## 8. B-Spline surfaces

We can create a B-Spline surface using a similar method to
the Bézier surface. For B-Spline curves, we used two phantom
knots to clamp the ends of the curve. For a surface, we will have
phantom knots all around the eal knots as shown below for an M+1 by N+1
knot surface.

There are two extra rows and two extra columns of knots in
parametric space surrounding the real knots. Where we place these knots
determines the shape of the surface at the edges. The method described
here gives similar results to the method used for Bézier
surfaces; that is, the edges of the surface form a B-Spline curve of
the edge knots. This means some of the boundary conditions are

for 0 <= m <= M and 0 <= n <= N.
These conditions are essentially the same as the two-dimensional case.
They mean the weighting of a sample taken at the boundary m=0 is
dependent only on knots along the m=0 boundary (the phantom knots at
m=-1 balance out the real knots at m=1). The remaining boundary
conditions make the surface corners and the corner knots coincide. The
co-ordinate of the corner as set by P0,0 (and hence the parametric knot
at {-1,-1}) is

This gives us a surface that interpolates the corner knots and forms
B-Spline curves down each side. To explore the B-Spline surface
further, see the surface created from a 4x4 or a 5x5 mesh.

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