# CylindricalDecompositionFunction

CylindricalDecompositionFunction[data][x1,x2,]

represents a cylindrical algebraic formula in x1,x2,.

# Examples

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## Basic Examples(2)

Find a cylindrical algebraic formula representation of the open unit ball:

Represent the single-cell solution as explicit inequalities:

Find a cylindrical algebraic formula representation of the solution set of a quantified formula:

Visualize the result:

Compute the boundary of the solution set:

Visualize the result:

Solve an equation over the solution set:

Visualize the result:

## Scope(23)

### Basic Uses(7)

Solution of polynomial equations and inequalities:

Coefficients can include real algebraic numbers:

Coefficients can include real exact transcendental numbers:

Solution of a quantified system of polynomial equations and inequalities:

CylindricalDecompositionFunction with a constant truth value simplifies to True or False:

CylindricalDecompositionFunction with real numeric initial arguments simplifies automatically:

Use CylindricalDecompositionFunction objects in further computation:

Compose a CylindricalDecompositionFunction with an algebraic mapping:

Use a Boolean combination of CylindricalDecompositionFunction objects and polynomial inequalities:

### Boolean Operations(2)

Compute a CylindricalDecompositionFunction solution of a quantified system:

Find the negation of :

Show that is true:

Compute CylindricalDecompositionFunction solutions of two different quantified systems:

Compute :

### Solving Equations and Inequalities(2)

Compute a CylindricalDecompositionFunction solution of a quantified system:

Solve an equation over the solution set of :

Solve a Boolean combination of CylindricalDecompositionFunction objects, equations and inequalities:

### Quantifier Elimination(3)

Eliminate quantifiers from quantified CylindricalDecompositionFunction objects:

Eliminate quantifiers when the variables are quantified in the same order they appear in the formula:

Eliminate quantifiers from systems involving CylindricalDecompositionFunction objects:

The cost of quantifier elimination depends on the order of variables:

When the quantified variables appear in the formula after the free variables, elimination is fast:

Elimination of quantifiers from formulas constructed in a different variable order is much slower:

Check that the obtained results are equivalent:

### Topological Operations(5)

Find CylindricalDecompositionFunction representations of the solution region and its boundary:

Find CylindricalDecompositionFunction representations of the solution region and its closure:

Find CylindricalDecompositionFunction representations of the solution region and its interior:

Find CylindricalDecompositionFunction representations of the solution region and its exterior:

Find CylindricalDecompositionFunction representations of the solution region and its connected components:

### CylindricalDecompositionFunction as Input to Solvers(4)

Use CylindricalDecompositionFunction to specify optimization constraints:

Solve systems involving equations, inequalities and CylindricalDecompositionFunction constraints:

Use FindInstance to find solution instances for systems involving CylindricalDecompositionFunction:

Find at least one point in each connected component of the solution set of a CylindricalDecompositionFunction:

## Applications(4)

Compute an intersection of projections of two surfaces:

Optimize over the projection of a solid:

Find the connected components of an algebraic curve:

Find the connected components of a region:

## Properties & Relations(5)

This gives a cylindrical algebraic formula in an encapsulated form:

By default, CylindricalDecomposition gives the formula written as explicit equations and inequalities:

Normal of the CylindricalDecompositionFunction object is equivalent, and often identical, to the formula computed directly:

Cylindrical algebraic formulas in encapsulated form are often more efficient for further computation:

Operations on CylindricalDecompositionFunction objects are fast:

Operations on cylindrical algebraic formulas written as explicit equations and inequalities often are much slower:

Use FindInstance to find points that satisfy a CylindricalDecompositionFunction object:

SemialgebraicComponentInstances gives at least one point in each connected component of a semialgebraic set:

CylindricalDecompositionFunction merges several cells to get a more compact representation:

SemialgebraicComponentInstances now knows that fewer points suffice:

GenericCylindricalDecomposition computes representations of semialgebraic sets set up to lower-dimensional parts:

Use Method{"CylindricalDecompositionFunctionOutput"True} to get CylindricalDecompositionFunction results:

## Possible Issues(1)

When given inexact arguments, CylindricalDecompositionFunction uses the inexact values of arguments for comparisons, but produces an exact result:

## Neat Examples(1)

Semialgebraic sets are quite general:

Wolfram Research (2020), CylindricalDecompositionFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/CylindricalDecompositionFunction.html.

#### Text

Wolfram Research (2020), CylindricalDecompositionFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/CylindricalDecompositionFunction.html.

#### CMS

Wolfram Language. 2020. "CylindricalDecompositionFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CylindricalDecompositionFunction.html.

#### APA

Wolfram Language. (2020). CylindricalDecompositionFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CylindricalDecompositionFunction.html

#### BibTeX

@misc{reference.wolfram_2021_cylindricaldecompositionfunction, author="Wolfram Research", title="{CylindricalDecompositionFunction}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/CylindricalDecompositionFunction.html}", note=[Accessed: 27-January-2022 ]}

#### BibLaTeX

@online{reference.wolfram_2021_cylindricaldecompositionfunction, organization={Wolfram Research}, title={CylindricalDecompositionFunction}, year={2020}, url={https://reference.wolfram.com/language/ref/CylindricalDecompositionFunction.html}, note=[Accessed: 27-January-2022 ]}