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PolygonDecomposition[poly]

decomposes the polygon poly into a disjoint union of simpler polygons.

PolygonDecomposition[poly,"type"]

decomposes into polygons of the specified "type".

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Basic Uses  
Convex Decomposition  
Simple Decomposition  
Triangle Decomposition  
Properties & Relations  
See Also
Related Guides
History
Cite this Page

PolygonDecomposition

PolygonDecomposition[poly]

decomposes the polygon poly into a disjoint union of simpler polygons.

PolygonDecomposition[poly,"type"]

decomposes into polygons of the specified "type".

Details

  • PolygonDecomposition is also known as tessellation, triangulation or partition.
  • PolygonDecomposition is typically used to represent a polygon as a union of simpler objects for which a problem may be easier to solve.
  • PolygonDecomposition gives a Polygon consisting of a union of polygons with disjoint interiors, but boundaries may overlap.
  • Possible "type" specifications:
  • "Simple"simple polygons
    "Convex"convex polygons
    "Triangle"triangles

Examples

open all close all

Basic Examples  (2)

Decompose a Polygon into a union of simpler polygons:

Wolfram Language code: 𝒫 = DynamicModule[«3»];
Wolfram Language code: PolygonDecomposition[𝒫]

Triangulate a rectangle:

Wolfram Language code: PolygonDecomposition[Rectangle[], "Triangle"]
Wolfram Language code: Graphics[{EdgeForm[Gray], %}]

Scope  (15)

Basic Uses  (5)

Decompose polygons:

Wolfram Language code: 𝒫 = DynamicModule[«3»];
Wolfram Language code: PolygonDecomposition[𝒫]
Wolfram Language code: Graphics[{EdgeForm[Gray], %}]

Decompose into polygons of a specific type:

Wolfram Language code: 𝒫 = Polygon[{{0.776550531261063, 0.8718175094875975}, {0.21915439771041, 0.5135066427407438}, {0.20458351808417796, 0.26349912396990893}, {0.45039826006328854, 0.5050901720622678}, {0.2565027386056602, 0.02008003822175297}, {0.6444665281388617, 0.15148472373175426}}, {1, 2, 3, 4, 5, 6}];
Wolfram Language code: PolygonDecomposition[𝒫, "Convex"]

PolygonDecomposition works on polygonal regions:

Wolfram Language code: PolygonDecomposition[Parallelogram[], "Triangle"]
Wolfram Language code: PolygonDecomposition[ImplicitRegion[0 ≤ x ≤ 1 && 0 ≤ y ≤ 1, {x, y}]]

PolygonDecomposition works on polygons with GeoPosition:

Wolfram Language code: Polygon[GeoPosition[{{{40.083441, -88.235716}, {40.083607, -88.257488}, {40.082603, -88.257149}, {40.076136999999996, -88.25740499999999}, {40.076178, -88.270888}, {40.076516, -88.271558}, {40.083686, -88.271512}, {40.083659999999995, -88.267046}, ... 33323}, {40.098112, -88.228687}, {40.095216, -88.228627}, {40.095179, -88.238547}, {40.094480999999995, -88.238546}, {40.094508999999995, -88.23267}, {40.094106, -88.232556}, {40.090666999999996, -88.232477}, {40.090741, -88.235745}}}]];
Wolfram Language code: PolygonDecomposition[%]//Short

Polygons with GeoGridPosition:

Wolfram Language code: Polygon[GeoGridPosition[{{{-0.9950503945490105, 1.2366760550756015}, {-0.9952074890903578, 1.2369207053693891}, {-0.9952196732768064, 1.2369073327446167}, {-0.9953160063787643, 1.236848436956935}, {-0.9954141759436825, 1.2369993898475449}, {-0. ... 197645333103}, {-0.9949098578570917, 1.2368130881428654}, {-0.9948663952535768, 1.2367477711687371}, {-0.9948714472169538, 1.2367426500757825}, {-0.9949211061652593, 1.2367089232486177}, {-0.9949439717990124, 1.236746107097628}}}, "Bonne"]];
Wolfram Language code: PolygonDecomposition[%, "Triangle"]//Short

