Research On Deformation Technology Of Curves And Surfaces Based On Subdivision

Geometric modeling is the kernel of CAD. Deformation technology is an important component of geometric modeling. Subdivision modeling has become the important content in the domain of geometric modeling because of its good-sized advantages such as the arbitrary topological adaptability, the continuity of spline curves and surfaces, and so on. Deformation based on subdivision also becomes an important direction of deformation research. The main research contents and creative achievements are as follows:A subdivision-based curve deformation method under basic functions and an improved arc-length preserving curve deformation algorithm based on subdivision are presented. The former combines the free-form deformation under basic functions with the interpolatory subdivision curve"skinning", and resolves the fast solution to deformation in the case of multiple intersectant curves with multiple constrained points. The latter applies a curve simplification method which can add the arc-length and an approximating subdivision scheme with an adjustable parameter to the arc-length preserving curve deformation process, and selects directly the adjustable parameter according to the constant global arc-length. Compared with the existing arc-length preserving curve deformation algorithm, the improved algorithm can achieve better fairness, especially when a curve is simplified highly.A subdivision surface-based Poisson mesh editing approach is proposed, and based on it, a mesh deformation method based on subdivision surface control is further proposed. The former constructs the deformation control surface using the subdivision surface determined by a bounding mesh of the deformable model. When the bounding mesh is modified, the change information of the corresponding subdivision surface is transformed into the alteration of the mesh gradient field. The latter designs a subdivision surface attached to a specified mesh deformation region as the deformation control surface. A gradient field modification is performed for the deformation region mesh according to the subdivision control surfaces before and after editing as well as the reference and target controlling curves designed for need. Both can effectively preserve the geometric details of an object. The former has the advantage of the FFD method using the subdivision surface spanning an intermediate deformation space, and the latter overcomes the shortcoming that traditional parametric spline surfaces as the deformation controlling surfaces are difficult to attach those objects with arbitrary topologies. Two free-form deformation algorithms of subdivision surfaces are put forward. One is named as the free-form deformation algorithm of subdivision surfaces under Deformation Reference Curve (DRC), and the other is named as the free-form deformation algorithm of subdivision surfaces under field functions. The former applies the simple geometric constrained deformation under DRC to the shape editing of subdivision surfaces. According to the subdivision algorithm, the image of the constrained points, constrained curves, constrained surfaces, and the deformation region determined interactively at successive subdivision levels is resolved. For each subdivision mesh vertex within the deformation region, its new position after deformation is obtained according to the corresponding DRC. The latter applies the mesh constrained deformation under field functions to the shape editing of subdivision surfaces. The geodesic-based field value of each vertex after subdivision as the deformation weight of this vertex is updated according to the equal subdivision mesh. The former has the advantage of fast solution to deformation, and the latter outgoes the former in terms of the deformation quality and the deformation stability.Two deformation algorithms of non-uniform Doo-Sabin subdivision surfaces with interpolated curve constraints are presented. One is based on least-square, and the other is based on discrete PDE. Acting on curve interpolation surfaces constructed based on the non-uniform Doo-Sabin subdivision scheme, both algorithms follow the basic idea of interpolated curve driving deformation. The former establishes constraint equations according to symmetric polygonal complexes, and resolves them to minimize the total disturbance of control vertices of symmetric polygonal complexes. It is suitable for local deformation and has the advantage of fast calculation speed. The latter is based on the former, and fit for the non-uniform Doo-Sabin subdivision mesh at a deep level. For each free vertex, the ideal average curvature value as the criterion of adjusting its position is obtained by establishing and resolving the discrete PDE equation. The latter is suitable for global deformation and the resulting deformation surface has better fairness.