In brief. A randomly generated Moebius transform is applied to the a sphere, which is freely moving in three dimensions, and the resulting four-dimensional object is projected onto three-dimensional space. By dragging the mouse, you can rotate both the untransformed object (on the left), and the transformed/projected object in three dimensions (on the right).
In detail. The (quaternion) Moebius transformation is nonlinear. Thus if we consider the two objects formed by (1) applying the transform to a given object and (2) applying the transform to a translate or a scaling of the original object, the second object will not simply be a translation or a scaling of the first. This is illustrated above two windows: on the left, an object moves freely in three-dimensions, and on the right, the transform of the object is shown. Note that the transformed object is actually a four-dimensional object; what is shown is an orthogonal projection of the object onto three dimensions.
By clicking on the type button above, you can cycle through each of the eight different Moebius transformations. See the document Quaternion Moebius transforms for a description of these transformations.
The reset button will randomly generate another quaternion Moebius transformation.
The sphere/cube button selects the object to apply the quaternion Moebius transform to: either a sphere, or a cube.
The scroll bar at the bottom of the window can be used to control the size of the sphere/cube before the quaternion Moebius transformation is applied: sliding the bar to the left decreases the size, and sliding to the right increases the size.
The translation on/off button is used to pause the translation of the sphere/cube in three dimensional space. If if the motion is paused, you can still rotate both the untransformed and transformed object in three-dimensional space.
The rotation on/off button pauses and resumes the rotation of the transformed object in four dimensions before it is projected onto three dimensions.