Alexander A. Pasko

Function representation in geometric modeling : concepts, implementation and applications

Concepts of functionally based geometric modeling including sets of objects, operations, and relations are discussed. Transformations of a defining real function are described for set-theoretic operations, blending, offsetting, bijective mapping, projection, cartesian products, and metamorphosis. Inclusion, point membership, and intersection relations are also described. We use a high-level geometric language that can extend the interactive 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.

Function Representation of Solids Reconstructed from Scattered Surface Points and Contours

This paper presents a novel approach to the reconstruction of geometric models and surfaces from given sets of points using volume splines. It results in the representation of a solid by the inequality f(x,y,z) ≥ 0. The volume spline is based on use of the Greens function for interpolation of scalar function values of a chosen “carrier” solid. Our algorithm is capable of generating highly concave and branching objects automatically. The particular case where the surface is reconstructed from cross‐sections is discussed too. Potential applications of this algorithm are in tomography, image processing, animation and CAD for bodies with complex surfaces.

Function representation for sweeping by a moving solid

Studies a function representation of point sets swept by moving solids. The original solid-generator is defined by an inequality f(x,y,z,t)/spl ges/0 where x, y, z are Cartesian coordinates and t is treated as the time. 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. In the paper, we did it with superposition of time-dependent affine transformations. To get the function representation F(x,y,z)/spl ges/0 of the swept solid, we apply the concept of an envelope, previously used basically for boundary represented objects. We have reduced the problem of swept solid description to a global extremum search by the t variable. The algorithm for procedural swept solid modeling is discussed. The benefit of our model is that it is applied not only for visualization but allows one to use the swept solid as an argument for other operations. For example, the swept solid can be intersected with other ones that are useful for the implementation of such operations as cutting and drilling. Ordinary texture mapping and hypertexturing can also be applied to it. The possibility of using a functionally defined generator with variable shape allows us to achieve a complexity of swept solids which was hardly possible before.

Procedural function-based modelling of volumetric microstructures

We propose a new approach to modelling heterogeneous objects containing internal volumetric structures with size of details orders of magnitude smaller than the overall size of the object. The proposed function-based procedural representation provides compact, precise, and arbitrarily parametrized models of coherent microstructures, which can undergo blending, deformations, and other geometric operations, and can be directly rendered and fabricated without generating any auxiliary representations (such as polygonal meshes and voxel arrays). In particular, modelling of regular lattices and cellular microstructures as well as irregular porous media is discussed and illustrated. We also present a method to estimate parameters of the given model by fitting it to microstructure data obtained with magnetic resonance imaging and other measurements of natural and artificial objects. Examples of rendering and digital fabrication of microstructure models are presented.

Bounded blending for function-based shape modeling

New analytical formulations of bounded blending operations can enhance function-based constructive shape modeling. In this article, we introduce bounded-blending operations that we define using R-functions and displacement functions with the localized area of influence. We define the shape and location of the blend by control points on the surfaces of two solids or by an additional bounding solid. We can apply our proposed blending using a bounding solid to a single selected edge or vertex. We also introduce new multiple blends and a partial edge blend. Our description supports set-theoretic operations on blends and blends on blends - that is, recursive blends. In this sense, our proposed operations could replace pure set-theoretic operations in the construction of a solid without rebuilding the entire construction tree data structure. Our proposed blending method can have application in interactive design.

Geometric modeling in the analysis of trivariate functions

Abstract A use of the integrated computer geometry/graphics systems in the analysis of functions of three variables (trivariate functions) is discussed. A method is obtained for the piecewise analytical description of the hypersurface defined by a trivariate function and of the isosurface of such a function. Algorithms for graphic presentation of the isosurface with hidden lines and surfaces removal are described. These algorithms are implemented at the SM-4 computer using the SAGRAF program complex.

Dynamic meshes for accurate polygonization of implicit surfaces with sharp features

The paper presents a novel approach for accurate polygonization of implicit surfaces with sharp features. The approach is based on mesh evolution towards a given implicit surface with simultaneous control of the mesh vertex positions and mesh normals.

Ridges and ravines on implicit surfaces

Surface creases provide us with important information about the shapes of objects and can be intuitively defined as curves on a surface along which the surface bends sharply. Our mathematical description of such surface creases is based on a study of extrema of the principal curvatures along their curvature lines. On a smooth generic surface we define ridges to be the local positive maxima of the maximal principal curvature along its associated curvature line and ravines to be the local negative minima of the minimal principal curvature along its associated curvature line. The ridges and ravines are important for shape analysis and possess remarkable mathematical properties. For example, they correspond to the end points of shape skeletons. In this paper we derive formulas to detect the ridges and ravines on a surface given in implicit form. We also propose an algorithm for obtaining piecewise linear approximation of ridges and ravines as intersection curves of two implicit surfaces.

Transformation of functionally defined shapes by extended space mappings

We present a general mathematical framework for transforming functionally defined shapes. The proposed model of extended space mappings considers transformations of a hypersurface in coordinate-function space with its projection onto geometric space. This model covers coordinate space mappings, metamorphosis, and algebraic operations on defining functions, and introduces several new types of transformations, such as function-dependent space mappings and combined mappings. The approach is illustrated by new local deformations created by means of function mappings, feature-based space mapping, offsetting along the normal, thin shell generation, 2D shape blending, and collision-free metamorphosis.

Bounded blending operations

New analytical formulations of bounded blending for functionally defined set-theoretic operations are proposed. The blending set operations are defined using R-functions and displacement functions with the localized area of influence. The shape and location of the blend is defined by control points on the surfaces of two solids or by an additional bounding solid. The proposed blending using a bounding solid can be applied to a single selected edge or a vertex. We introduce new types of blends such as a multiple blend with the disconnected bounding solid and a partial edge blend. It is shown to have versatile applications in interactive design.