Polynomial spline interpolation of incompatible boundary conditions with a single degenerate surface

• Propose a method to interpolate a four-sided region with incompatible boundary.
• Achieve G1G1 continuity with the boundary except for incompatible corner points.
• Utilize the property of multi-valued normal vectors at degenerate points.
• The proposed method is constructive and straightforward.

Coons’ construction generates a surface patch that interpolates four groups of specified boundary curves and the corresponding cross-boundary derivative curves. This constructive method is simple and widely used in computer aided design. However, at the corner points, it requires compatibility of the boundary conditions, which is usually difficult to satisfy in practice. In order to handle the incompatible case where the normal directions respectively indicated by two adjacent boundaries do not agree with each other at the common corner point, we utilize the property of degenerate parametric surfaces that the normal directions can converge to multiple values at degenerate points, and therefore the local degenerate geometry can satisfy conflicting conditions simultaneously. Following this idea, we use a single patch of (2(p+2),2(p+2))(2(p+2),2(p+2))-degree polynomial spline surface with four degenerate corners to interpolate incompatible boundary conditions, which are represented by pp-degree polynomial spline curves with G1G1 continuity. This method is based on symbolic operations and polynomial reparameterizations for polynomial splines, and without introducing any theoretical errors, it achieves G1G1 continuity on the boundary except for the four corner points.