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In mathematics, a subpaving is a set of nonoverlapping
''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
boxes
A box (plural: boxes) is a container used for the storage or transportation of its contents. Most boxes have flat, parallel, rectangular sides. Boxes can be very small (like a matchbox) or very large (like a shipping box for furniture), and can ...
of R⁺. A subset
In mathematics, set ''A'' is a subset of a set ''B'' if all 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 are unequal, then ''A'' is a proper subset o ...
''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
Computation is any type of arithmetic or non-arithmetic calculation that follows a well-defined model (e.g., an algorithm).
Mechanical or electronic devices (or, historically, people) that perform computations are known as '' computers''. An esp ...
, a well-known application of subpaving in R² is the Quadtree data structure. In image tracing context and other applications is important to see ''X⁻'' as topological interior, 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:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
is path connected,
to provide topological properties of such sets,
to solve piano-mover's problems
or to implement set computation.
References
{{Reflist, 2 Topology Geometry