Reasons for single span, higher degree surfaces/patches - not what frequently thought

Multiple single span (Bezier) surfaces, typically of higher degree are frequently preferred over multi-span degree 3 surfaces when surface quality is important. I believe single-span, higher degree surfaces can provide some significant benefits but the benefits may not be the ones frequently cited. I’ll discuss how single spans can simplify match between surfaces, and why higher degree surfaces beyond what is actually required for smoothness and flow may be needed. Only matching across untrimmed edges is considered. Non-rational (all weights = 1) surfaces will be assumed in the interest of simplification.

Let’s start with matching surface edges for position continuity (G0). Quality surfacing generally requires edges of adjoining surface to coincide within a small tolerance, and coinciding exactly (within round-off error) is preferred. Single span surface have a key property

The untrimmed edge of a single span surface can have exact C0 and G0 continuity with all or any portion of the untrimmed edge of another single span surface provided the first surface is of the same or higher degree in the edge direction compared to the second surface. Exact G0 matching is possible without adding control points to either surface.

This means that a surface of degree 3 or higher in the edge direction can exactly match the position of any portion the edge of a degree 3 surface. A tolerance is not needed and no refinement or adding control points is needed.

Perhaps it should be noted that when a single span surface is split along an isocurve the resulting surfaces can be shrunk into two single span surfaces.

To be continued …