# RegionProduct

RegionProduct[reg1,reg2]

represents the Cartesian product of the regions reg1 and reg2.

RegionProduct[reg1,reg2,]

represents the Cartesian product of the regions reg1, reg2, .

# Details

• RegionProduct is also known as outer product region.
• RegionProduct[reg1,reg2] represents the region .
• The embedding dimension of the product region is the sum of embedding dimensions, and the geometric dimension is the sum of geometric dimensions.

# Examples

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## Basic Examples(3)

The product of two line segments:

Visualize it:

The product of a disk and a line segment:

Visualize it:

The product of two BoundaryMeshRegion objects:

## Scope(7)

A product of Disk and Line regions:

The RegionDimension is 3D, so compute its Volume:

Visualize it:

A product of a Polygon and a multiline:

Compute its Volume:

A product of a Ball and a Cuboid:

The dimension is 6:

Regions of dimension higher than 3 cannot be discretized, but can still be computed with:

### Formula Regions(3)

A product of an ImplicitRegion and a ParametricRegion in 1D:

Compute its Area:

Visualize it:

A product of two ImplicitRegion objects:

The region is unbounded, so its volume is infinite:

A product of two ParametricRegion objects:

Compute its Volume:

### Mesh Regions(4)

A product of two BoundaryMeshRegion objects in 1D:

The result is a MeshRegion, not a BoundaryMeshRegion:

A product of two MeshRegion objects in 1D:

Compute the Area:

A product of 1D and 2D BoundaryMeshRegion objects:

Compute the Volume:

A product of 1D and 2D MeshRegion objects:

Compute the Volume:

## Applications(2)

Define a tensor product mesh as the product of several 1D meshes:

Define a 1D mesh from a list of points:

Define a 2D tensor product mesh:

Define a 3D tensor product mesh:

Directly construct a MeshRegion representing a stage for the Cantor set. The set is defined by starting with the interval {0,1} and at each step removing the middle third:

The first few steps:

Use RegionProduct to produce Cantor dust:

Find the length of the Cantor set at each stage:

Guess the general formula:

Find the measure of the Cantor dust in 2D at each stage:

Find the measure of the Cantor dust in 3D at each stage:

## Properties & Relations(10)

The RegionEmbeddingDimension of a product is the sum of input embedding dimensions:

The RegionDimension of a product is the sum of input dimensions:

A RegionProduct of special regions is left unevaluated:

It is a region and can be used for computation:

A RegionProduct of formula regions is left unevaluated:

It is a region and can be used for computation:

A product of MeshRegion or BoundaryMeshRegion objects is itself a MeshRegion:

The RegionMeasure of a product is the product of the input measures:

The RegionCentroid of a product is the input centroids joined together:

A Rectangle is a product of two Line objects:

Show membership is equivalent:

A Cuboid is a product of three Line objects:

Show membership is equivalent:

A Cylinder is a product of a Disk and a Line:

Show membership is equivalent:

## Neat Examples(1)

Define Cantor dust-like regions as products of Cantor sets:

Products of Cantor sets in 2D:

Products of Cantor sets in 3D:

Wolfram Research (2014), RegionProduct, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionProduct.html.

#### Text

Wolfram Research (2014), RegionProduct, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionProduct.html.

#### BibTeX

@misc{reference.wolfram_2021_regionproduct, author="Wolfram Research", title="{RegionProduct}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/RegionProduct.html}", note=[Accessed: 01-August-2021 ]}

#### BibLaTeX

@online{reference.wolfram_2021_regionproduct, organization={Wolfram Research}, title={RegionProduct}, year={2014}, url={https://reference.wolfram.com/language/ref/RegionProduct.html}, note=[Accessed: 01-August-2021 ]}

#### CMS

Wolfram Language. 2014. "RegionProduct." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RegionProduct.html.

#### APA

Wolfram Language. (2014). RegionProduct. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RegionProduct.html