/ V. V. Savchenko / Professor
/ Kenjiro T. Miura / Associate Professor
/ Mark Sh. Levin / Visiting Research
/ Alexander A. Pasko / Assistant Professor
The Shape Modeling Laboratory is concerned with creating new geometric techniques and algorithms. Computer simulation and modeling are essential components of our research.
Various kinds of Gregory-type patches have been developed as free-form surfaces which can be easily connected to each other with G$^1$ or G$^2$ continuity. Although their importance has been recently recognized in the field of CAD/CAGD, many CAD/CAM systems are using NURBS (non-uniform B-spline) surfaces as the representation of free-from surfaces. To share geometric data with these CAD/CAM systems, it is desirable to have a method to convert Gregory-type patches to NURBS surfaces. It is possible to do it exactly, but it requires considerable increases of their degrees. Hence, we have proposed a new method to convert approximately any Gregory-type patches to NURBS surfaces. The method is that first, the patch is subdivided into several subpatches as specified and they are approximated with suitable Bezier-type patches. Then, their control points are combined to generate ones of the NURBS surface.
In the framework of the function representation (F-rep) of geometric models we have included two and three-dimensional objects defined by parametric patches and volumes. A new type of operations has been introduced: a non-linear deformation controlled by arbitrary points linked to the features of the object. We applied the proposed algorithms, for instance, for medical use to reconstruct a human femur and to operations on it. We have implemented and tested a system parallel animation of volumetric objects on the base of the workstations network. The first version of the HyperJazz environment for empirical geometric modeling has been implemented. We considered a long-standing problem, hair modeling, and showed how easy and effectively our functionally based model can be applied there.
Refereed Journal Papers
A new type of free-form surface patch called NURBS boundary C${\rm ^2}$ Gregory patch(NBC${\rm ^2}$G patch) is introduced. An NBC$^{\rm 2}$G patch, whose boundary is defined by four NURBS curves, is an extension of both the C${\rm ^2}$ Gregory patch developed by Miura and Wang, which gives users the capability of designing curvature-continuous(${\rm G^2}$ continuous) surfaces with reasonable flexibilities, and also that of NURBS boundary Gregory patch proposed by Sone et al., which is surrounded by NURBS curves and can be interpolated by specifications of its cross-boundary first derivatives. This new type of surface patch inherits advantages of both the C$^{\rm 2}$ Gregory patch and the NURBS boundary Gregory patch. It is defined so as to connect it with a rational B\'ezier patch and with a rational boundary C$^{\rm 2}$ Gregory patch with G$^{\rm 2}$ continuity when its boundary can be expressed as rational B\'ezier curves. Derivation, properties, and examples of the new type of surface patch are also presented.
A mesh generation algorithm based on a new label-driven subdivision technique is presented. The algorithm generates 2D mesh of quadrilaterals/triangles on a regular quadrilateral network such as the parameter space of a piecewise surface. Each face (patch) of the quadrilateral network has to be assigned a subdivision level (mesh density) first, which is in tern used to assign labels to the vertices of the quadrilateral network. The mesh is generated on the basis of individual faces by performing label-driven subdivision on each face separately. Parallel processing is achieved by performing label-driven subdivision on all the faces simultaneously. The new label-driven subdivision technique improves a previous label-driven subdivision technique in that it does not require the labels of the vertices to satisfy certain requirement and, consequently, the generation of an admissible extension of the given label assignment is not necessary.
We propose a new method to convert approximately any Gregory-type patches to NURBS surfaces. The method is that first, the patch is subdivied into several subpatches as specified and they are approximated with suitable B\'ezier-type patches. Then, their control points are combined to generate ones of the NURBS surface.
$G^2$ continuity of free-form surfaces is sometimes very important in engineering applications. The conditions for $G^2$ continuity to connect two B\'ezier patches were studied and methods have been developed to ensure it. However, they have some restrictions on the shapes of patches of the B\'ezier-patch formulation. Gregory patch is a kind of free-form surface patch which is extended from B\'ezier patch so that four first derivatives on its boundary curves can be specified without restrictions of the compatibility condition. Several types of Gregory patches have been developed for integral, rational, and NURBS boundary curves. In this paper, we propose some integral boundary Gregory-type patches bounded by cubic and quartic curves for $G^2$ continuity.
This paper studies a function representation of point sets swept by moving solids. The original solid-generator is defined by a real function f(x,y,z,t). This definition allows us to include solids which change their shapes in time. Constructive solids can be used as generators also when described by R-functions. The trajectory of the generator can be defined in parametric form as movement of its local coordinate system. To get the function representation F(x,y,z)=>0 of the swept solid we apply the concept of envelope used before basically for boundary represented objects. We have reduced the problem of swept solid description to global extremum search by t variable.
