In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an R-function, or Rvachev function, is a
real-valued function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
Real-valued functions of a real variable (commonly called ''real ...
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.
Interpreting positive values as ''true'' and negative values as ''false'', an R-function is transformed into a "companion"
Boolean function
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth functi ...
(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
Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. ...
and
geometric modeling
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Geometric modeling is a branch of applied mathematics and computational geometry that studies methods and algorithms for the mathematical description of shapes.
The shapes studied in geometric modeling are mostly two- or three-dimensi ...
in the context of
implicit surface
In mathematics, an implicit surface is a Surface (geometry), surface in Euclidean space defined by an equation
: F(x,y,z)=0.
An ''implicit surface'' is the set of Zero of a function, zeros of a Function of several real variables, function of ...
s and the
function representation. They also appear in certain
boundary-value problems, and are also popular in certain
artificial intelligence
Artificial intelligence (AI) is the capability of computer, computational systems to perform tasks typically associated with human intelligence, such as learning, reasoning, problem-solving, perception, and decision-making. It is a field of re ...
applications, where they are used in
pattern recognition
Pattern recognition is the task of assigning a class to an observation based on patterns extracted from data. While similar, pattern recognition (PR) is not to be confused with pattern machines (PM) which may possess PR capabilities but their p ...
.
R-functions were first proposed by
75 years to Vladimir L. Rvachev
(75th anniversary biographical tribute) () in 1963, though the name, "R-functions", was given later on by Ekaterina L. Rvacheva-Yushchenko, in memory of their father, Logvin Fedorovich Rvachev ({{langx, ru, Логвин Фёдорович Рвачёв).
See also
* Function representation
Notes
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
Non-classical logic
Real analysis
Types of functions
Boolean algebra