Interactive Shape Editing for Subdivision Surfaces

Abstract

This chapter firstly provides two free-form deformation algorithms of subdivision surfaces by combining mesh deformation with shape editing under simple geometrical constraints or potential function constraints.

Appendices

Remarks

Geometrical modeling is usually a repeated process in which interactive shape modification is inevitable. Therefore, shape editing is an important research field in surface modeling. T Shape editing of free-form surfaces is an important topic in surface modeling. NURBS surface is a classical free-form surface, and there are many publications on shape editing of NURBS surfaces [233–236]. Methods in these literatures modify shapes of NURBS surfaces by using knot vectors, control vertices, and weights of NURBS surfaces. Wang [235] present a method to modify shapes of B-spline surfaces by using geometric constraints, such as points, normals, curves, and faces. In shape modification algorithms presented in this chapter, we also use these geometric constraints to modify shapes of subdivision surfaces. Since multiresolution is an important property of subdivision surfaces, many shape modification methods [199, 238] use the property. So do methods presented in this chapter. Surface shape modification is also the surface deformation. The FFD method is an extensively applied method. It can be classified into two types: the medium mapping deformation [239–247] and the direct constraint deformation [248–252]. The former insert the model to be deformed into a parameter volume space (e.g., a Bézier volume with 3 parameters). By adjusting the shape of the volume (i.e., medium), the shape of the model is adjust. The latter deforms surfaces by solving linear systems or optimization models that are constructed based on given geometric constraints. These methods given in this chapter belong to the direct constraint deformation.

Exercises

• (1) What is the DRC deformation? For a point in the deformation region of a surface, how does the DRC deformation define its new position?

• (2) Give the basic process of the deformation under the potential function. What is the expression used in this chapter?

• (3) For the Catmull–Clark subdivision surface, see Fig. 7.7. Assume that there is a quadrilateral in the initial mesh. Draw the limit patch of the quadrilateral.