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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)
This paper presents a project devoted to developing an open system architecture for functionally based (implicit or more generally F-rep) shape modeling and its applications. The software tools are built around the shape models written in a high-level programming language called HyperFun. A model in HyperFun can serve as a protocol for exchanging F-rep(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)
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,(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)