Non-Uniform Subdivisions And Corner-Cutting Subdivisions

As an important representation method of geometric models,subdivision is widely used in computer animation,industrial modeling design,iso-geometric analysis and other fields.Non-uniform subdivisions can edit flexibly and conveniently geometric models by changing their knot intervals so that they have an indispensable position in practical applications.And the geometrical visualization of corner-cutting subdivision makes it an important subdivision scheme.The surface quality of the geometric models is very important for the application fields related to computer aided design.However,it is generally very complicated and difficult to design a non-uniform subdivision scheme that meets the specified require-ments due to the arbitrariness of knot intervals.Current non-uniform subdivision sur-faces can be at most G1-continuous at the extraordinary points,and their fairness qual-ities are always insufficient to some certain extent.So the result surfaces of existing non-uniform schemes are often unremarkable at extraordinary points.Due to the complexity of constructing non-uniform subdivision schemes,it's more diffi cult to predict and control the outcome performance when the non-uniform schemes are applied to the generation of sharp features.In this thesis,a noval non-uniform scheme for generating sharp features is proposed by extending the eigen-polyhedron technology.The new scheme allow arbitrary designation of sharp edges on the con-trol meshes and generate non-uniform cubic B-spline curves to represent these complex sharp features.The generating surfaces are G1 non-uniform Catmull-Clark surfaces outside the sharp features.Eigen-polyhedron technology is effective in improving the quality of non-uniform subdivision surfaces,one of current non-uniform subdivision schemes with best sur-face quality is obtained by eigen-polyhedron technology.This thesis continues to use the technology to further improve the continuity and fairness of non-uniform subdi-vision surfaces.Firstly,this thesis further improves the quality of subdivision sur-faces(expecially the surface fairness)by modifing the angle of eigen polyhedron and proposes also a systematic method for designing the angle.The proposed method has a certain geometric intuitiveness,and its actual effectiveness has also been confirmed by numerical results.Secondly,In order to fill the gap of the second-order or higher-order continuity of non-uniform subdivision surfaces,this thesis also generalize the concept of eigen polyhedron to the three-dimensional eigen paraboloid,and further builds a noval algorithm for mainly designing non-uniform subdivision surfaces with bounded curvature.Then based on the new algorithm,a preliminary solver for obtaining specific non-uniform surfaces is given to illustrate the effectiveness of the framework.As a corner-cutting subdivision scheme,Lane-Riesenfeld algorithm of uniform B-spline curves possesses a concise and unified form,vertex splitting plus several arith-metic averages.This thesis changes its second arithmetic average into weighted average by introducing a parameter,which control the size of cutting corners.The same strategy is also generalized to general-degree surfaces of arbitrary topology so the surface shape can be adjusted flexibly by altering the parameter value.In a word,this thesis aims to improve the quality of non-uniform subdivision sur-faces at extraordinary points and allow non-uniform subdivisions to generate complex sharp features on surface.In addition,the weighted corner-cutting subdivision scheme of general degrees proposed in this thesis allows uniform subdivision to adjust the shape of the surface more flexibly.