- an ascending sequence of points, or
*knot values*, which are abscissas for the spline function - the B-spline degree,
*N*.

In periodic boundary conditions, these equations may be written in matrix form as

For fixed first or second-derivative boundary condtions, they take the form

The one-dimensional B-spline can be generalized to two or more
dimensions. To do this, we can construct a two-dimensional basis
consisting of the tensor product of one-dimensional basis functions in
each direction. For example, for a 1D cubic B-spline, their are four
non-zero basis functions at each point,
The great advantage of B-splines is that the number of floating point
value which needs to be stored per mesh point does not increase with
dimensionality. In contrast, the more commonly used splines required 2 |
A multi-dimensional interpolating B-spline can be constructed solving
the interpolating equations for each direction in sequence. That is,
for a 2D spline, we first solve the interpolating equations in the
^{x}(x_{i}, y_{j}). We then
solve the interpolating equations in the y direction, using
these F^{x} coefficients as the RHS, yielding the final 2D
B-spline coefficients. |