
The theory of Rfunctions was developed
in Ukraine by Rvachev and his students [1].
A realvalued function f(x1,x2,...,xn)
is called an Rfunction if its sign is completely determined
by the signs of its arguments xi. If the sign of a
function is considered to be a logical property, negative
values of a function can be considered to correspond with
logical FALSE, and positive values can be considered to correspond
with logical TRUE. In other words, an Rfunction f works as a Boolean switching function, changing its sign only
when its arguments change their signs; it can be regarded
as ?on? or ?off? (or ?true? or ?false?) depending on the values
of the input variables. For example, the function xyz
can be negative only when the number of its negative arguments
is odd. As another example, min(x_{1}, x_{2})
is an Rfunction whose companion Boolean function is logical
?and? (/\) (logical conjunction), and max(x_{1},
x_{2}) is an Rfunction whose companion Boolean
function is logical ?or? (\/) (logical disjunction). This is seen in that function min(x_{1}, x_{2})
takes on positive values only when x_{1} AND
x_{2} are positive; similarly function max(x_{1}, x_{2}) takes on positive
values when x_{1} OR x_{2} are
positive. Except the functions min(x_{1}, x_{2})
and max(x_{1}, x_{2}) there are
infinitely many Rfunctions with different differential properties
[1, 2, 3].
Here is the most popular system of Rfunctions:
Similar to Boolean functions, Rfunctions
are closed under composition, which means that a combination
of several Rfunctions is another Rfunction which corresponds
to a more complex logical expression. Thus, just as
any logical function can be written using only three operations
Â (logical negation or NOT), \/ (logical disjunction or OR),
and /\ (logical conjunction or AND), three corresponding Rfunctions
can be combined into a corresponding Rfunction. Expressed
another way, for every formal logical sentence (i.e., for
every Boolean function), one may construct a corresponding
Rfunction using Rconjunction, Rdisjunction, and Rnegation,
whose sign is determined by the truth table of the logical
sentence. For nonzero arguments, the negation (Â) operation
is usually accomplished by changing the sign of the Rfunction.
The logical disjunction \/ and conjunction /\ operations can
respectively be accomplished in the usual case by performing
intersection and union operations. Depending on the
particular form of the Rconjunction, Rdisjunction, and Rnegation
chosen to construct the corresponding Rfunction (i.e., depending
on the ?family? from which the Rfunctions are chosen), a
rich variety of differential properties may be obtained.
Rfunctions are a toolkit for construction
of normalized implicit functions with desirable differential
properties from any logical geometric representation. 


References 
[1] V. L. Rvachev. Theory
of Rfunctions and Some Applications. Naukova Dumka, 1982.
In Russian.
[2]
Theory of Rfunctions and applications: A primer. V. Shapiro,
Technical Report, Cornell University.
[3]
V. Shapiro and I. Tsukanov, Implicit Functions with Guaranteed
Differential Properties, In Proceedings of the Fifth ACM Symposium
on Solid Modeling and Applications, June 1999, Ann Arbor,
MI
[4]
V. Shapiro and I. Tsukanov, Meshfree Simulation of Deforming
Domains, ComputerAided Design , Vol. 31, No. 7, 1999, pp.
459471
[5]
V.L.Rvachev, T.I.Sheiko, V.Shapiro and I.Tsukanov, On Completeness
of RFM Solution Structures, Computational Mechanics, special
issue on meshfree methods, Vol. 25, 2000, pp. 305316
[6]
V.L. Rvachev, T.I. Sheiko, V. Shapiro, I. Tsukanov, Transfinite
Interpolation Over Implicitly Defined Sets, Computer Aided
Geometric Design, Vol. 18, No. 4, 2001, pp.195220 
