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

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.

T-Junction

By adding a special "T-Point" on one edge of a polygon and extruding, a "T-Junction" is formed.

The internal influence region of a T-Junction is limited to the touching quadrant of the T-Point inside the polygon. The two control wires on the facing direction of the T-Point is subdivided and aligned to be interpolated in patches.

Currently unclear about the situation when T-point lands on a polygon where the facing edge is the border of manifold. From preliminary analysis it seems to be directly compatible.

T-Junction can be used to achieve local refinement of geometry features and restricting region of influence, and avoid breaking existing features of interpolated surface around the connecting regions.

Layered topology representation

Allows partial reconstruction of topology on top of existing control mesh and granular refinement of geometry features.

This works by plannarly or cylindrically projecting portions of interpolated surface patches into projected space, and connect new topologies on this projected structure. Patches generated from this structure in projected space is treated as following the original patch pre-projection spatially. Afterwards, the geometries spatially connected to this structure will generate interpolated patches based on the control of positions from the original patch but weights specified in the form of this newly connected topology.

This method achieves:

• Cutting curved surface without breaking surface features (trimming).
• Connecting new structures from the cutting positions and achieve smooth transition.
• Granular refinement of geometry features.
• upper level refined geometry features will follow adjustments of base surfaces.

Projection regions from one layer of geometry can/should be stitched together. Additional tangent control handles would be generated on connected points on the original geometry situated along stitching seams to provide directional information for interpolation to reduce the discontinuity caused by segmented projection.

In theory the top layer is not strictly required to connect all points from original geometry, although not doing this will lead to watertight issues that requires further treatments or additional rules to be applied.