S.E Weinstein

Degree reduction of Bézier curves by uniform approximation with endpoint interpolation

The approximation of a given Bezier curve Pn of degree n by another of degree m < n is called degree reduction. The paper presents two algorithms, which, for 1-degree reduction, produce an approximation which, componentwise, is the best uniform approximation to Pn from the set of all polynomials of degree less than n which interpolate Pn at the endpoints. Algorithms are also presented for multiple degree reductions. In each case, implementation of the algorithm entails multiplying the matrix of control points by a reduction matrix that depends only on the degrees n and m, and not on the coefficients of Pn. Error bounds and the results of computational experiments are presented.

On extremal sets and strong unicity constants for certain C∞ functions

Abstract For each f continuous on the interval I , let B n ( f ) denote the best uniform polynomial approximation of degree less than or equal to n . Let M n ( f ) denote the corresponding strong unicity constant. For a certain class of nonrational functions F , it is shown that there exist positive constants α and β and a natural number N such that αn ⩽ M n ( f ) ⩽ βn for n ⩾ N . The results of the present paper also provide concise estimates to the location of the extreme points of f − B n ( f ). The set F includes the functions f α ( x ) = e αx , α ≠ 0.

Orders of strong unicity constants

Abstract Given f ϵ C ( I ), the growth of the strong unicity constant M n ( f ) for changing dimension is considered. Under appropriate hypotheses it is shown that 2 n + 1 ⩽ M n ( f ) ⩽ βn 2 . Furthermore, relationships between certain Lebesgue constants and M n ( f ) are established.

Best piecewise monotone uniform approximation

Abstract The problem considered is that of finding a best uniform approximation to a real function f ∈ C [ a , b ] from the class of piecewise monotone functions. The existence, characterization, and nonuniqueness of best approximations are established.

Distances between oriented curves in geometric modeling

We consider the choice of a functional to measure the distance between two parametric curves. We identify properties of such a distance functional that are important for geometric design. Several popular definitions of distance are examined, and new functionals are presented which satisfy the desired properties.

Best L p approximation with multiple constraints for 1≤p≤∞

The problem considered in this paper is best Lp approximation with multiple constraints for 1 ⩽ p < ∞. Characterizations of best Lp approximations from multiple n-convex splines and functions are established and the relationship between them is investigated. Applications to best monotone convex approximation are studied.