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 .

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.