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CylindricalDecompositionFunction
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CylindricalDecompositionFunction

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represents a cylindrical algebraic formula in x1,x2,….

Details and Options

Examples

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Basic Examples  (2)Summary of the most common use cases

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

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-txqtp0
Out[1]=1

Represent the single-cell solution as explicit inequalities:

In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-szc91n
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-bs8q93
Out[1]=1

Visualize the result:

Compute the boundary of the solution set:

In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-jv2fs6
Out[2]=2

Visualize the result:

Solve an equation over the solution set:

In[3]:=3
✖
https://wolfram.com/xid/0tp61ztgwipg8y-jfgi86
Out[3]=3

Visualize the result:

Scope  (23)Survey of the scope of standard use cases

Basic Uses  (7)

Solution of polynomial equations and inequalities:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-4ka8we
Out[1]=1

Coefficients can include real algebraic numbers:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-610w84
Out[1]=1

Coefficients can include real exact transcendental numbers:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-lytng
Out[1]=1

Solution of a quantified system of polynomial equations and inequalities:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-u74z9h
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-p4m37k
Out[1]=1

CylindricalDecompositionFunction with real numeric initial arguments simplifies automatically:

In[1]:=1
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https://wolfram.com/xid/0tp61ztgwipg8y-vahvqq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-wzr26b
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tp61ztgwipg8y-l6tcwq
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0tp61ztgwipg8y-6qorrm
Out[4]=4

Use CylindricalDecompositionFunction objects in further computation:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-h4zr8k
Out[1]=1

Compose a CylindricalDecompositionFunction with an algebraic mapping:

In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-noc5yx
Out[2]=2

Use a Boolean combination of CylindricalDecompositionFunction objects and polynomial inequalities:

In[3]:=3
✖
https://wolfram.com/xid/0tp61ztgwipg8y-h6htvg
Out[3]=3

Boolean Operations  (2)

Compute a CylindricalDecompositionFunction solution of a quantified system:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-7buclv
Out[1]=1

Find the negation of :

In[2]:=2
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https://wolfram.com/xid/0tp61ztgwipg8y-m551qu
Out[2]=2

Show that is true:

In[3]:=3
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https://wolfram.com/xid/0tp61ztgwipg8y-b2fp2x
Out[3]=3

Compute CylindricalDecompositionFunction solutions of two different quantified systems:

In[1]:=1
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https://wolfram.com/xid/0tp61ztgwipg8y-otuq92
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0tp61ztgwipg8y-my8umm
Out[2]=2

Compute :

In[3]:=3
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https://wolfram.com/xid/0tp61ztgwipg8y-1oe6zp
Out[3]=3

Solving Equations and Inequalities  (2)

Compute a CylindricalDecompositionFunction solution of a quantified system:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-jeynzi
Out[1]=1

Solve an equation over the solution set of :

In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-0ktnf
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0tp61ztgwipg8y-587kk0
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0tp61ztgwipg8y-8s3r7o
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0tp61ztgwipg8y-e5i5cc
Out[3]=3

Quantifier Elimination  (3)

Eliminate quantifiers from quantified CylindricalDecompositionFunction objects:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-pxai50
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0tp61ztgwipg8y-4eub1v
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tp61ztgwipg8y-oripu5
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0tp61ztgwipg8y-ujqmbg
Out[4]=4

Eliminate quantifiers from systems involving CylindricalDecompositionFunction objects:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-b7ueq6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-jh3dy1
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tp61ztgwipg8y-1hq9cu
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-6ywg4s
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-6oq0ze
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tp61ztgwipg8y-oivpf3
Out[3]=3

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

In[4]:=4
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https://wolfram.com/xid/0tp61ztgwipg8y-w24x3r
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0tp61ztgwipg8y-men2kl
Out[5]=5

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

In[6]:=6
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https://wolfram.com/xid/0tp61ztgwipg8y-oy34tt
Out[6]=6
In[7]:=7
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https://wolfram.com/xid/0tp61ztgwipg8y-h85073
Out[7]=7

Check that the obtained results are equivalent:

In[8]:=8
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https://wolfram.com/xid/0tp61ztgwipg8y-g7xyyz
Out[8]=8
In[9]:=9
✖
https://wolfram.com/xid/0tp61ztgwipg8y-ip67h4
Out[9]=9

Topological Operations  (5)

Find CylindricalDecompositionFunction representations of the solution region and its boundary:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-dm2euj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-fstovq
Out[2]=2

