 |
(11.107) |
is a rational unit normal of the tangent plane
and 
is a rational distance function from the origin. The
rest of the discussions are analogous to the curve case.
A developable surface has a constant
tangent plane along a generator. Therefore its tangent plane depends
on only one parameter, say 
. In other words, a developable surface
can be considered as the envelope of a one parameter family of planes
. The cross product of the normal vectors of the two planes
and
provides a vector of the generator at
parameter
. An explicit representation of rational developable
surfaces with rational offsets has also been given in
[324].
Lü [240] studied the rationality of offsets of quadrics. The key idea is to transform the problem of rational offsets of quadrics to a simple problem on the rationality of a cubic algebraic surface and use existing results in algebraic geometry [362]. He showed that the offsets of paraboloids, ellipsoids and hyperboloids can be rationally parametrized, while cylinders and cones except for parabolic cylinders, cylinders of revolution and cones of revolution do not possess any rational offset.
Pottmann et al. [327] proved that offsets of a nondevelopable rational ruled surface in the whole space always admit a rational parametrization. Even though the offsets to ruled surfaces are rational in the whole space where they are defined, the offset patch to a rational patch may not be expressible as a rational patch. Therefore, further research is needed for applying this technique to a finite patch.
Peternell and Pottmann [308] construct 
surfaces from
arbitrary rational surfaces with the aid of a geometric transformation
which describes a change between two models of Laguerre geometry. The
two fundamental elements of Laguerre geometry are oriented planes and
cycles. A cycle represents an oriented sphere or a point which is a
degenerate sphere with zero radius. The orientation of the fundamental
elements is determined by a unit normal vector field or equivalently
by a signed radius for spheres. An oriented sphere and an oriented
plane are said to be in oriented contact, if they are tangent to each
other and their unit normals coincide at the point of
contact. Laguerre geometry studies properties which are invariant
under Laguerre transformations. If we consider a surface as an
envelope of its oriented tangent planes, a dilatation, which is a
Laguerre transformation that adds a constant 
to the signed
radius of each cycle without moving its center, maps the surface onto
its offset at distance 
.