98.27.21.67 at 17:39, 4 July 2022

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'''Function Representation''' (FRep<ref>Shape Modeling and Computer Graphics with Real Functions, [http://www.hyperfun.org/F-rep.html FRep Home Page]</ref> 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" <ref>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.</ref> 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 <math>f(X)</math> of point coordinates <math>X[x_1,x_2, ..., x_n]</math> 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 <math>f(x_1,x_2, ..., x_n) \ge 0</math> belong to the object, and the points with <math>f(x_1,x_2, ..., x_n) < 0</math> are outside of the object. The point set with <math>f(x_1,x_2, ..., x_n)=0</math> is called an [[isosurface]].

== Geometric domain ==
The geometric domain of FRep in 3D space includes solids with [[non-manifold model]]s 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 surface]]s, 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-function]]s originally introduced in V.L. Rvachev's "On the analytical description of some geometric objects",<ref>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).</ref> provide <math>C^k</math> [[Smooth function|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 modeling is supported by the special-purpose language [[HyperFun]].

== Shape Models ==
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"<ref>A. Pasko, V. Adzhiev, B. Schmitt, C. Schlick, "Constructive hypervolume modelling", Graphical Models, 63(6), 2001, pp. 413-442.</ref> 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"<ref>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, Saarbrücken, Germany, ACM Press, 2002, pp. 192-203. 3-540-65620-0</ref> 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.

==See also==

* [[Boundary representation]]
* [[Constructive Solid Geometry]]
* [[Solid modeling]]
* [[Isosurface]]
* [[Signed distance function]]
* [[HyperFun]]
* [[Digital materialization]]

==References==
{{Reflist}}

== External links ==
* http://hyperfun.org/FRep/
* https://github.com/cbiffle/ruckus
* http://libfive.com/
* http://www.implicitcad.org/

[[Category:Geometric algorithms]]
[[Category:Computer graphics]]
[[Category:3D computer graphics]]

Function representation - Revision history

98.27.21.67 at 17:39, 4 July 2022