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.