The theory of R-functions was developed
in Ukraine by Rvachev and his students .
A real-valued function f(x1,x2,...,xn)
is called an R-function 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 R-function 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(x1, x2)
is an R-function whose companion Boolean function is logical
?and? (/\) (logical conjunction), and max(x1,
x2) is an R-function whose companion Boolean
function is logical ?or? (\/) (logical disjunction). This is seen in that function min(x1, x2)
takes on positive values only when x1 AND
x2 are positive; similarly function max(x1, x2) takes on positive
values when x1 OR x2 are
positive. Except the functions min(x1, x2)
and max(x1, x2) there are
infinitely many R-functions with different differential properties
[1, 2, 3].
Here is the most popular system of R-functions:
Similar to Boolean functions, R-functions
are closed under composition, which means that a combination
of several R-functions is another R-function 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 R-functions
can be combined into a corresponding R-function. Expressed
another way, for every formal logical sentence (i.e., for
every Boolean function), one may construct a corresponding
R-function using R-conjunction, R-disjunction, and R-negation,
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 R-function.
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 R-conjunction, R-disjunction, and R-negation
chosen to construct the corresponding R-function (i.e., depending
on the ?family? from which the R-functions are chosen), a
rich variety of differential properties may be obtained.
R-functions are a toolkit for construction
of normalized implicit functions with desirable differential
properties from any logical geometric representation.
 V. L. Rvachev. Theory
of R-functions and Some Applications. Naukova Dumka, 1982.
Theory of R-functions and applications: A primer. V. Shapiro,
Technical Report, Cornell University.
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,
V. Shapiro and I. Tsukanov, Meshfree Simulation of Deforming
Domains, Computer-Aided Design , Vol. 31, No. 7, 1999, pp.
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. 305-316
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.195-220