# TransformedRegion

TransformedRegion[reg,f]

represents the transformed region , where reg is a region and f is a function.

# Details and Options • TransformedRegion is also known as the image of a region.
• # Examples

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## Basic Examples(2)

A rotated rectangle:

A disk transformed by :

## Scope(24)

### Special Regions(10)

Some transformed regions are computed explicitly:

Visualize the transformation:

A linear-fractional transformation of the unit ball:

Visualize the transformation:

Translate a unit Disk:

Point membership test:

Conditions for point membership:

Shear a unit Rectangle:

Compute the RegionBounds:

Rotate a standard Triangle:

Region remains constant and bounded:

Scale a Circle:

Compute the ArcLength:

A disk transformed by a nonlinear transformation :

Express the same point mapping using Indexed:

Compute the RegionMeasure:

Map points from a 3D Cuboid into 2D by a nonlinear transformation :

Geometric and embedding dimension:

Compute the RegionCentroid:

Visualize:

Signed distance from a point:

Rotate a 3D Cuboid:

Distance from a point:

Nearest point in the region:

Nearest points:

Integrate over a rotated unit cube:

Optimize:

Solve equations:

### Formula Regions(6)

Shear a ParametricRegion:

Compute the ArcLength:

Nearest point on the region from a given point:

A rotated ParametricRegion:

Compute the Area:

Compute the RegionBounds:

Shear an ImplicitRegion:

Region remains unbounded:

Point membership test:

Conditions for point membership:

Scale an ImplicitRegion:

Compute the Volume:

Integrate over the region:

Optimize:

Solve equations:

An ImplicitRegion transformed by a nonlinear transformation :

Express the same point mapping using Indexed:

Compute the RegionMeasure:

Map points from a 3D ball into 2D by a nonlinear transformation :

Geometric and embedding dimension:

Compute the RegionCentroid:

Visualize:

### Mesh Regions(3)

Rotate a BoundaryMeshRegion:

Transformed region is still BoundaryMeshRegionQ:

Point membership test:

Compute the RegionCentroid:

Visualize it:

Compute the Area:

Integrate over the region:

Shear a MeshRegion:

Transformed region is still MeshRegionQ:

Compute the Volume:

Compute the RegionBounds:

RegionDistance from a point:

Integrate over the region:

Scale a lower-dimensional MeshRegion:

Compute the ArcLength:

Nearest point on the region from a given point:

### Derived Regions(5)

Transform a TransformedRegion:

Compute the Volume:

Integrate over the region:

Optimize:

Solve equations:

Transform a RegionDifference:

Compute the Area:

Compute the RegionBounds:

SignedRegionDistance from a given point:

Transform a RegionBoundary:

Compute the ArcLength:

Nearest point on the region from a given point:

Transform a RegionProduct:

Compute the Volume:

Compute the RegionBounds:

RegionDistance from a point:

Transform a RegionUnion by a nonlinear transformation :

Express the same point mapping using Indexed:

Compute the RegionBounds:

## Applications(2)

Any triangle is an affine transformation of the standard triangle:

The transformation is given by , where :

Compare original and transformed unit triangle:

Find the perspective transformation of a unit Cuboid with center :

Visualize the region:

Compute the Volume: