Preliminary research progress report of "Wu-Surface" and its manipulation tool (1)

Main problem

A new kind of interpolated surface/boundary representation structure is needed in Cartesian space that has similar operability as subdivision surface, and it should have following additional features:

• Ability to restrict and adjust the range of influence of control points.
• Allowing constraints on control points to alleviate labour associated with adjustments.
• The topology should support layered detail levels to allow gradual refinements of geometry features.
• Gradient between different layers of geometry features should be smooth and with restricted and controllable border of influence.
• Allow adjustments to any layer of geometry features to affect later layers which have based their shapes upon the layer being adjusted.
• Intersection features be not represented as fit/approximation of intersection lines from discretized geometries.
• Not allowing topologies that are generated procedurally. If necessary, then the result is always editable.

Main Definitions

"Wu-Surface" is a type of free form manifold mesh topology data structure that is researched in current project that 1) allows curved surface interpolation with at least G1/C1 continuity, 2) can process control point constraints at topology level, 3) uses T-junction to represent spatial transition on level of detail of the control mesh and 4) has dependency based layered topology representation.

"T-junction" is a certain type of topology that describes the process of adding control point(s) that are special on one edge of a polygon and connecting them to 2 or multiple rows of additional faces, the subdivision center of the polygon is kept and the smooth interpolation between original polygon and new geometries are achieved.

Geometric workings

Base surface

The basic subdivision surface is pieced together with bicubic bezier patches. The geometric center of each face is the corner control point of actual beizer patches. The base mesh is not the control point of the resulting bezier surface, it works by controlling the control points of the bezier surface, thus gives similar experiences as operating a Catmull-Clark control mesh. Edges of the base mesh can record the extent of push-pull of control points and also curvature of interpolation. By default the control points are pushed half way and the curvature is at circular approximation.

Benefits:

• Allows visual squeezing of the curved surface without introducing more control points.
• If a few more additional arguments are added to represent the separation of adjacent bezier corner points, then any one face in the base mesh can represent a combined flat and curved portion.
• Adjusting the curvature control can degrade the surface elegantly back to biquadratic.