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ParametricRegion—Wolfram Language Documentation

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BUILT-IN SYMBOL

ParametricRegion[{f₁,…,f_n},{u₁,…,u_m}]

represents a region in given by the points {f₁,…,f_n} for parameters u_i∈.

ParametricRegion[{f₁,…,f_n},{{u₁,a₁,b₁},…}]

constrains parameters to an interval a₁≤u₁≤b₁ etc.

ParametricRegion[{{f₁,…,f_n},cond},…]

constrains parameters to satisfy the condition cond.

Details

ParametricRegion is also known as parametric curve, surface, volume, etc.
The value of u_i in ParametricRegion[{{f₁,…,f_n},cond},{u₁,…,u_m}] is taken to be localized, as in Block.
ParametricRegion[{funs,cond},{{u₁,a₁,b₁},…}] is equivalent to ParametricRegion[{funs,cond∧a₁≤u₁≤b₁∧⋯},{u₁,…}].
ParametricRegion can be used in functions such as RegionDistance, Reduce, and Integrate.

Examplesopen allclose all Basic Examples (2)

Specify a circle as a ParametricRegion:

Specify a disk:

The region between two parabolas:

Scope (20) Purely Parametric Description (10)

Parametrically described region:

Embedding dimension:

Geometric dimension:

Point membership test:

Get conditions for point membership:

Area and centroid:

Distance from a point:

Signed distance from a point:

Nearest point in the region:

Nearest points:

The set of points above a parabola is not bounded:

Integrate over an implicitly defined region:

Optimize over an implicitly defined region:

Solve equations in an implicitly defined region:

Mixed Implicit and Parametric Description (10)

A region given in a mixed implicit and parametric description:

Embedding dimension:

Geometric dimension:

Point membership test:

Get conditions for point membership:

Area and centroid:

Distance from a point:

Signed distance from a point:

Nearest point in the region:

Nearest points:

ℛ is bounded but not convex:

Integrate over ℛ:

Optimize over ℛ:

Solve equations and inequalities in ℛ:

Wolfram Research (2014), ParametricRegion, Wolfram Language function, https://reference.wolfram.com/language/ref/ParametricRegion.html. Text Wolfram Research (2014), ParametricRegion, Wolfram Language function, https://reference.wolfram.com/language/ref/ParametricRegion.html.CMS Wolfram Language. 2014. "ParametricRegion." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ParametricRegion.html.APA Wolfram Language. (2014). ParametricRegion. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ParametricRegion.htmlBibTeX @misc{reference.wolfram_2025_parametricregion, author="Wolfram Research", title="{ParametricRegion}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/ParametricRegion.html}", note=[Accessed: 08-July-2025 ]}BibLaTeX @online{reference.wolfram_2025_parametricregion, organization={Wolfram Research}, title={ParametricRegion}, year={2014}, url={https://reference.wolfram.com/language/ref/ParametricRegion.html}, note=[Accessed: 08-July-2025 ]}

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