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Geometric modeling using continuous real functions of several variables is discussed. Modeling concepts include sets of objects, operations and relations. An object is a closed point set of n-dimensional Euclidean space with a defining inequality f x x x n 1 2 0 ≥. Transformations of a defining function are described for the set-theoretic operations,(More)
The paper presents an approach to modeling heterogeneous objects as multidimensional point sets with multiple attributes (hypervolumes). A theoretical framework is based on a hybrid model of hypervolumes combining a cellular representation and a constructive representation using real-valued functions. This model allows for independent but unifying(More)
Shape transformation between objects of different topology and positions in space is an open modeling problem. We propose a new approach to solving this problem for two given 2D or 3D shapes. The key steps of the proposed algorithm are: increase dimension by converting two input kD shapes into half-cylinders in (k+1)D space-time, applying bounded blending(More)
Distribution of material density and other properties of heterogeneous objects can be parametrized by the Euclidean distance function from the object boundary or from special material features. For objects constructed using geometric primitives and set-theoretic operations , an approximation of the distance function can be obtained in a constructive manner(More)
We propose a new operation to locally control the metamorphosis of two functionally defined shapes. To implement this operation, a set of non-overlapping space partitions is introduced, where the metamorphosis occurs locally. The sequence of local metamorphosis processes is controlled by a specific time schedule. The definitions of the partitions, of the(More)
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 The volume spline is based on use of the Green's function for interpolation of scalar function values of a chosen " carrier " solid. Our algorithm is capable(More)
This paper deals with modeling point sets with attributes. A point set in a geometric space of an arbitrary dimension is a geometric model of a real/abstract object or process under consideration. An attribute is a mathematical model of an object property of arbitrary nature (material, photometric, physical, statistical, etc.) defined at any point of the(More)
This paper 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)≥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(More)
The paper presents a novel approach for accurate polygo-nization 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. Given an initial polygonization of an implicit surface, a mesh evolution process initialized by the(More)