Electrical Engineering
      and Computer Sciences

Electrical Engineering and Computer Sciences

COLLEGE OF ENGINEERING

UC Berkeley

Beta Continuity and Its Application to Rational Beta-splines

Ronald N. Goldman and Brian A. Barsky

EECS Department
University of California, Berkeley
Technical Report No. UCB/CSD-88-442
August 1988

http://www.eecs.berkeley.edu/Pubs/TechRpts/1988/CSD-88-442.pdf

This paper provides a rigorous mathematical foundation for geometric continuity of rational Beta-splines of arbitrary order. A function is said to be n^th order Beta-continuous if and only if it satisfies the Beta-constraints for a fixed value of Beta = (Beta1, Beta2, ... Beta n). Sums, differences, products, quotients, and scalar multiples of Beta-continuous scalar-valued functions are shown to also be Beta-continuous scalar-valued functions (for the same value of Beta). Using these results, it is shown that the rational Beta-spline basis functions are Beta-continuous for the same value of Beta as the corresponding integral basis functions. It follows that the rational Beta-spline curve and tensor product surface are geometrically continuous.


BibTeX citation:

@techreport{Goldman:CSD-88-442,
    Author = {Goldman, Ronald N. and Barsky, Brian A.},
    Title = {Beta Continuity and Its Application to Rational Beta-splines},
    Institution = {EECS Department, University of California, Berkeley},
    Year = {1988},
    Month = {Aug},
    URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1988/5280.html},
    Number = {UCB/CSD-88-442},
    Abstract = {This paper provides a rigorous mathematical foundation for geometric continuity of rational Beta-splines of arbitrary order. A function is said to be <i>n^th</i> order Beta-continuous if and only if it satisfies the Beta-constraints for a fixed value of Beta = (Beta1, Beta2, ... Beta<i>n</i>). Sums, differences, products, quotients, and scalar multiples of Beta-continuous scalar-valued functions are shown to also be Beta-continuous scalar-valued functions (for the same value of Beta). Using these results, it is shown that the rational Beta-spline basis functions are Beta-continuous for the same value of Beta as the corresponding integral basis functions. It follows that the rational Beta-spline curve and tensor product surface are geometrically continuous.}
}

EndNote citation:

%0 Report
%A Goldman, Ronald N.
%A Barsky, Brian A.
%T Beta Continuity and Its Application to Rational Beta-splines
%I EECS Department, University of California, Berkeley
%D 1988
%@ UCB/CSD-88-442
%U http://www.eecs.berkeley.edu/Pubs/TechRpts/1988/5280.html
%F Goldman:CSD-88-442