I am interested in studying the class surfaces defined by:

Take an arbitrary parametric curve
f: {0..1} > ℝ3
. 
Choose an arbitrary radio function
r: {0..1} > ℝ
. 
A point
P
inℝ3
is part of this surface if, for somet
,dist (f (t), p) = r (t)
.
Where dist
It is the usual vector distance function. dist (a, b) = (b  a) · (b  a)
.
I will call that type of surface a tube
for lack of a better name. Given a tuple (f, r)
representing a tube and a point P
in ℝ3
, there is a function D ((f, r), p)
which returns the minimum distance between P
and any point in (f, r)
?