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RegionConvert

RegionConvert

RegionConvert[reg,form]

converts the region representation reg to the specified form.

Details

RegionConvert will produce an expression equivalent to expr.
RegionConvert is typically used for comparing regions with different representations or to get a representation suitable for different functions, e.g. optimization expects implicit forms of region constraints, whereas integration over a region expects the parametric form of a region.
The following forms can be used for converting between region representations:
"Implicit" implicit representation

"Mesh" mesh representation

"Parametric" parametric representation

"Spline" spline representation

Examples

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Basic Examples (2)

Convert the unit disk to the implicit form:

Wolfram Language code: RegionConvert[Disk[], "Implicit"]

Parametric form:

Wolfram Language code: RegionConvert[Disk[], "Parametric"]

Convert a cube to the implicit form:

Wolfram Language code: RegionConvert[Cube[], "Implicit"]

Parametric form:

Wolfram Language code: RegionConvert[Cube[], "Parametric"]

Scope (11)

Special Regions (5)

Convert Rectangle to the implicit form:

Wolfram Language code: RegionConvert[Rectangle[{1, 2}, {10, 8}], "Implicit"]

Parametric form:

Wolfram Language code: RegionConvert[Rectangle[{1, 2}, {10, 8}], "Parametric"]

Visualize it:

Wolfram Language code: Region[Rectangle[{1, 2}, {10, 8}]]

Convert DiskSegment to the implicit form:

Wolfram Language code: RegionConvert[DiskSegment[{0, 0}, 1, {0, 3Pi / 2}], "Implicit"]

Parametric form:

Wolfram Language code: RegionConvert[DiskSegment[{0, 0}, 1, {0, 3Pi / 2}], "Parametric"]

Visualize it:

Wolfram Language code: Region[DiskSegment[{0, 0}, 1, {0, 3Pi / 2}]]

Convert CapsuleShape to the implicit form:

Wolfram Language code: RegionConvert[CapsuleShape[], "Implicit"]

Parametric form:

Wolfram Language code: RegionConvert[CapsuleShape[], "Parametric"]

Visualize it:

Wolfram Language code: Region[CapsuleShape[]]

Convert Hyperplane to the implicit form:

Wolfram Language code: RegionConvert[Hyperplane[{-1, -1, 1}, {1, 2, 3}], "Implicit"]

Parametric form:

Wolfram Language code: RegionConvert[Hyperplane[{-1, -1, 1}, {1, 2, 3}], "Parametric"]

Visualize it:

Wolfram Language code: Region[Hyperplane[{-1, -1, 1}, {1, 2, 3}]]

Represent Sphere as a B-spline surface:

Wolfram Language code: bs = RegionConvert[Sphere[], "Spline"]

Visualize it, along with the control points:

Wolfram Language code:

cpts = First[bs];
Show[Graphics3D[{PointSize[Medium], Red, Map[Point, cpts], Gray, Line[cpts], Line[Transpose[cpts]]}], Graphics3D[bs]]

Formula Regions (2)

Convert an ImplicitRegion object to the implicit form:

Wolfram Language code: ℛ = ImplicitRegion[(x - 5)^2 + (y - 1)^2 ≤ 1, {x, y}];

Wolfram Language code: RegionConvert[ℛ, "Implicit"]

Convert a ParametricRegion object to the implicit form:

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

Wolfram Language code: RegionConvert[ℛ, "Implicit"]

Mesh Regions (2)

Convert a MeshRegion object to the implicit form:

Wolfram Language code: ℛ = ConvexHullMesh[RandomReal[1, {50, 2}]]

Wolfram Language code: RegionConvert[ℛ, "Implicit"]

A mesh region with disconnected components:

Wolfram Language code: ℛ = MeshRegion[{{0, 0}, {1, 0}, {0, 1}, {1, 1}, {2, 1}, {2, 0}}, Polygon[{{1, 2, 3}, {4, 5, 6}}]]

Wolfram Language code: RegionConvert[ℛ, "Implicit"]

Convert a BoundaryMeshRegion object to the implicit form:

Wolfram Language code:

ℛ = BoundaryMeshRegion[{{0, 0, 0}, {4, 0, 0}, {0, 4, 0}, {0, 0, 4}}, Polygon[{{1, 2, 3}, {1, 2, 4}, {2, 3, 4}, {1, 3, 4}}]]

Wolfram Language code: RegionConvert[ℛ, "Implicit"]

A boundary mesh region with disconnected components:

Wolfram Language code:

ℛ = BoundaryMeshRegion[{{0, -1}, {-1, 0}, {0, 0}, {1, 0}, {0, 1}, {-1 / 2, 1 / 2}, {-1, 1 / 2}, {-1 / 2, 1}}, Line[{1, 3, 2, 1}], Line[{3, 4, 5, 3}], Line[{6, 7, 8, 6}]]

Wolfram Language code: RegionConvert[ℛ, "Implicit"]

Derived Regions (2)

Convert a BooleanRegion object to the implicit form:

Wolfram Language code: ℛ = BooleanRegion[Or, {Disk[{1, 1}, 1], Disk[{0, 0}, 1]}];

Wolfram Language code: RegionConvert[ℛ, "Implicit"]

Visualize it:

Wolfram Language code: Region[%]

Convert a TransformedRegion object to the implicit form:

Wolfram Language code: ℛ = TransformedRegion[Rectangle[], RotationTransform[π / 4]];

Wolfram Language code: RegionConvert[ℛ, "Implicit"]

Visualize it:

Wolfram Language code: Region[%]

Applications (3)

Compute the area of an annulus using its implicit form:

Wolfram Language code: RegionConvert[Annulus[], "Implicit"]

Wolfram Language code: NIntegrate[Boole[(1/4) ≤ x ^ 2 + y ^ 2 ≤ 1], {x, -1, 1}, {y, -1, 1}]

Compare:

Wolfram Language code: N[Area[Annulus[]]]

Compute the volume of a ball using its implicit form:

Wolfram Language code: RegionConvert[Ball[], "Implicit"]

Wolfram Language code: NIntegrate[Boole[x^2 + y^2 + z^2 ≤ 1], {x, -1, 1}, {y, -1, 1}, {z, -1, 1}]

Compare:

Wolfram Language code: N[Volume[Ball[]]]

Find the projection of an implicit region on the plane in :

Wolfram Language code: ℛ = RegionConvert[Icosahedron[], "Implicit"];

Wolfram Language code: prj = Reduce[Subscript[∃, z]{x, y, z}∈ℛ, Reals];

Visualize the projection:

Wolfram Language code: RegionPlot[prj, {x, -1, 1}, {y, -1, 1}]

Properties & Relations (3)

Use CanonicalizePolygon to convert a region representation to a polygon form:

Wolfram Language code: CanonicalizePolygon[Triangle[]]

Use CanonicalizePolyhedron to convert a region representation to a polyhedron form:

Wolfram Language code: CanonicalizePolyhedron[Dodecahedron[]]

Use DiscretizeRegion to convert a region into a mesh:

Wolfram Language code: DiscretizeRegion[Triangle[], MaxCellMeasure -> Infinity]

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