Find CylindricalDecompositionFunction representations of the solution region and its closure:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-kq94g
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-h68469
Out[2]=2

Find CylindricalDecompositionFunction representations of the solution region and its interior:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-zbz1z
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-fzy3az
Out[2]=2

Find CylindricalDecompositionFunction representations of the solution region and its exterior:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-iufpj0
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-nww150
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-h81845
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-dldfgw
Out[2]=2

CylindricalDecompositionFunction as Input to Solvers  (4)

Use CylindricalDecompositionFunction to specify optimization constraints:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-69wsye
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-6w4bl9
Out[2]=2

Solve systems involving equations, inequalities and CylindricalDecompositionFunction constraints:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-5822m2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-yx0p7p
Out[2]=2

Use FindInstance to find solution instances for systems involving CylindricalDecompositionFunction:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-pruiwr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-0axenw
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-4z9e7c
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-591jt8
Out[2]=2

Applications  (4)Sample problems that can be solved with this function

Compute an intersection of projections of two surfaces:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-pn5up9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-3es10r
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tp61ztgwipg8y-t3oz2b
Out[3]=3

Optimize over the projection of a solid:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-2qmc4w
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-y32dg9
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tp61ztgwipg8y-fx4e8u
Out[3]=3

Find the connected components of an algebraic curve:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-h9j2el
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-nf7ezg
Out[2]=2

Find the connected components of a region:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-cle32
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-gxpixa
Out[2]=2

Properties & Relations  (5)Properties of the function, and connections to other functions

This gives a cylindrical algebraic formula in an encapsulated form:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-k6s8ju
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-kp87e9
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0tp61ztgwipg8y-pj8stk
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-l0vshb
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-iu0wkb
In[3]:=3
✖
https://wolfram.com/xid/0tp61ztgwipg8y-bj6p5r

Operations on CylindricalDecompositionFunction objects are fast:

In[4]:=4
✖
https://wolfram.com/xid/0tp61ztgwipg8y-3aqxv2
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0tp61ztgwipg8y-qnhkrn
Out[5]=5

Use FindInstance to find points that satisfy a CylindricalDecompositionFunction object:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-kcmahm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-m248tf
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-b4wd8
Out[1]=1

CylindricalDecompositionFunction merges several cells to get a more compact representation:

In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-f7q2be
Out[2]=2

SemialgebraicComponentInstances now knows that fewer points suffice:

In[3]:=3
✖
https://wolfram.com/xid/0tp61ztgwipg8y-zqwj2s
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-fg39i
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-cf0dgm
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0tp61ztgwipg8y-7ufw6a
Out[3]=3

Possible Issues  (1)Common pitfalls and unexpected behavior

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

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-ccflnb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-3f8nuc
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tp61ztgwipg8y-7csxqf
Out[3]=3

Neat Examples  (1)Surprising or curious use cases

Semialgebraic sets are quite general:

In[1]:=1
✖
https://wolfram.com/xid/0tp61ztgwipg8y-bsypdu
In[2]:=2
✖
https://wolfram.com/xid/0tp61ztgwipg8y-gp03o
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tp61ztgwipg8y-bu0e2r
Out[3]=3
Wolfram Research (2020), CylindricalDecompositionFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/CylindricalDecompositionFunction.html.
✖
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.

✖
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.

✖
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

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

BibTeX

@misc{reference.wolfram_2025_cylindricaldecompositionfunction, author="Wolfram Research", title="{CylindricalDecompositionFunction}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/CylindricalDecompositionFunction.html}", note=[Accessed: 18-May-2025 ]}

✖
@misc{reference.wolfram_2025_cylindricaldecompositionfunction, author="Wolfram Research", title="{CylindricalDecompositionFunction}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/CylindricalDecompositionFunction.html}", note=[Accessed: 18-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_cylindricaldecompositionfunction, organization={Wolfram Research}, title={CylindricalDecompositionFunction}, year={2020}, url={https://reference.wolfram.com/language/ref/CylindricalDecompositionFunction.html}, note=[Accessed: 18-May-2025 ]}

✖
@online{reference.wolfram_2025_cylindricaldecompositionfunction, organization={Wolfram Research}, title={CylindricalDecompositionFunction}, year={2020}, url={https://reference.wolfram.com/language/ref/CylindricalDecompositionFunction.html}, note=[Accessed: 18-May-2025 ]}