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9.7.1 Differential geometry of developable surfaces
A ruled surface is a curved surface which can be generated by
the continuous motion of a straight line in space along a space curve
called a directrix.
This straight line is called a generator, or ruling, of the
surface. A
book by Pottmann and Wallner [330] studies line geometry
from the viewpoint of scientific computation and shows the interplay
between theory and applications.
Any point on a parametric ruled surface can be expressed as
(9.74)
where
is a
directrix or base curve of the ruled surface and
is a unit vector which gives the direction of the ruling
at each point on the directrix. Alternatively, the surface can be
represented as a ruling joining corresponding points on two space
curves. This is represented by
(9.75)
where
and
are directrices, as shown in
Fig. 9.10. The two representations are identical if
and
(9.76)
Figure 9.10:
A ruled surface
A developable surface is a special ruled surface which
has the same tangent plane at all points along a generator [13,222,120,32,326,252,329].
Since surface normals
are orthogonal to the tangent plane and the
tangent plane along a generator is constant,
all normal vectors along a generator are parallel.
This is shown in Fig. 9.11.
A developable surface has following differential geometry properties
[412]:
A developable surface can be mapped isometrically
onto a plane.
Isometric surfaces have the same Gaussian curvature
at corresponding points.
Corresponding curves on isometric surfaces
have the same geodesic curvature at corresponding points.
Every
isometric mapping is conformal; i.e. the angle of intersection of
every arbitrary pair of intersecting arcs on a developable surface is
the same as that of the corresponding inverse image in the plane at
the corresponding points.
A geodesic on a developable surface maps
to a straight line in the plane.
Figure 9.11:
A developable surface with its tangent plane along a ruling
A developable surface can be formed by bending or rolling a planar
surface without stretching or tearing; in other words, it can be
developed or unrolled isometrically onto a plane. Developable
surfaces are
also known as singly curved surfaces, since one of their principal
curvatures is zero. Developable
surfaces are widely used with materials that are not amenable to
stretching. Applications include the formation of ship hulls, ducts,
shoes, clothing and automobile parts such as upholstery, body panels
and windshields [120].
As indicated by Munchmeyer and Haw [281], a developable
surface can be shaped purely by rolling and should be fed to the
roller so that the direction of the zero principal curvature is
parallel to the rolls. However, when the sheet reaches a line of
inflection, it can no longer be fed into the roller in the same
direction because the direction of bending changes. Therefore, it is
beneficial for planning the fabrication process to determine the lines
of inflection prior to such a process.
Surface inflection of a developable surface was studied by Hoitsma
[172] who showed that a surface has an inflection
at a point
if and only if its mean curvature changes sign in the
neighborhood of
. Maekawa and Chalfant [251] further
extended this result and derived two theorems (Theorem 9.7.1 and 9.7.2).
Since the Gaussian curvature of a developable surface is zero
everywhere [412,76], the maximum and minimum
principal curvatures (3.49) and (3.50)
of a developable surface can be written as
(9.77)
The principal curvatures
reduce to
(9.78)
(9.79)
(9.80)
It is clear from (9.78) through (9.80)
that at least one of the principal curvatures is zero at each point on a
developable
surface, which agrees with the fact that the Gaussian curvature is
zero everywhere (see (3.61)).
in (9.78) and
in (9.80)
are termed the nonzero
principal curvature,
, where
.
In the following we establish some elementary differential geometry
properties of developable surfaces. We
assume that the developable surface is regular and the
isoparametric
line corresponds to the generator of the developable surface or, in
other words, the straight line ruling is in the
direction. With
this assumption,
, and hence the second
fundamental form coefficient (see (3.28))
vanishes. From (3.46), since
Gaussian curvature of a developable surface is zero,
(9.81)
and hence we have
. Therefore, the mean curvature
(3.47) reduces to
(9.82)
Recall that the nonzero principal curvature is given by
and since
,
and hence
become zero if and only
if
, otherwise
.
Next, we will show that the
parametric straight lines become
the lines of zero curvature. This can be seen from the fact that i)
the
isoparametric straight lines have zero normal curvature,
and ii) no other direction has zero normal curvature. The second fact
comes from Euler's theorem introduced in
Sect. 3.6. When
, (3.87) reduces to
, which becomes zero only when
or
, corresponding to the direction of
.
Similarly, when
,
only when
or
, corresponding to the direction of
.
Theorem 9.7.1.A developable surface does not possess
generic isolated flat points ^{9.5}
but rather may contain a line of nongeneric flat points along
a generator [251].
Proof:
From (9.79) and (9.82),
vanishes
at a flat point
where both principal curvatures are zero.
Therefore from the first equation of (3.27) we have
(9.83)
From (9.81),
on a developable surface.
