The purpose of our work was to develop a prototype CAD system that would implement some new ideas that might also find their way into software provided by Boeing's CAD vendors. More typically, splines that occur in practical applications are piecewise polynomials of degree 3, and usually they are constructed so that their first and second derivatives are continuous everywhere.
Splines are a useful form for the representation of curves and surfaces.
Since splines are formed by piecing together polynomials, one can represent, or closely approximate, many entities without the need to use high-degree polynomials. One of the surfaces that presented a challenge was the Coons Patch , a surface that looks like a deformed rectangle, bounded by four arbitrary curves that met at the corners. The above formula, while a bit daunting looking, says that a Coons patch can be constructed by adding together the ruled surface between the Left and Right curves and the ruled surface between the Bottom and Top curves and then subtracting off a surface formed by bilinear blending of the four corner points. Similarly, any rational spline curve and any polynomial or rational spline surface can be written as a linear combination of such basis functions. In the rational case, we can view the B-spline coefficients as 4D points.