A canal surface is the envelope of a one-parameter family of spheres. The centers follow a space curve 
, called the spine curve, and the radius 
may vary. For differentiable
and 
, the envelope equations
and 
, where 
, give
 |  |  |
(1)
|
 |  |  |
(2)
|
For fixed 
with 
, these are a sphere
and a plane perpendicular to the tangent
vector. If 
, they meet
in a nondegenerate circle with center 
and radius 
given by
 |  |  |
(3)
|
 |  |  |
(4)
|
This is the characteristic circle of the family. Equality gives a single point, while 
gives no real
intersection. The nondegeneracy condition on the circles does not by itself exclude
singularities or self-intersections of the whole envelope.
The illustrated family has spine curve and radius
 |  |  |
(5)
|
 |  |  |
(6)
|
where 
. At 
, the highlighted sphere
has radius 
, its characteristic plane
is 
, and the contact circle
has radius 
. The right panel shows
their intersection in the plane 
.
For constant radius, the characteristic circles lie in the normal planes of the spine curve and generate a tube. A straight spine gives a circular cylinder, and a circular spine gives a torus. Cyclides admit two such families of spheres.
When the center coordinates and squared radius are rational functions, implicitization
using a resultant to eliminate 
gives a polynomial condition
for the envelope. Extraneous factors can occur, so the
resultant alone need not be its defining equation (Dohm
and Zube 2009).
