# 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