Homeomorphism of a Disk Mapping the Origin to Another Interior Point
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]
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 .
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.[less]
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
 Mathematics Stack Exchange. "Conformal Automorphism of Unit Disk That Interchanges Two Given Points." (Mar 4, 2022). math.stackexchange.com/a/3093167.
 J. M. Lee, Introduction to Topological Manifolds, 2nd ed., New York: Springer, 2011.
 Mathematics Stack Exchange. "A Homeomorphism of Fixing the Boundary?" (Mar 4, 2022). math.stackexchange.com/a/1517119.
 Mathematics Stack Exchange. "  Is a Strongly Locally Homogeneous Space." (Mar 4, 2022). math.stackexchange.com/a/4066088.
 M. Eisenberg, Topology, New York: Holt, Rinehart and Winston, 1974.
 E. W. Weisstein. "Linear Fractional Transformation" from MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/LinearFractionalTransformation.html (Wolfram MathWorld).