Function Representation (FRep[1] or F-Rep) is used in solid modeling, volume modeling and computer graphics. FRep was introduced in "Function representation in geometric modeling: concepts, implementation and applications" [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
f(X)
X[x1,x2,...,xn]
f(x1,x2,...,xn)\ge0
f(x1,x2,...,xn)<0
f(x1,x2,...,xn)=0
The geometric domain of FRep in 3D space includes solids with non-manifold models 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 V.L. Rvachev's "On the analytical description of some geometric objects",[3] provide
Ck
FRep combines and generalizes different shape models like
A more general "constructive hypervolume"[4] allows for modeling multidimensional point sets with attributes (volume models in 3D case). Point set geometry and attributes have independent representations 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, medicine, etc.). The concept of "implicit complex" proposed in "Cellular-functional modeling of heterogeneous objects"[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.