156 4 Surfaces of Gaussian curvature zero
Definition 4.1 (Ruled surface).
A surface is called ruled surface if there exists a parameterization of the continuity class C
2 of type
X(U, V ) = x(U ) + V Y (U ), where x(U ) is a differentiable curve and Y (U ) is a vector field along the
curve x(U ) which vanishes nowhere.
End of Definition.
The matrix of the metric, G, associated with a ruled surface is a typical element of this kind. Compare
with Box 4.2, where we have computed the matrix G.
Box 4.2 (The matrix of the metric of a ruled surface).
G 11 = E = D U x
2 +V
2 D U Y
2 = D U X (U, V )
2 ,
(4.10)
G 12 = G 21 = F = D U x(U ) Y (U ) = D U X (U, V ) D V X (U, V ) ,
(4.11)
G 22 = G = Y (U )
2 = D V X (U, V )
2 ,
(4.12)
G =
"
E F
F G
#
=
"
D U x
2 +V
2 D U Y
2 D U x(U ) Y (U )
D U x(U ) Y (U )
Y (U )
2
#
.
(4.13)
4-2 Developable surfaces
Developable surfaces (equivalence theorem for ruled surfaces, Gauss flat surfaces, tangent developable:
developable helicoid.)
A ruled surface is called developable if it can be locally mapped to the plane, preserving the metric of
the surface and the generating lines. One of the lines that lies in the plane and afterwards strips of
the surface is developed on both sides of the plane, preserving both angles and lengths.
Theorem 4.2 (Equivalence theorem for ruled surfaces).
For a ruled surface, the following conditions are equivalent. (i) The surface is developable. (ii) The
surface is Gauss flat: k = 0. (iii) Along each of the straight lines, the surface normales are parallel.
End of Theorem.
A ruled surface which satisfies one of the conditions (i), (ii), or (iii) is also called a torse or a developable.
Every surface element without planar points which is Gauss flat (k = 0) is a ruled surface.
Theorem 4.3 (Torse, developable).
An open and dense subset of every torse consists of (i) planes, (ii) cylinders, (iii) cones, and (iv)
tangent developables, namely ruled surfaces for which the vector Y is tangent to the directrix x.
End of Theorem.
A detailed proof of Theorem 4.3 is given by W. S. Massey (1962) as well as by W. Kuehnel (2002,
pp. 86–89). Here, we illustrate Gauss flat surfaces of type (i) plane, (ii) cylinder, (iii) cone, and (iv)
tangent developable (“developable helicoid”) in Figs. 4.3–4.6. In contrast, Fig. 4.7 illustrates a Gauss
flat surface which is not a ruled surface based upon two segments of a cone.
Definition 4.1 (Ruled surface).
A surface is called ruled surface if there exists a parameterization of the continuity class C
2 of type
X(U, V ) = x(U ) + V Y (U ), where x(U ) is a differentiable curve and Y (U ) is a vector field along the
curve x(U ) which vanishes nowhere.
End of Definition.
The matrix of the metric, G, associated with a ruled surface is a typical element of this kind. Compare
with Box 4.2, where we have computed the matrix G.
Box 4.2 (The matrix of the metric of a ruled surface).
G 11 = E = D U x
2 +V
2 D U Y
2 = D U X (U, V )
2 ,
(4.10)
G 12 = G 21 = F = D U x(U ) Y (U ) = D U X (U, V ) D V X (U, V ) ,
(4.11)
G 22 = G = Y (U )
2 = D V X (U, V )
2 ,
(4.12)
G =
"
E F
F G
#
=
"
D U x
2 +V
2 D U Y
2 D U x(U ) Y (U )
D U x(U ) Y (U )
Y (U )
2
#
.
(4.13)
4-2 Developable surfaces
Developable surfaces (equivalence theorem for ruled surfaces, Gauss flat surfaces, tangent developable:
developable helicoid.)
A ruled surface is called developable if it can be locally mapped to the plane, preserving the metric of
the surface and the generating lines. One of the lines that lies in the plane and afterwards strips of
the surface is developed on both sides of the plane, preserving both angles and lengths.
Theorem 4.2 (Equivalence theorem for ruled surfaces).
For a ruled surface, the following conditions are equivalent. (i) The surface is developable. (ii) The
surface is Gauss flat: k = 0. (iii) Along each of the straight lines, the surface normales are parallel.
End of Theorem.
A ruled surface which satisfies one of the conditions (i), (ii), or (iii) is also called a torse or a developable.
Every surface element without planar points which is Gauss flat (k = 0) is a ruled surface.
Theorem 4.3 (Torse, developable).
An open and dense subset of every torse consists of (i) planes, (ii) cylinders, (iii) cones, and (iv)
tangent developables, namely ruled surfaces for which the vector Y is tangent to the directrix x.
End of Theorem.
A detailed proof of Theorem 4.3 is given by W. S. Massey (1962) as well as by W. Kuehnel (2002,
pp. 86–89). Here, we illustrate Gauss flat surfaces of type (i) plane, (ii) cylinder, (iii) cone, and (iv)
tangent developable (“developable helicoid”) in Figs. 4.3–4.6. In contrast, Fig. 4.7 illustrates a Gauss
flat surface which is not a ruled surface based upon two segments of a cone.