Research On Blending Free-Form Surfaces With Constraints Of Geometric Continuities

Manufacturing is a pillar industry of the national economy. Geometric modeling isthe source of it and occupies its core position. The continuity and the precision directlyimpact on the mechanical performance and the aesthetic quality of the modern indus-trial products, and therefore, they have become the hot spots of the competition of theindustrial modeling systems. The geometric continuity overcomes the drawback that thetraditional parametric continuity depends on the parameterization and the mathematicalrepresentation. It is also the bottleneck of increasing the modeling quality because ofits theoretical difculty and the complicated representations. This thesis focuses on itsurgent and essential problems in industrial applications, such as the compatibility prob-lems and the continuity conditions on vertices. It presents the following items of progresson the continuity theory and the blending methods of free-form surfaces constrained bygeometric continuities.1N-sided hole filling is widely used in geometric modeling such as vertex blend-ing and complicated filleting. The relevant problem of multi-sided surfaces is also anunsolved academic issue discussed for a long time. A method using the crescent-shapedextended surfaces and a method using triangular Coons surfaces are respectively proposedto solve the tangential and the twist compatibility problems in the corners of the bound-ary conditions, which restrain the elevation of the continuity order. These two methodsutilize the singular properties of the degenerated points to handle the G~1compatibilityproblems, and they achieve G~1continuity.2This thesis corrects a conceptual mistake in previous papers on G~1Coons surfaceconstruction, which is a commonly-supported method in modeling systems. It proposesa solution of the G~2compatibility problems on the corner points, and the continuity orderof the created surface can be then elevated from G~1to (ε|â†')-G~2.3The thesis provides the criterion, the construction method and the adjustmentalgorithms of G~2continuity on vertices. It also presents the proof of the conjecture onthe condition number of G~2continuity on vertices. Based on that theory, it also proposesan interpolation method of meshes, which is often used in shipbuilding. Compared withprevious methods, not only does it adapt to arbitrary closed quad-dominant manifolds, italso elevates the continuity order up to G~2. 4Three methods on surface bridging, filling and blending are respectively proposedto achieve G~ncontinuity. First, the regular curved-knot B-spline surface generalizes theoriginal constant knot-vectors of B-spline surfaces to sequences of Hermite polynomialinterpolants. It is well-suited for bridging incompatible opposite boundaries and repre-senting the continuity transition from a sharp point to rounded features. Second, the peri-odic B-spline surface is used to form a cap-shaped filling surface. It can be converted toa standard B-spline surface without introducing errors. Third, the polar blending surfaceblends N parametric surfaces simultaneously. That method reparameterizes the ordinaryrectangular domain into a disc-shaped polar domain and then sums up all phase-shiftedpolar parametric surfaces by a G~nblending function. It is G~ncontinuous inside and onthe boundary. All of these methods benefit high-performance implementation since noiteration nor large-scale equation-solving are required.