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In
''X⁻'' ⊂ ''X'' ⊂ ''X⁺''. In R¹ the boxes are line segments, in R² rectangles and in Rⁿ hyperrectangles. A R² subpaving can be also a " non-regular tiling by rectangles", when it has no holes.
Boxes present the advantage of being very easily manipulated by computers, as they form the heart of interval analysis. Many interval algorithms naturally provide solutions that are regular subpavings.
In
''X'' =
with different accuracies. The set ''X⁻'' corresponds to red boxes and the set ''X⁺'' contains all red and yellow boxes.
Combined with interval-based methods, subpavings are used to approximate the solution set of non-linear problems such as set inversion problems.
Subpavings can also be used to prove that a set defined by nonlinear
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 ...
, a subpaving is a set of nonoverlapping boxes of R⁺. A subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
''X'' of Rⁿ can be approximated by two subpavings ''X⁻'' and ''X⁺'' such that''X⁻'' ⊂ ''X'' ⊂ ''X⁺''. In R¹ the boxes are line segments, in R² rectangles and in Rⁿ hyperrectangles. A R² subpaving can be also a " non-regular tiling by rectangles", when it has no holes.
Boxes present the advantage of being very easily manipulated by computers, as they form the heart of interval analysis. Many interval algorithms naturally provide solutions that are regular subpavings.
In computation
A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computation are mathematical equation solving and the execution of computer algorithms.
Mechanical or electronic devices (or, hist ...
, a well-known application of subpaving in R² is the Quadtree data structure. In image tracing
In computer graphics, image tracing, raster-to-vector conversion or raster vectorization is the conversion of raster graphics into vector graphics.
Background
An image does not have any structure: it is just a collection of marks on paper, grai ...
context and other applications is important to see ''X⁻'' as topological interior
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of t ...
, as illustrated.
Example
The three figures on the right below show an approximation of the set''X'' =
with different accuracies. The set ''X⁻'' corresponds to red boxes and the set ''X⁺'' contains all red and yellow boxes.
Combined with interval-based methods, subpavings are used to approximate the solution set of non-linear problems such as set inversion problems.
Subpavings can also be used to prove that a set defined by nonlinear inequalities
Inequality may refer to:
* Inequality (mathematics), a relation between two quantities when they are different.
* Economic inequality, difference in economic well-being between population groups
** Income inequality, an unequal distribution of i ...
is path connected
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Conne ...
,
to provide topological
Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...
properties of such sets,
to solve piano-mover's problems
or to implement set computation.
References
{{Reflist, 2 Topology Geometry