Energy Optimization Method and Subdivision Surfaces

Abstract

An important research content of surface modeling is to construct fairing curves and surfaces, and the optimization modeling is an important method to construct fairing curves and surfaces, so which has been always being focused by many researchers.

Appendices

Remarks

This chapter constructs blending patches and interpolatory surfaces by using physical PDE. In view of the relationship between the optimization model and the control mesh, the former can be divided into the global optimization models and the local optimization models. In the global optimization model, all vertices of the considered control mesh are variables of the optimization model; while in the local optimization model, some vertices of the meshes are the variables of the optimization model. It is needed to solve multiple local optimization model in order to determine vertices of the control mesh; in the local optimization model, only some vertices of a considered control mesh are variables of the optimization model. We only have to solve many optimization models in order to determine vertices of the control mesh. Generally, the surfaces constructed by using the global optimization method have better fairness than that of the surfaces constructed by using the local optimization method. However, the local optimization method has higher computation speed than that of the global optimization method. This chapter uses the global optimization method. Though we only use the optimization models to construct surfaces, the optimization method is still the effective method to fair surfaces. In the last several chapters in this book, the optimization method is used for constructing surfaces. For example, the fairness of the subdivision surfaces interpolated the curve nets is improved by adopting the geometric PDE and the fairing deformation surfaces are constructed by using the physical PDE.

Exercises

• 1. Let ${\varvec{p}}(u)=\displaystyle \sum ^8_{i=0}P_i{\varvec{N}}_{i, 3}(u),{\varvec{p}}(u)=\displaystyle \sum ^8_{i=0}{\varvec{P}}_iN_i$ is the uniform cubic B-spline curve. ${\varvec{P}}_i(i=0,1,2,6,7,8)$ are given. Compute the vertices ${\varvec{P}}_i(i=3,4,5)$ using the optimization model (6.1).

• 2. Assume that there are three patches on a plane as shown in following figure. They have the same shapes and symmetric positions. These patches are uniform cubic B-spline surfaces and three matrix lattices are their control polygons. Now, we have to construct a Catmull–Clark subdivision surface to blend the three patches. The topology structure is shown in the following figure. Compute the black point coordinates according to the optimization model (6.11).

• 3. Assume that there is a Catmull–Clark subdivision surface interpolating all vertices of a unit cubic. The initial control mesh of the subdivision surface has the same topology as the unit cubic. Compute the initial control vertices of the subdivision surface.