier curve ''B'' defined by the control polygon P, the curve will have no more intersection with any plane as that plane has with P. This may be generalised into higher dimensions. This property was first studied by

Isaac Jacob Schoenberg Isaac Jacob Schoenberg (April 21, 1903 – February 21, 1990) was a Romanian-American mathematician, known for his invention of splines. Life and career Schoenberg was born in Galați. He studied at the University of Iași, receiving his ...

in his 1930 paper, . He went on to derive it by a transformation of

Descartes' rule of signs In mathematics, Descartes' rule of signs, first described by René Descartes in his work ''La Géométrie'', is a technique for getting information on the number of positive real roots of a polynomial. It asserts that the number of positive roots i ...

Proof

The proof uses the process of repeated degree elevation of

. The process of degree elevation for

s can be considered an instance of piecewise

linear interpolation In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points. Linear interpolation between two known points If the two known poin ...

. Piecewise linear interpolation can be shown to be variation diminishing. Thus, if R₁, R₂, R₃ and so on denote the set of polygons obtained by the degree elevation of the initial control polygon R, then it can be shown that *

\mathbf = \mathbf

* Each R_r has fewer intersections with a given plane than R_r-1 (since degree elevation is a form of linear interpolation which can be shown to follow the variation diminishing property) Using the above points, we say that since the Bézier curve ''B'' is the limit of these polygons as ''r'' goes to

\infty

, it will have fewer intersections with a given plane than R_i for all ''i'', and in particular fewer intersections that the original control polygon R. This is the statement of the variation diminishing property.

Totally positive matrices

The variation diminishing property of totally positive matrices is a consequence of their decomposition into products of Jacobi matrices. The existence of the decomposition follows from the

Gauss–Jordan triangulation In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...

algorithm. It follows that we need only prove the VD property for a Jacobi matrix. The blocks of Dirichlet-to-Neumann maps of

planar graph In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross ...

s have the variation diminishing property.

References

{{reflist Curves Interpolation Splines (mathematics) Matrices