Towards the curvature cuff of a canal surface

A canal surface is the envelope of a family of spheres,
whose centers move on a curve while their radii vary.
By means of a stereographic mapping, a sphere corresponds to
a point in 4-space. The canal surface then corresponds to a curve
in 4-space.
The paper gives insights into this mapping and presents
a construction of an osculating Dupin cyclide, which
provides the principal curvatures for each generating circle of
a curvature continuous canal surface.