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
.
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