Direct manipulation of free-form deformation using curve-pairs

Curve-pair based deformation is an extension of axial deformation for providing more intuitive and convenient control of object deformation. In this paper, we investigate the problem of direct manipulation of curve-pair based deformation with geometric constraints. With different objective functions, two kinds of constrained optimization problems are derived from the direct manipulation problem, the first one is based on the minimization of the changes in the control polygon of the curve pair, and the second one is based on the minimization of the length change of the object to be deformed. The corresponding nonlinear constrained optimization problems are solved by using the Uzawa method. In order to preserve geometric detail in a deformation, we propose a detail-preserving direct manipulation approach by using Laplacian coordinates. Users are only required to specify some constraint points to control the deformation. Experimental results and comparisons with other approaches are presented to demonstrate the effectiveness and stability of the proposed methods.