# Homeomorphism of a Disk Mapping the Origin to Another Interior Point

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

This Demonstration shows the action of a homeomorphism of the closed unit disk in the plane that maps the origin 0 to a selected point in the open unit disk , while keeping each point on the boundary of fixed. It also shows the action of the inverse of as well as of the compositions and .

[more]
Contributed by: Murray Eisenberg (June 13)

With additional contributions by: Mark D. Normand

Open content licensed under CC BY-NC-SA

## Details

Snapshot 1: image under the homeomorphism of another point in the open disk for the same point seen in the Thumbnail image

Snapshot 2: image under of a point in the open disk but for a different given point

Snapshot 3: image under of the origin is the given point

Snapshot 4: image under of a point on the bounding circle is the same as

Snapshot 5: image under the inverse homeomorphism of a point in the open disk for a given point

Snapshot 6: image under of another point in the open disk for the same point

Snapshot 7: image under of a point in the open disk but for a different point

Snapshot 8: image under of the given point is the origin

Snapshot 9: image under of a point on the bounding circle is the same as

References

[1] Mathematics Stack Exchange. "Conformal Automorphism of Unit Disk That Interchanges Two Given Points." (Mar 4, 2022). math.stackexchange.com/a/3093167.

[2] J. M. Lee, *Introduction to Topological Manifolds*, 2nd ed., New York: Springer, 2011.

[3] Mathematics Stack Exchange. "A Homeomorphism of Fixing the Boundary?" (Mar 4, 2022). math.stackexchange.com/a/1517119.

[4] Mathematics Stack Exchange. " Is a Strongly Locally Homogeneous Space." (Mar 4, 2022). math.stackexchange.com/a/4066088.

[5] M. Eisenberg, *Topology*, New York: Holt, Rinehart and Winston, 1974.

[6] E. W. Weisstein. "Homeomorphism" from MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/Homeomorphism.html (Wolfram *MathWorld*).

[7] E. W. Weisstein. "Disk" from MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/Disk.html (Wolfram *MathWorld*).

[8] E. W. Weisstein. "Linear Fractional Transformation" from MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/LinearFractionalTransformation.html (Wolfram *MathWorld*).

## Snapshots

## Permanent Citation