A spline is a piecewise polynomial function that can have a locally simple form while remaining globally flexible and smooth. Splines are useful for modeling arbitrary functions and are used extensively in computer graphics.
Cubic splines are implemented in the Wolfram Language as BSplineCurve[pts, SplineDegree -> 3] (red), Bézier curves as BezierCurve[pts] (blue), and B-splines as BSplineCurve[pts].
For a triangulation 
of a polygonal disk, let 
be the vector
space of functions with continuous partial
derivatives through order 
whose restriction to each triangle is a polynomial
of total degree at most 
. No boundary conditions are imposed.
For 
and 
,
Wu and Deng (2026) proved that the lower bound of
Schumaker (1979) is exact for every nondegenerate triangulation
of the disk. The dimension
is
 |
where 
and 
are the numbers of interior edges
and interior vertices, respectively, and the outer
sum is over interior vertices 
.
The number 
counts distinct unoriented lines supporting the edges
incident with 
.
Splines at Degree 
." 5 Nov 2026.