Hence from the second equation of (3.27) we have
(9.84)
Since
is a unit vector, we also have
(9.85)
If
is not zero, then
from (9.83), (9.84)
and (9.85)
must be perpendicular to
,
and
. This is impossible because
is perpendicular to both
and
, and
is not parallel to
. Thus,
must
equal zero. For developable surfaces, the unit normal vector
is constant along a generator. Therefore, the rate of change of
the unit normal vector in the
direction must also be constant
along a generator. This leads us to the fact that
is not
only zero at
but also zero along the
isoparametric line. Therefore, for a given
,
(9.83) becomes
(9.86)
for
. Consequently,
the entire generator consists of a
line of flat points.
For a developable surface, the inflection line is a generator
which consists of a line of flat points and the nonzero principal
curvature changes sign. The inflection line can be detected by
finding
such that
where
is an arbitrary
constant between 0 and 1.
can be
written as
(9.87)
Since we are assuming a regular surface such that
, we only need to set the numerator of
(9.87) equal to zero. Thus,
. For a polynomial surface with degree
in the
direction this results in a univariate polynomial equation of degree
in
(9.88)
If the surface is
expressed in a piecewise polynomial form such as a Bspline
representation,
(9.88) must be applied to each polynomial
segment separately.
The univariate polynomial equation can be
robustly and efficiently solved by the Interval Projected Polyhedron
algorithm described in Chap. 4.
The local approximation (8.73) will now be applied to
developable surfaces.
Lemma 9.7.1.A developable surface is, in general, locally a parabolic cylinder except at
an inflection line, where it becomes a cubic cylinder, provided that
[251].
Proof: Let us consider an orthogonal Cartesian reference frame

attached to the surface
at an
arbitrary point
with
being
. We
choose unit vectors
,
and
at
as the directions
of
,
and
axes such that the
axis coincides with the
generator
, the
axis coincides with the surface normal vector and
the
axis is orthogonal to both axes. Therefore the local
coordinates
,
and
are given by
(9.89)
(9.90)
(9.91)
where subscript
denotes that the expressions are evaluated at
(
,
). In (9.89) through
(9.91) all terms involving
,
and
are evaluated at (
,
), so
is
the only term that is a function of
and
. If we consider
and
as functions of
and
, i.e.
and
,
then the height function
can be represented as a function of
and
through intermediate variables
and
, i.e.
.
The second fundamental form coefficient
in terms of the height
function
is given in (3.65) as
. Since
on a
developable surface (see (9.81)), we have
. Furthermore, all the second and higher order partial derivatives
with respect to
vanish, since the
axis corresponds to the
generator, which is linear in
. Thus
(8.73) reduces to
(9.92)
The second fundamental form coefficient
in terms of the height function is given in (3.65) as
.
Since
is zero along the entire generator line,
and its variation in the
direction
become zero at a line of inflection. Thus
(9.92) further reduces to
(9.93)
provided that
. Here
is obtained
by using the inverse function theorem [76] (see Sect.
9.3) and it can be shown that in general
(9.94)
From (9.92) and (9.93) it is apparent
that for small
, the quadratic term dominates except at an inflection
line where the surface will become locally a cubic cylinder.
When
becomes zero, the higher order partial derivatives must be
considered, and this is studied in the following.
A developable surface is said to have contact of order
with the
tangent plane along the generator if the Taylor expansion for
starts with terms of degree
. The ordinary inflection line
(see (9.93)) thus has
a contact of order
. If the tangent plane has contact of order
with the surface along a generator, a developable
surface may not look like a cubic cylinder at an inflection line.
If the developable surface has a contact of order
with the tangent
plane,
is zero or equivalently
is zero for
along the entire generator. Accordingly its variation
in
also vanishes; hence
for
along the
entire generator. Since all the second and higher order partial
derivatives with respect to
vanish, the Taylor expansion of the
height function along the higher order contact line reduces to
(9.95)
We can observe that for an even
the
height function
changes sign when
moves across the
inflection line, while for an odd
it maintains the same sign. In
other words, for an even
the height function passes through the
tangent plane along the inflection line, whereas for an odd
, it lies
entirely on one side of the tangent plane. Therefore inflection lines exist only for even
.
The order of contact can be detected by first solving
(9.88), and substituting the solution into
(9.96)
to find
such that (9.96) is zero for
but nonzero for
. The integer
found by this process gives
the order of contact (see (9.88),
(9.94)). A simple example for the higher order odd case is
given by
. Since
and
, we can
easily see that the line of flat points is located at
from
(9.88). And the order of contact is found to be
, since
and
. In this case the line of flat points is not an inflection
line, since
is odd.