In mathematics, an R-function, or Rvachev function, is a real-valued function whose sign does not change if none of the signs of its arguments change; that is, its sign is determined solely by the signs of its arguments.[1][2]
Interpreting positive values as true and negative values as false, an R-function is transformed into a "companion" Boolean function (the two functions are called friends). For instance, the R-function ƒ(x, y) = min(x, y) is one possible friend of the logical conjunction (AND). R-functions are used in computer graphics and geometric modeling in the context of implicit surfaces and the function representation. They also appear in certain boundary-value problems, and are also popular in certain artificial intelligence applications, where they are used in pattern recognition.
R-functions were first proposed by Vladimir Logvinovich Rvachev [ru][3] (Russian: Влади́мир Логвинович Рвачёв) in 1963, though the name, "R-functions", was given later on by Ekaterina L. Rvacheva-Yushchenko, in memory of their father, Logvin Fedorovich Rvachev (Russian: Логвин Фёдорович Рвачёв).
See also
Function representation
Slesarenko function (S-function)
Notes
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)
V. Shapiro, Semi-analytic geometry with R-Functions, Acta Numerica, Cambridge University Press, 2007, 16: 239-303
75 years to Vladimir L. Rvachev (75th anniversary biographical tribute)
References
Meshfree Modeling and Analysis, R-Functions (University of Wisconsin)
Pattern Recognition Methods Based on Rvachev Functions (Purdue University)
Shape Modeling and Computer Graphics with Real Functionsa
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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