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

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

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]:

1. A developable surface can be mapped isometrically onto a plane.
2. Isometric surfaces have the same Gaussian curvature at corresponding points.
3. Corresponding curves on isometric surfaces have the same geodesic curvature at corresponding points.
4. 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.
5. 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

The principal curvatures reduce to

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 iso-parametric 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,

and hence we have . Therefore, the mean curvature (3.47) reduces to

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 iso-parametric 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 non-generic 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

From (9.81), on a developable surface. Hence from the second equation of (3.27) we have

Since is a unit vector, we also have

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 iso-parametric line. Therefore, for a given , (9.83) becomes

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

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

If the surface is expressed in a piecewise polynomial form such as a B-spline 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

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

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

provided that . Here is obtained by using the inverse function theorem [76] (see Sect. 9.3) and it can be shown that in general

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

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

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.

Footnotes

... points^9.5

A developable surface cannot possess spherical umbilics since one of the principal curvatures is always zero.