Convex Decomposition  (4)

Decompose polygons:

Wolfram Language code: 𝒫 = DynamicModule[«3»];
Wolfram Language code: PolygonDecomposition[𝒫, "Convex"]
Wolfram Language code: Graphics[{EdgeForm[Gray], %}]

Polygons with holes:

Wolfram Language code: 𝒫 = DynamicModule[«3»];
Wolfram Language code: PolygonDecomposition[𝒫, "Convex"]
Wolfram Language code: Graphics[{EdgeForm[Gray], %}]

Three-dimensional polygons:

Wolfram Language code: 𝒫 = DynamicModule[«3»];
Wolfram Language code: PolygonDecomposition[𝒫, "Convex"]
Wolfram Language code: Graphics3D[{LightGray, EdgeForm[{Thick, Gray}], %}]

-dimensional polygons:

Wolfram Language code: 𝒫 = DynamicModule[«3»];
Wolfram Language code: PolygonDecomposition[𝒫, "Convex"]

Simple Decomposition  (2)

Decompose polygons:

Wolfram Language code: 𝒫 = DynamicModule[«3»];
Wolfram Language code: PolygonDecomposition[𝒫, "Simple"]
Wolfram Language code: Graphics[{EdgeForm[Gray], %}]

-dimensional polygons:

Wolfram Language code: 𝒫 = DynamicModule[«3»];
Wolfram Language code: PolygonDecomposition[𝒫, "Simple"]

Triangle Decomposition  (4)

Decompose polygons:

Wolfram Language code: 𝒫 = DynamicModule[«3»];
Wolfram Language code: PolygonDecomposition[𝒫, "Triangle"]
Wolfram Language code: Graphics[{EdgeForm[Gray], %}]

Polygons with holes:

Wolfram Language code: 𝒫 = DynamicModule[«3»];
Wolfram Language code: PolygonDecomposition[𝒫, "Triangle"]
Wolfram Language code: Graphics[{EdgeForm[Gray], %}]

Three-dimensional polygons:

Wolfram Language code: 𝒫 = DynamicModule[«3»];
Wolfram Language code: PolygonDecomposition[𝒫, "Triangle"]
Wolfram Language code: Graphics3D[{LightGray, EdgeForm[{Thick, Gray}], %}]

-dimensional polygons:

Wolfram Language code: 𝒫 = DynamicModule[«3»];
Wolfram Language code: PolygonDecomposition[𝒫, "Triangle"]

Properties & Relations  (2)

Use SimplePolygonQ to test whether a polygon is simple:

Wolfram Language code: 𝒫 = Polygon[{{0, 0}, {1, 0}, {0.2, 1}, {0.9, 1}}];
Wolfram Language code: SimplePolygonQ[𝒫]

Decompose into simple polygons:

Wolfram Language code: PolygonDecomposition[𝒫, "Simple"]

Use ConvexPolygonQ to test whether a polygon is convex:

Wolfram Language code: 𝒫 = Polygon[{{0, 0}, {3, 0}, {3, 1}, {1, 1}, {1, 2}, {3, 2}, {3, 3}, {0, 3}}];
Wolfram Language code: ConvexPolygonQ[𝒫]

Decompose into convex polygons:

Wolfram Language code: PolygonDecomposition[𝒫, "Convex"]
Wolfram Research (2019), PolygonDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/PolygonDecomposition.html.

Text

Wolfram Research (2019), PolygonDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/PolygonDecomposition.html.

CMS

Wolfram Language. 2019. "PolygonDecomposition." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PolygonDecomposition.html.

APA

Wolfram Language. (2019). PolygonDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PolygonDecomposition.html

BibTeX

@misc{reference.wolfram_2026_polygondecomposition, author="Wolfram Research", title="{PolygonDecomposition}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/PolygonDecomposition.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_polygondecomposition, organization={Wolfram Research}, title={PolygonDecomposition}, year={2019}, url={https://reference.wolfram.com/language/ref/PolygonDecomposition.html}, note=[Accessed: 12-June-2026]}