In brief. A randomly generated Moebius transform is applied to the unit sphere; the resulting four-dimensional object is rotated freely (in four dimensions) and projected onto three-dimensional space. By dragging the mouse, you can rotate the projected object in three dimensions.
In detail. Since quaternion multiplication is noncommutative, the four quaternions A,B,C,D determine eight different quaternion Moebius transforms:
Each of these transforms will give quantitatively and qualitatively different results. By clicking on the type button above, you can cycle through each of the above eight Moebius transformations. Clicking the reset button will randomly generate another set of values for the coefficients A,B,C,D.
By clicking on the sphere/cube button, you can select the object to apply the quaternion Moebius transform to: either a unit sphere, or a unit cube.
By clicking on the rotation on/off button, you can stop the rotation of the transformed sphere/cube in four-dimensional space; however, you can still rotate the projected object in three-dimensional space by dragging the mouse over the object.
The wire/solid frame/ribbon button toggles between a wire frame representation of the quaternion blob, a solid representation, and a semi-solid representation.
The scroll bar at the bottom of the above window can be used to control the size of the sphere/cube: sliding the bar to the left decreases the size, and sliding to the right increases the size. Since a quaternion Moebius transformation is nonlinear (in general), increasing or decreasing the radius of the sphere/cube before the transformation is applied will not simply scale the resulting object.
Question: as indicated in the Quaternion Moebius transforms document, the transformations numbered 4 and 5 in the above list are special in that they arise naturally from considering quaternion projective transformations. Indeed, the reader will note that in the above applet these two transformations always yield "squashed" spheres (when the sphere/cube button is set to sphere). Now, a sphere in four dimensional space will appear squashed when projected onto three dimensions; but are the quaternion blobs in these cases actually the projections of spheres in four dimensions? For a complex Moebius transformation, a circle is mapped to a circle (or a straight line in some degenerate cases). So the question is this: is the image of a sphere under a quaternion Moebius transformation necessarily a sphere? [I do not know the answer to this, but I suspect it is actually true.]