Function Repository Resource:

# InversionTransform

Generate a geometric transformation function for inversion about a point

Contributed by: Robert Nachbar
 ResourceFunction["InversionTransform"][p] gives a TransformationFunction that represents an inversion through the point p.

## Details

In an inversion, all points of a geometric object are projected through the point of inversion, and out to the same distance on the opposite side.
ResourceFunction["InversionTransform"] gives a TransformationFunction that can be applied to vectors.
ResourceFunction["InversionTransform"] works in any number of dimensions. The dimensions of the point p give the dimensions of the TransformationFunction.

## Examples

### Basic Examples (1)

Invert through the point {1,2}:

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### Scope (3)

Inversion transform for a symbolic point {u,v,w}:

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Apply an inversion to a 2D shape:

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Apply an inversion to a 3D shape:

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### Applications (4)

Invert a graphic:

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Invert a 2D image:

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Inversion transform of a 3D image with respect to the origin:

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Invert the geometry and stereochemistry of a chiral molecule:

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Show the two enantiomers:

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### Properties and Relations (3)

The inversion transformation is an isometric transform—that is, it preserves distances:

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The inversion transformation is its own inverse:

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The determinant of the transformation matrix is –1 for odd-order dimensions and +1 for even-order dimensions:

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Robert Nachbar

## Version History

• 1.0.0 – 25 April 2022