Real-time surface manipulation with C 1 continuity through simple and efficient physics-based deformations

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.