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ParametricRegion[{f1,…,fn},{u1,…,um}]

表示对于参数 ui∈，由点 {f1,…,fn} 给出的 中的区域.

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

约束参数位于区间 a1≤u1≤b1 内.

ParametricRegion[{{f1,…,fn},cond},…]

约束参数以满足条件 cond.

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Details and Options Details and Options
范例  
基本范例  
范围  
纯参数化描述  
混合隐式和参数化描述  
参见
相关指南
历史
按以下格式引用：

ParametricRegion

ParametricRegion[{f1,…,fn},{u1,…,um}]

表示对于参数 ui∈，由点 {f1,…,fn} 给出的 中的区域.

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

约束参数位于区间 a1≤u1≤b1 内.

ParametricRegion[{{f1,…,fn},cond},…]

约束参数以满足条件 cond.

更多信息

范例

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基本范例  (2)

以 ParametricRegion 指定一个圆：

Wolfram Language code: ℛ = ParametricRegion[{Cos[θ], Sin[θ]}, {{θ, 0, 2π}}]
Wolfram Language code: Region[ℛ]

指定一个圆盘：

Wolfram Language code: Region[ParametricRegion[{r Cos[θ], r Sin[θ]}, {{θ, 0, 2π}, {r, 0, 1}}]]

两条抛物线之间的区域：

Wolfram Language code: ℛ = ParametricRegion[{{s, (1 + t) s ^ 2 - t}, -1 ≤ s ≤ 1 && 0 ≤ t ≤ 1}, {s, t}];
Wolfram Language code: Region[ℛ]

范围  (20)

纯参数化描述  (10)

用参数描述的区域：

Wolfram Language code: ℛ = ParametricRegion[{{s ^ 2t ^ 2, s t ^ 3}, -1 ≤ s ≤ 1 && -1 ≤ t ≤ 1}, {s, t}];
Wolfram Language code: Region[ℛ]

嵌入维度：

Wolfram Language code: RegionEmbeddingDimension[ℛ]

几何维度：

Wolfram Language code: RegionDimension[ℛ]

判断点是否为区域的成员：

Wolfram Language code: ℛ = ParametricRegion[{{s ^ 2, s ^ 3}, -1 ≤ s ≤ 1}, {s}];
Wolfram Language code: RegionMember[ℛ, {1, 1}]
Wolfram Language code: RegionMember[ℛ, {2, 2}]

获取点为成员的条件：

Wolfram Language code: RegionMember[ℛ, {x, y}]

面积和形心：

Wolfram Language code: ℛ = ParametricRegion[{{s ^ 2t ^ 2, s t ^ 3}, -1 ≤ s ≤ 1 && -1 ≤ t ≤ 1}, {s, t}];
Wolfram Language code: RegionMeasure[ℛ]
Wolfram Language code: RegionCentroid[ℛ]

到一个点的距离：

Wolfram Language code: ℛ = ParametricRegion[{{t, s t ^ 2}, 0 ≤ s ≤ 1}, {s, t}];
Wolfram Language code: RegionDistance[ℛ, {1, 2}]
Wolfram Language code: RegionDistance[ℛ, {2, 1}]

到一个点的有符号距离：

Wolfram Language code: ℛ = ParametricRegion[{{t, s t ^ 2}, 0 ≤ s ≤ 1}, {s, t}];
Wolfram Language code: SignedRegionDistance[ℛ, {1, 2}]
Wolfram Language code: SignedRegionDistance[ℛ, {2, 1}]

区域中最近的点：

Wolfram Language code: ℛ = ParametricRegion[{{t, s (1 - t ^ 2)}, -1 ≤ s ≤ 1 && -1 ≤ t ≤ 1}, {s, t}];
Wolfram Language code: RegionNearest[ℛ, {2, 1}]
Wolfram Language code: N[%]
Wolfram Language code: RegionNearest[ℛ, {1, 0}]

最近的点：

Wolfram Language code: pts = Table[2{Cos[k 2 π / 16], Sin[k 2π / 16]}, {k, 0., 15}];
Wolfram Language code: npts = RegionNearest[ℛ, pts];
Wolfram Language code: Show[{BoundaryDiscretizeRegion[ℛ], Graphics[{{Dashed, Line[Transpose[{pts, npts}]]}, {Red, Point[npts]}, {Blue, Point[pts]}}]}, PlotRange -> All]

抛物线以上的点集不是有界的：

Wolfram Language code: ℛ = ParametricRegion[{{t, s + t ^ 2}, s ≥ 0}, {s, t}];
Wolfram Language code: BoundedRegionQ[ℛ]

在隐式定义的区域上积分：

Wolfram Language code: ℛ = ParametricRegion[{{t, s (1 - t ^ 2)}, -1 ≤ s ≤ 1 && -1 ≤ t ≤ 1}, {s, t}];
Wolfram Language code: Integrate[x ^ 2 + y ^ 2, {x, y}∈ℛ]

在隐式定义的区域上优化：

Wolfram Language code: ℛ = ParametricRegion[{{t, s (1 - t ^ 2)}, -1 ≤ s ≤ 1 && -1 ≤ t ≤ 1}, {s, t}];
Wolfram Language code: MinValue[{x y - x, {x, y}∈ℛ}, {x, y}]

