Freeform Surfaces:

Here we illustrate how the same four techniques we just studied for shape approximating curves can be applied to shape approximating tensor product surfaces.

1. Piecewise construction: Creating an extended surface by repeatedly grafting new surfaces onto an edge of another. The example here illustrates construction of a piecewise Bezier surface.

   The 4x3 Bezier surface from the previous page	The new (red) degree 4x3 surface was grafted onto the previous with C1 continuity

2. Subdivision: Creating two independent surfaces by splitting a given surface into two independent pieces.

   We start again with our original degree 4x3 surface	Now we subdivide in the s parametric direction; the red and cyan pieces, taken together, exactly reproduce the original surface
   Now we subdivide the red piece in the t parametric direction; the red piece and the cyan piece without grid lines exactly reproduce the red piece in the previous step	Finally we subdivide the cyan piece with grid lines in the t parametric direction; the resulting four patches (two with grid lines and two without) exactly reproduce the original degree 4x3 surface; as we have seen with curves, any of the pieces can now be independently adjusted; C1 is preserved as long as we don't adjust points along or adjacent to the seams

3. Degree raising: Re-represent a surface of degree (n x m) as either a surface of degree ((n+1) x m), or a surface of degree (n x (m+1)).

   Back to our original degree 4x3 Bezier surface	The degree 4x4 surface that results from a degree raise
   Another degree raise in the other parametric direction gives us an equivalent degree 5x4 surface	Finally we make some control point adjustments to the degree 5x4 surface