View source for Variation diminishing property

[[File:Bezier curves.png|thumb|right|Sample curves (red) with their polygons (grey).]]
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]]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.<ref>{{citation |chapter=Variation-diminishing Property |page=298 |title=Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable |author=Rida T. Farouki |publisher=Springer |year=2007 |isbn=9783540733973}}</ref>

This property was first studied by [[Biography:Isaac Jacob Schoenberg|Isaac Jacob Schoenberg]] in his 1930 paper, {{lang|de|Über variationsvermindernde lineare Transformationen}}.<ref>Schoenberg, I. J.. “Über variationsvermindernde lineare Transformationen.” ''Mathematische Zeitschrift'' 32 (1930): 321-328.</ref>  He went on to derive it by a transformation of [[Descartes' rule of signs]].<ref>{{citation |page=62 |author=T. N. T. Goodman |title=Shape Preserving Representations in Computer-Aided Geometric Design |year=1999 |isbn=9781560726913 |chapter=Shape properties of normalized totally positive bases|publisher=Nova Publishers }}</ref>

===Proof===

The proof uses the process of repeated degree elevation of [[Bézier curve#Repeated degree elevation|Bézier curve]]. The process of degree elevation for [[Bézier curve#Degree elevation|Bézier curve]]s can be considered an instance of piecewise [[Linear interpolation|linear interpolation]]. Piecewise linear interpolation can be shown to be variation diminishing.<ref>{{Cite book
|title=Curves and surfaces for computer-aided geometric design
|first=Gerald
|last=Farin
|publisher=[[Company:Elsevier|Elsevier]] Science & Technology Books
|year=1997
|isbn=978-0-12-249054-5
|edition=4
}}</ref>
Thus, if '''R'''<sub>1</sub>, '''R'''<sub>2</sub>, '''R'''<sub>3</sub> 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
* <math>\mathbf{\lim_{r \to \infty}R_r} = \mathbf{B}</math>
* Each '''R'''<sub>r</sub> has fewer intersections with a given plane than '''R'''<sub>r-1</sub> (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 <math>\infty</math>, it will have fewer intersections with a given plane than '''R'''<sub>i</sub> 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 matrix|totally positive matrices]] is a consequence of their decomposition into products of [[Jacobi operator|Jacobi matrices]].

The existence of the decomposition follows from the Gauss–Jordan triangulation 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|planar graph]]s have the variation diminishing property.

==References==
{{reflist}}

[[Category:Curves]]
[[Category:Interpolation]]
[[Category:Splines (mathematics)]]
[[Category:Matrices]]

{{Sourceattribution|Variation diminishing property}}