Geometric modeling using a continuous real functions of several variables is discussed. Modeling concepts include sets of objects, operations and relations. Transformations of a defining function are described for set-theoretic operations, blending, offsetting, bijective mapping, projection, cartesian product and metamorphosis. Inclusion, point membership and intersection relations are described. We introduce high-level geometric language that provides extendibility of a modeling system by input symbolic descriptions of primitives, operations and predicates. This approach supports combinations of representational styles, including constructive geometry, sweeping, soft objects, voxel-based objects, deformable and other animated objects. Application examples of aesthetic design, collisions simulation, NC machining, range data processing, and 3D texture generation are given.
A mesh generation algorithm based on a new label-driven subdivision technique is presented. The algorithm generates 2D mesh of quadrilaterals/triangles on a regular quadrilateral network such as the parameter space of a piecewise surface. Each face (patch) of the quadrilateral network has to be assigned a subdivision level (mesh density) first, which is in tern used to assign labels to the vertices of the quadrilateral network. The mesh is generated on the basis of individual faces by performing label-driven subdivision on each face separately. Parallel processing is achieved by performing label-driven subdivision on all the faces simultaneously. The new label-driven subdivision technique improves a previous label-driven subdivision technique in that it does not require the labels of the vertices to satisfy certain requirement and, consequently, the generation of an admissible extension of the given label assignment is not necessary.
This paper proposed a predefined graphical structural syntax checker called Model Hierarchy Definition (MHD) which provides interactive hierarchy syntax checking during modeling the structured requirement specifications and predefined lowest level graphical formalisms called Model Hierarchy Model (MHM) which defining the meta rules, the graphical diagramming entity and the hierarchical relationship for MHD embedding in the prototyping models. With the MHD and the MHM in the prototyping environment, which can yield a higher quality structured requirement specifications when building hierarchical multi-model-based prototype for complex system analysis, and greatly reduce the prototyping time and costs.
We propose a new method to convert approximately any Gregory-type patches to NURBS surfaces. The method is that first, the patch is subdivied into several subpatches as specified and they are approximated with suitable B\'ezier-type patches. Then, their control points are combined to generate ones of the NURBS surface.
The paper presents a project called `HyperJazz' devoted to development and implementation of the `empirical' approach to solid modelling, which stresses the importance of interactive experiments of the user with the model. The approach is based on two foundations: geometric modelling based on the function representation and definitive programming environment. We propose a technology of user's interactive work with the modeller and a technology of creating interactive geometric modelling systems that can be adapted to different application domains and can be customized for needs of particular users. The basic set of API-procedures and the Kernel Geometric Interactive Environment are introduced. Their charasteristic feature called ``inherent symbolic interactivity" is based on a high-level definitive language of point-sets. The technology of building a particular CSG system as well as an example of behaviour modelling are presented.
This paper presents practical applications of volume spline techniques for interpolating scattered data. The surface of the solid is represented by the equation f(x,y,z)=0. Reconstructed objects can be easily modified by further application of sculpting. A femur has been reconstructed from parallel cross-sections to illustrate the capability of the presented approach.
Generation of animation sequence of a deformed volumetric object on networked workstations is discussed. We use several types of deformation: space mapping controlled by points linked to features of an object, deformation with an algebraic sum, and metamorphosis. These deformations are directly applied to interpolated volume data with following polygonization of an isosurface for visualization.
We present an approach to sculpting of functionally defined (or implicit) 3D geometric objects with arbitrary control points linked to features of an object. Displacements of these control points defineglobal space mapping. To iterpolate displacements we use a volume spline based on the Green's function. We apply this technique to objects defined by implicit functions constructed in different ways: set-theoretic operations with R-functions, volume data interpolation, and depth data conversion. A splitting operation by a deformed halfspace is introduced to show the benefits of combining the implicit representation and set-theoretic modeling with sculpting.
We take an approach to specify and control implicitly defined n-dimensional free-form solid primitives using parametric functions of n variables. Outside of the domain, the parametric function value increases or decreases rapidly that is not suitable to design geometric objects. In this paper, we introduce a functional clipping operation which allows a free-form primitive to be treated as traditional implicit primitives in set-theoretic modelling based on R-functions. We also introduce an extended B\'ezier clipping operation for fast inverse mapping in the case of the changed domain. Examples of set-theoretic, offsetting, hypertexturing and metamorphosis operations on free-form primitives are given. Tessellation of trimmed parametric patches is explained as an application of our new geometric modelling method.