We present a novel but simple physics-based method to interactively manipulate surface shapes of 3D models with \( C^1 \) continuity in real time. A fourth-order partial differential equation involving a sculpting force originating from elastic bending of thin plates is proposed to define physics-based deformations and achieve \( C^1 \) continuity at the boundary of deformation regions. In order to obtain real-time physics-based surface manipulation, we construct a mapping relationship between a deformation region in a 3D coordinate space and a unit circle on a 2D parametric plane, formulate corresponding \( C^1 \) continuous boundary conditions for the unit circle, and obtain a simple analytical solution to describe the physics-based deformation in the unit circle caused by a sculpting force. After that, the obtained physics-based deformation is mapped back to the 3D coordinate space, and added to the original surface to create a new surface shape with \( C^1 \) continuity at the boundary of the deformation region. We also develop an interactive user interface as a plug-in of the 3D modelling software package Maya to achieve real-time surface manipulation. The effectiveness, easiness, real-time performance, and better realism of our proposed method is demonstrated by testing surface deformations on several 3D models and comparing with other methods and ground-truth deformations.

我们提出了一种新颖但简单的基于物理的方法，以交互方式实时操作具有\( C^1 \)连续性的 3D 模型的表面形状。提出了一个涉及源自薄板弹性弯曲的雕刻力的四阶偏微分方程来定义基于物理的变形并在变形区域的边界实现\( C^1 \)连续性。为了获得实时的基于物理的表面处理，我们构建了 3D 坐标空间中的变形区域与 2D 参数平面上的单位圆之间的映射关系，公式化了对应的\( C^1 \)单位圆的连续边界条件，并获得一个简单的解析解来描述单位圆中由雕刻力引起的基于物理的变形。之后，将获得的基于物理的变形映射回 3D 坐标空间，并添加到原始曲面以在变形区域的边界处创建具有\( C^1 \)连续性的新曲面形状。我们还开发了交互式用户界面作为 3D 建模软件包 Maya 的插件，以实现实时表面处理。通过在几个 3D 模型上测试表面变形并与其他方法和地面真实变形进行比较，证明了我们提出的方法的有效性、简便性、实时性和更好的真实性。