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Homeomorphism of a Disk Mapping the Origin to Another Interior Point

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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 .

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Drag the point to the desired location; that determines the homeomorphism . Drag the point around inside the disk to see the corresponding point given by the homeomorphism.

Similarly, select the button to see the locator for the point . Drag around inside the disk to see the corresponding point given by the inverse of the homeomorphism .

To say that is an homeomorphism of means that it is a one-to-one continuous function that maps onto and whose inverse function is also continuous.

For a point , the function maps the radial line segment from 0 to linearly onto the line segment from to . Such an is given by the formula , where denotes the Euclidean norm of , that is, or, what is the same thing, is the modulus of the complex number .

A consequence of what is shown here is that the open unit disk is homogeneous in the sense that given any two points in , there is a homeomorphism of mapping one of the points to the other. An alternative way to obtain the same result is to use a Möbius transformation, that is, a linear fractional transformation, as indicated in [1].

The method used here generalizes to any number of dimensions: given a point in the open unit -ball in -dimensional Euclidean space , there is a homeomorphism of the corresponding closed unit -disk that maps the origin to while keeping each point of the bounding -sphere fixed. Consequently, the open unit -ball is homogeneous.

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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).

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