CAGD Handbook

Abstract

References (p. 23) 22. Index (p. 25) This chapter promotes, details and exploits the fact that (univariate) splines, i.e., smooth piecewise polynomial functions, are weighted sums of B-splines. 1. Piecewise polynomials A piecewise polynomial of order k with break sequence # # # # # (necessarily strictly increasing) is, by definition, any function f that, on each of the half-open intervals [# j . . # j+1 ), agrees with some polynomial of degree < k. The term `order' used here is not standard but handy. Note that this definition makes a piecewise polynomial function right-continuous, meaning that, for any x, f(x) = f(x+) := lim h#0 f(x+h). This choice is arbitrary, but has become standard. Keep in mind that, at its break # j , the piecewise polynomial function f has, in e#ect, two values, namely its limit from the left, f(# j -), and its limit from the right, f(# j +) = f(# j ). The set of all piecewise polynomial functions of order k with break sequence # # # # # is denote