Visual Comput (2007) 23: 119–131 DOI 10.1007/s00371-006-0033-3 ORIGINAL ARTICLE Qiang Liu Edmond C. Prakash Mandayam A. Srinivasan Interactive deformable geometry maps Efficient modeling for interactive deformation of non-rigid 3D objects Published online: 11 July 2006  Springer-Verlag 2006 Q. Liu (✉) · E.C. Prakash School of Computer Engineering, Nanyang Technological University, Singapore 639798 {pg04825433, asprakash}@ntu.edu.sg M.A. Srinivasan MIT Touch Lab, Room 36-791, 77 Massachusetts Avenue, Cambridge MA 02139, USA srini@mit.edu Abstract Haptics on 3D deformable models is a challenge because of the inevitable and expensive 3D deformation computation. In this paper, we propose a new technique that extends the conventional rigid geometry images approach proposed by Gu et al. [9]. Our approach not only flattens the geometry, but also helps to accomplish deformation in an effective and efficient manner. Our approach is suitable for hap- tics computing, as it performs the deformation on the geometry map itself thereby avoiding the expensive 3D deformation computation. We demonstrate construction of the deformable geometry map represen- tation and its application utilizing practical methods for interactive surgery simulation and interactive textile simulation. Keywords Deformable geom- etry · Surgery simulation · Haptics computing 1 Introduction Traditional model representations, such as complex trian- gle meshes or NURBS surfaces, pose a variety of prob- lems when it comes to interactive deformation. Some of the problems encountered are: determining the contact point, application of textures, difficulty in deformation, non-real-time response for interaction. – Mesh primitives: Traditionally the input data for mod- eling 3D deformable objects have been 3D meshes and primitives which are represented as a collection of triangles. This representation seriously limits the pick- ing of one or more triangles out of a large mesh. It is also extremely difficult to determine the neighbor- hood points or triangles associated with a given contact point. – Contact point: Determining the contact point in 3D space is tedious. This requires ray intersection with the object of interest to find the closest intersection point. – Deformation footprint: Even though the footprint ker- nel of shape deformation in picking or pinching is quite uniform for most objects, representing the orientation of the shape of the kernel in 3D is difficult. – Interactive deformation: In order to visualize the de- formation, we need to have immediate feedback to facilitate responsive user interaction with the 3D de- formable object. In addressing these problems, this paper makes two major contributions. First, it introduces the deformable geometry map (DGM), a multi-layer parameterized repre- sentation of 3D meshes (for example, the representation of human organs for the purpose of virtual surgery simu- lation). An unstructured 3D input mesh is parameterized and resampled into a regular 2D parameterized model. The