Summary: THE GENERIC DIMENSION OF THE SPACE OF C 1 SPLINES
OF DEGREE d – 8 ON TETRAHEDRAL DECOMPOSITIONS
PETER ALFELDy, LARRY L. SCHUMAKER z AND WALTER WHITELEYx
Abstract. We consider the linear space of globally differentiable piecewise poly
nomial functions defined on a threedimensional polyhedral domain which has been
partitioned into tetrahedra. Combining BernsteinB'ezier methods and combinato
rial and geometric techniques from rigidity theory, we give an explicit expression
for the generic dimension of this space for sufficiently large polynomial degrees
(d – 8). This is the first general dimension statement of its kind.
Key words. multivariate splines, piecewise polynomial functions, triangulations,
projection, edge contracting, generic dimensions.
AMS(MOS) subject classifications. 65D07, 41A63, 41A15
x1. Introduction. Multivariate spline spaces are linear spaces of smooth piece
wise polynomial functions defined on a suitable partition of an underlying domain
\Omega ae IR Ÿ . They are important in a variety of areas including finite elements, data
fitting, and computeraided geometric design. In the past few years, consider
able effort has gone into the development of a constructive theory of multivariate
splines analogous to the univariate theory. Such a theory would ideally include
results on dimension, locally supported basis elements, approximation power, etc.
This paper is devoted to the dimension problem, which has a rich history going