Function representation

The function representation FRep [1] was introduced in [2] as a uniform representation of multidimensional geometric objects (shapes). An object as a point set in multidimensional space is defined by a single continuous real-valued function of point coordinates F(X) which is evaluated at the given point by a procedure traversing a tree structure with primitives in the leaves and operations in the nodes of the tree. The points with F(X) >= 0 belong to the object, and the points with F(X) < 0 are outside of the object.

The geometric domain of FRep in 3D space includes solids with nonmanifold boundaries and lower dimensional entities (surfaces, curves, points) defined by zero value of the function. A primitive can be defined by an equation or by a "black box" procedure converting point coordinates into the function value. Solids bounded by algebraic surfaces, skeleton-based implicit surfaces, and convolution surfaces, as well as procedural objects (such as solid noise), and voxel objects can be used as primitives (leaves of the construction tree). In the case of a voxel object (discrete field), it should be converted to a continuous real function, for example, by applying the trilinear or higher-order interpolation. Many operations such as set-theoretic, blending, offsetting, projection, non-linear deformations, metamorphosis, sweeping, hypertexturing, and others, have been formulated for this representation in such a manner that they yield continuous real-valued functions as output, thus guaranteeing the closure property of the representation. R-functions originally introduced in [3] provide Ck continuity for the functions exactly defining the set-theoretic operations (min/max functions are a particular case). Because of this property, the result of any supported operation can be treated as the input for a subsequent operation; thus very complex models can be created in this way from a single functional expression.
FRep combines and generalizes different shape models like
• algebraic surfaces
• skeleton based "implicit" surfaces
• set-theoretic solids or CSG (Constructive Solid Geometry)
• sweeps
• volumetric objects
• parametric models
• procedural models

A more general “constructive hypervolume” [4] allows for modeling multidimensional point sets with attributes. Point set geometry and attributes have an independent representation but are treated uniformly. A point set in a geometric space of an arbitrary dimension is an FRep based geometric model of a real object. An attribute that is also represented by a real-valued function (not necessarily continuous) is a mathematical model of an object property of an arbitrary nature (material, photometric, physical etc.). The concept of “implicit complex” proposed in [5] provides a framework for including geometric elements of different dimensionality by combining polygonal, parametric, and FRep components into a single cellular-functional model of a heterogeneous object.

References
[1] Shape Modeling and Computer Graphics with Real Functions, FRep Home Page, http://cis.k.hosei.ac.jp/~F-rep/
[2] A. Pasko, V. Adzhiev, A. Sourin, V. Savchenko, "Function representation in geometric modeling: concepts, implementation and applications", The Visual Computer, vol.11, no.8, 1995, pp.429-446.
[3] V.L. Rvachev, “On the analytical description of some geometric objects”, Reports of Ukrainian Academy of Sciences, vol. 153, no. 4, 1963, pp. 765-767 (in Russian).
[4] A. Pasko, V. Adzhiev, B. Schmitt, C. Schlick, “Constructive hypervolume modelling”, Graphical Models, 63(6), 2001, pp. 413-442.
[5] V. Adzhiev, E. Kartasheva, T. Kunii, A. Pasko, B. Schmitt, "Cellular-functional modeling of heterogeneous objects", Proc. 7th ACM Symposium on Solid Modeling and Applications, Saarbrucken, Germany, ACM Press, 2002, pp. 192-203.

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