====== What is FRep? ======

The Function Representation (or FRep) defines a geometric object by a single real continuous function of point coordinates as 

<m>F(X) >= 0</m>

It combines many different models like 
  * algebraic surfaces 
  * skeleton based "implicit" surfaces 
  * set-theoretic solids or CSG (Constructive Solid Geometry) 
  * sweeps 
  * volumetric objects 
  * parametric models 
  * procedural models 

Main features of FRep: 
  * Defining the function evaluation procedure traversing the constructive tree structure 
  * Leaves of the constructive tree are primitives with known types of defining functions 
  * Nodes of the tree contain operations and relations 
  * Extensibility of primitives, operations, and relations 

The main goal of this project is to develop a rich system of geometric operations and relations. Our approach is based on the R-functions - <m>C^k</m> continuous definitions of set-theoretic operations. 

FRep can also be used to model point-wise object's properties such as material, density, color. This is supported by the [[hypervol|constructive hypervolume model]].

We apply theoretic results in different areas:  
  * computer-aided design
  * rapid prototyping and digital fabrication 
  * computer animation 
  * physically based simulation 
  * artificial life 
  * computer art 

Frequently asked question: [[why|Why not just "implicit" surfaces?]] 






===== Surveys =====

  * A. Pasko and V. Adzhiev, "Function-based shape modeling: mathematical framework and specialized language", Automated Deduction in Geometry, Lecture Notes in Artificial Intelligence 2930, Ed. F. Winkler, Springer-Verlag, Berlin Heidelberg, 2004, pp. 132-160. \\ Electronic version: [[http://hyperfun.org/PA_ADG02.pdf|PDF (847K)]]

  * [[sig96|FRep in the SIGGRAPH'96 Course on Implicit Surfaces]]

  * [[http://www.osp.ru/os/1996/05/178962/|Modeling shapes using real functions]], Open Systems, No.5 (19), 1996, pp.14-18 (in Russian). 

  * Pasko A., Adzhiev V., Sourin A., Savchenko V. "Function representation in geometric modeling: concepts, implementation and applications", The Visual Computer, vol.11, No.8, 1995, pp.429-446. \\ Electronic version (scanned): [[http://hyperfun.org/TVC95.pdf|PDF (7 Mb)]]

  * Pasko A.,  Adzhiev V., Schmitt B., Schlick C., "Constructive hypervolume modeling", Graphical Models, special issue on Volume Modeling, vol. 63, No. 6, November 2001, pp. 413-442. \\ Electronic version: [[http://hyperfun.org/GMOD02.pdf|PDF (1.69 Mb)]]
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This FRep Home Page represents the results of joint work by an international group of researchers (see the authors of [[publications|publications]]).