2D shape manipulation via topology‐aware rigid grid

This paper presents a new method which allows user to manipulate a two‐dimensional shape in an intuitive and flexible way. The shape is discretized as a regular grid. User places handles on the grid and manipulates the shape by moving the handles to the desired positions. To meet the constraints of the user's manipulation, the grid is then deformed in an as‐rigid‐as‐possible way. However, this straightforward approach tends to produce unnatural deformations when the grid resolution is not high enough to capture the topological structure of the shape. In the proposed method, the regular grid is trimmed and only the cells that are inside the fatty regions of the shape are preserved, namely “interior grid.” When user manipulates the shape, the interior grid and the shape boundary curve are deformed with minimum distortions. To make the deformations of the interior grid and the boundary curve consistent, a junction energy is introduced. In this way, the unnatural deformation effects could be effectively removed and the physically plausible results can be obtained. Meanwhile, the proposed approach provides user an intuitive and simple way to adjust the shape global and local stiffnesses. The deformation is formulated as an energy minimization problem. The energy function is non‐quadratic and could be efficiently solved using an iterative solver with the fast summation technique that exploits the interior grid and boundary curve regularities. In addition, the method could be easily extended to manipulate curves and stick figures. Experimental results demonstrate the capability and flexibility of the new method. Copyright © 2009 John Wiley & Sons, Ltd.