在隐式定义的区域中求解方程：

Wolfram Language code: ℛ = ParametricRegion[{{s ^ 2, s ^ 3}, -1 ≤ s ≤ 1}, {s}];
Wolfram Language code: Reduce[x^2 + y^2 == 1 && {x, y}∈ℛ, {x, y}]

混合隐式和参数化描述  (10)

以混合隐式和参数化描述给出的区域：

Wolfram Language code: ℛ = ParametricRegion[{{s, s t}, s ^ 2 + t ^ 2 ≤ 1}, {s, t}];
Wolfram Language code: Region[ℛ]

嵌入维度：

Wolfram Language code: RegionEmbeddingDimension[ℛ]

几何维度：

Wolfram Language code: RegionDimension[ℛ]

判断点是否为区域的成员：

Wolfram Language code: ℛ = ParametricRegion[{{s ^ 2, t ^ 3}, s t == 1}, {s, t}];
Wolfram Language code: {RegionMember[ℛ, {1, 1}], RegionMember[ℛ, {2, 2}]}

获取点为成员的条件：

Wolfram Language code: RegionMember[ℛ, {x, y}]

面积和形心：

Wolfram Language code: ℛ = ParametricRegion[{{s, s t}, s ^ 2 + t ^ 2 ≤ 1}, {s, t}];
Wolfram Language code: RegionMeasure[ℛ]
Wolfram Language code: RegionCentroid[ℛ]

到一个点的距离：

Wolfram Language code: ℛ = ParametricRegion[{{s + t, s - t}, s ≤ t ^ 2}, {s, t}];
Wolfram Language code: RegionDistance[ℛ, {1, 1}]
Wolfram Language code: RegionDistance[ℛ, {-1, 0}]

到一个点的有符号距离：

Wolfram Language code: ℛ = ParametricRegion[{{s + t, s - t}, s ≤ t ^ 2}, {s, t}];
Wolfram Language code: SignedRegionDistance[ℛ, {1, 1}]
Wolfram Language code: SignedRegionDistance[ℛ, {-1, 0}]

区域中最近的点：

Wolfram Language code: ℛ = ParametricRegion[{{s, s t}, s ^ 2 + t ^ 2 ≤ 1}, {s, t}];
Wolfram Language code: RegionNearest[ℛ, {0, 1}]//RootReduce
Wolfram Language code: N[%]
Wolfram Language code: RegionNearest[ℛ, {0, 0}]

最近的点：

Wolfram Language code: pts = Table[2{Cos[k 2 π / 16], Sin[k 2π / 16]}, {k, 0., 15}];
Wolfram Language code: npts = RegionNearest[ℛ, pts];
Wolfram Language code: Show[{BoundaryDiscretizeRegion[ℛ], Graphics[{{Dashed, Line[Transpose[{pts, npts}]]}, {Red, Point[npts]}, {Blue, Point[pts]}}]}, PlotRange -> All]

ℛ 是有界的，但不是凸的：

Wolfram Language code: ℛ = ParametricRegion[{{s, s t}, s ^ 2 + t ^ 2 ≤ 1}, {s, t}];
Wolfram Language code: BoundedRegionQ[ℛ]

在 ℛ 上积分：

Wolfram Language code: ℛ = ParametricRegion[{{s, s t}, s ^ 2 + t ^ 2 ≤ 1}, {s, t}];
Wolfram Language code: Integrate[x ^ 2 + y ^ 2, {x, y}∈ℛ]

在 ℛ 上优化：

Wolfram Language code: ℛ = ParametricRegion[{{s, s t}, s ^ 2 + t ^ 2 ≤ 1}, {s, t}];
Wolfram Language code: MinValue[{x y - x, {x, y}∈ℛ}, {x, y}]//RootReduce

在 ℛ 中求解方程和不等式：

Wolfram Language code: ℛ = ParametricRegion[{{s + t, s - t}, s ≤ t ^ 2}, {s, t}];
Wolfram Language code: Reduce[x^2 + y^2 == 1 && x ≥ y && {x, y}∈ℛ, {x, y}]
Wolfram Research (2014)，ParametricRegion，Wolfram 语言函数，https://reference.wolfram.com/language/ref/ParametricRegion.html.

文本

Wolfram Research (2014)，ParametricRegion，Wolfram 语言函数，https://reference.wolfram.com/language/ref/ParametricRegion.html.

CMS

Wolfram 语言. 2014. "ParametricRegion." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/ParametricRegion.html.

APA

Wolfram 语言. (2014). ParametricRegion. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/ParametricRegion.html 年

BibTeX

@misc{reference.wolfram_2026_parametricregion, author="Wolfram Research", title="{ParametricRegion}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/ParametricRegion.html}", note=[Accessed: 11-October-2026]}

BibLaTeX

@online{reference.wolfram_2026_parametricregion, organization={Wolfram Research}, title={ParametricRegion}, year={2014}, url={https://reference.wolfram.com/language/ref/ParametricRegion.html}, note=[Accessed: 11-October-2026]}