The theory of univariate splines began its rapid development in the early sixties, resulting in several thousand research papers and a number of books. This development was largely over by 1980, and the bulk of what is known today was treated in the classic monographs of deBoor [Boo78] and Schumaker [Sch81]. Univariate splines have become an essential tool in a wide variety of application areas, and are by now a standard topic in numerical analysis books.

If 1960–1980 was the age of univariate splines, then the next twenty years can be regarded as the age of multivariate splines. Prior to 1980 there were some results for tensor-product splines, and engineers were using piecewise polynomials in two and three variables in the finite element method, but multivariate splines had attracted relatively little attention. Now we have an estimated 1500 papers on the subject.

The purpose of this book is to provide a comprehensive treatment of the theory of bivariate and trivariate polynomial splines defined on triangulations and tetrahedral partitions. We have been working on this book for more than ten years, and initially planned to include details on some of the most important applications, including for example CAGD, data fitting, surface compression, and numerical solution of partitial differential equations. But to keep the size of the book manageable, we have reluctantly decided to leave applications for another monograph.

For us, a multivariate spline is a function which is made up of pieces of polynomials defined on some partition Δ of a set Ω, and joined together to ensure some degree of global smoothness.