The solution of the inverse problem of analytic geometry. Theory of R-Functions

Inspired by a recent post on the construction of various images using the Hilbert curve. There will be some theory and some pictures.

Bit of theory

The computer age has generated the theory of R-functions - functions with a "logical charge", which arose at the junction of discrete and continuous analysis, using the apparatus of Boolean algebra, which is also inherent in computers. Based on the theory of R-functions, the inverse problem of analytic geometry was solved, it became possible to construct the boundary equation of a complex object in the form of an elementary function, and moreover, an equation that would possess the necessary differential properties . V. L. RvachevUsing the constructive apparatus of the theory of R-functions, he developed a unified approach to the problem of constructing coordinate sequences for the main variational and projection methods. To date, the R-function method has been applied to solve a large number of problems in electrodynamics, mechanics of a deformable solid, the theory of plates and shells, hydrodynamics and magnetic hydrodynamics, thermophysics, etc.

Definition of R-functions and basic systems of R-functions

Denote by .

If we call the Boolean sign of a quantity , then we can give the following definition of R-functions: a function is called an R-function if the Boolean sign of this function is equal to the Boolean function of the Boolean signs of the arguments . Any Boolean function can be represented in terms of (in conjunctive and disjunctive normal forms). This fact means that the system is a complete system of Boolean functions (that is, the set of H-realizable functions (M (H)) is the set of all Boolean functions).
Consider the functions:

The functions of the first column are R-functions . Any continuous function of any number of arguments can be assigned to one of these columns. What feature separates R-functions from non-R-functions presented in the second column?
A person who does not have a preliminary acquaintance with R-functions is unlikely to be able to guess the "secret of R-functions." Meanwhile, this feature is very simple: R-functions have the property that specifying the signs of the arguments uniquely determines the sign of the R-function . And this is their genius. For this property is obvious. In order to prove its validity for , we consider a right-angled triangle with sides . If , then modules can not write, and then the sum of the legs more than the hypotenuse: . Ifhave different signs, that is, the difference of legs, and then . If negative, then even more so . The signs are the same as y , and the sign is the same as y . It is obvious. Thus, a character table can be compiled for these functions.

-	-	-	-	-	+
-	-	+	+	-	+
-	+	-	+	-	-
-	+	+	-	-	-
+	-	-	+	-	-
+	-	+	-	-	-
+	+	-	-	+	+
+	+	+	+	+	+

If in this table we replace "-" with "0" and "+" with "1", we get tables of three Boolean functions. For example, the function corresponds to the conjunction , the function corresponds to the Boolean function .

The most common and historically first is such a system of R-functions:

Actually an example

Let simple (supporting) areas be given
- a vertical strip between the lines ,
- a horizontal strip between the lines ,
- a vertical strip between the lines ,
- a horizontal strip between the lines ,
and a complex drawing is determined by the logical formula:
It is easy to notice that this drawing is a cross-shaped area shown on figure, provided that .
As a result, we get:

Some pictures

The apparatus of R-functions will allow us to build the objects shown below, while we always know the exact (analytical) expression for each geometric object, and in fact we can not lose accuracy on an approximate description of the geometry of the object.

And here is a program that can visualize and solve problems using R-functions

Source: Rvachev V.L. The theory of R-functions and some of its applications