Rvachev function

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^[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

1. ^ 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)
2. ^ V. Shapiro, Semi-analytic geometry with R-Functions, Acta Numerica, Cambridge University Press, 2007, 16: 239-303
3. ^ 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 Functions