Graphics Reference
In-Depth Information
Figure 9.29.
A tube surface.
C
It follows from Lemma 9.12.1 that canal surfaces can also be thought of as a union
of circles with centers on a curve. Note that since both S(t) and
S
are tangent on
S(t) «
S
, they have the same normals along their intersection. Therefore in the special
case of a tube surface, a characteristic circle is the boundary of a disk that intersects
the surface orthogonally. This leads to yet another view of tube surfaces, namely, as
the boundary of a solid that is obtained by sweeping a disk of constant radius orthog-
onally along a curve
C
. See Figure 9.29.
Before we state the next theorem, recall from Section 9.9 that the centers of cur-
vatures at a point
p
of a surface
S
lie along the normal line to the surface at that point.
These centers lie on segments in the normal line whose endpoints were called the
focal points of the surface
S
, and the set of these was called the surface of centers (or
the evolute).
9.12.2. Theorem.
(1) The characteristic circles of a canal surface are lines of curvature for the
surface.
(2) The center curve for a canal surface consists of centers of curvature for the
surface, so that one component of its surface of centers is a curve.
(3) Every surface with the property that one component of its surface of centers
is a curve is a canal surface.
Proof.
See [Gray98].
9.12.3. Theorem.
Let
S
be a canal surface with radius function r(t) and center curve
g : [a,b] Æ
R
3
whose tangent vectors g¢(t) have unit length and whose curvature is
nonzero. Then
S
admits a parameterization of the form
[
]
2
()
=
()
+
()
-¢
() ()
±-¢
()
(
()
+
()
)
fq g
t
,
t rt rtTt
1
rt
-
cos
q
Nt
sin
q
Bt
,
where (T(t),N(t),B(t)) is the Frenet frame of the curve g(t).
Proof.
See [Gray98].
If the radius of the spheres or disks for a tube surface is small enough we get an
immersed surface. We can relate the curvature of the center curve and the Gauss cur-
vature of the surface.
