4 Surfaces of Gaussian curvature zero
Classification of surfaces of Gaussian curvature zero (Gauss flat, two-dimensional Riemann manifolds) in
a two-dimensional Euclidean space, ruled surfaces, developable surfaces.
While in the first chapter we discuss the mapping of a left surface (two-dimensional Riemann manifold)
to a right surface (two-dimensional Riemann manifold), the second chapter specializes the right surface
to be a plane. In contrast, the third chapter answers the question of how to parameterize a surface (in
general, a Riemann manifold) in order to cover all points of such differentiable manifolds completely by
an atlas. Special attention is paid to the question of a minimal atlas. Here, we fill the gap between the
first and the second chapter. In particular, we introduce a special ruled surface of Gaussian curvature
zero which can be developed to a plane, a cylinder, a cone, or a “tangent developable”. Such Gauss flat
two-dimensional Riemann manifolds are fundamental for the classification of the right surface assumed
to be developable. All following chapters are based upon this classification scheme. First, we clarify
the notion of a ruled surface. Second, we specialize to developable surfaces, in short, “developables”.
The text is only explanatory and rich of illustrations. All proofs are referred to the literature.
4-1 Ruled surfaces
Ruled surfaces (circular cone of the sphere as a ruled surface, helicoid as a ruled surface, one-sheeted
hyperboloid of revolution as a ruled surface, directrix).
Let us make familiar with a special surface, usually called ruled surface. First, we introduce a curve
x(U ) ∈ {R
3 , I 3 } in a three-dimensional Euclidean space, called the directrix of the surface. U is the
parameter of the curve. Second, a ruler is moving along the directrix, generating the ruling of the
surface. Alternatively, we may say that a ruled surface results from the motion of a straight line
in space. Movements of this kind of surfaces or segments are found in many physical, in particular,
mechanical applications. For instance, the motion of a robot arm generates a ruled surface. Example 4.1
together with Fig. 4.1 illustrates such generators of a ruled surface, here the circular cone of the
sphere S
2
R . In contrast, Fig. 4.2 presents the helicoid and the one-sheeted hyberboloid of revolution as
alternative examples of a ruled surface.
Example 4.1 (Circular cone C
2
R cos Φ 0
of the sphere S
2
R ).
Let us construct a circular cone of the sphere S
2
R as a ruled surface. First, we choose the parallel
circle, also called small circle, of the parameterized sphere as the reference curve or directrix. Second,
we attach locally to any point of the directrix a vector field which is generated by a ruler moving
along the reference curve. Consult Box 4.1 and Fig. 4.1 for a more detailed analysis. In terms of
spherical coordinates {Λ, Φ, R}, we parameterize the “position vector” X(Λ, Φ, R) with respect to an
orthonormal frame of reference
E 1 , E 2 , E 3
O
, spanning a three-dimensional Euclidean space E
3 ,
attached to the origin O, which is the center of the sphere S
2
R of radius R.
C Λ , C Φ
Λ, Φ
is the local
frame of reference, i. e. Cartan’s moving frame (“rep´ ere mobile”), attached to a point {Λ, Φ} ∈ S
2
R . As
the reference curve, we have chosen the parallel circle Φ 0 = constant, namely the directrix x(U ), where
U = Λ is the parameter of the reference curve. The generator or the ruler of the surface is the vector
field Y (U ) := C Φ (U ), the unit vector which is normal to C Λ (Λ, Φ 0 ), directed towards North. The
linear manifold V Y (U ), also called the bundle of straight lines, is forming the circular cone C
2
R cos Φ 0
of radius R cos Φ 0 as soon as the ruler moves around the parallel circle. Finally, we have gained the
parameterized ruled surface X(U, V ). Its typical matrix of the metric, G, has been computed.
End of Example.
Additionally, let us more precisely define a ruled surface in Definition 4.1, which follows after Box 4.1
summarizing the vector definitions of Example 4.1.
Classification of surfaces of Gaussian curvature zero (Gauss flat, two-dimensional Riemann manifolds) in
a two-dimensional Euclidean space, ruled surfaces, developable surfaces.
While in the first chapter we discuss the mapping of a left surface (two-dimensional Riemann manifold)
to a right surface (two-dimensional Riemann manifold), the second chapter specializes the right surface
to be a plane. In contrast, the third chapter answers the question of how to parameterize a surface (in
general, a Riemann manifold) in order to cover all points of such differentiable manifolds completely by
an atlas. Special attention is paid to the question of a minimal atlas. Here, we fill the gap between the
first and the second chapter. In particular, we introduce a special ruled surface of Gaussian curvature
zero which can be developed to a plane, a cylinder, a cone, or a “tangent developable”. Such Gauss flat
two-dimensional Riemann manifolds are fundamental for the classification of the right surface assumed
to be developable. All following chapters are based upon this classification scheme. First, we clarify
the notion of a ruled surface. Second, we specialize to developable surfaces, in short, “developables”.
The text is only explanatory and rich of illustrations. All proofs are referred to the literature.
4-1 Ruled surfaces
Ruled surfaces (circular cone of the sphere as a ruled surface, helicoid as a ruled surface, one-sheeted
hyperboloid of revolution as a ruled surface, directrix).
Let us make familiar with a special surface, usually called ruled surface. First, we introduce a curve
x(U ) ∈ {R
3 , I 3 } in a three-dimensional Euclidean space, called the directrix of the surface. U is the
parameter of the curve. Second, a ruler is moving along the directrix, generating the ruling of the
surface. Alternatively, we may say that a ruled surface results from the motion of a straight line
in space. Movements of this kind of surfaces or segments are found in many physical, in particular,
mechanical applications. For instance, the motion of a robot arm generates a ruled surface. Example 4.1
together with Fig. 4.1 illustrates such generators of a ruled surface, here the circular cone of the
sphere S
2
R . In contrast, Fig. 4.2 presents the helicoid and the one-sheeted hyberboloid of revolution as
alternative examples of a ruled surface.
Example 4.1 (Circular cone C
2
R cos Φ 0
of the sphere S
2
R ).
Let us construct a circular cone of the sphere S
2
R as a ruled surface. First, we choose the parallel
circle, also called small circle, of the parameterized sphere as the reference curve or directrix. Second,
we attach locally to any point of the directrix a vector field which is generated by a ruler moving
along the reference curve. Consult Box 4.1 and Fig. 4.1 for a more detailed analysis. In terms of
spherical coordinates {Λ, Φ, R}, we parameterize the “position vector” X(Λ, Φ, R) with respect to an
orthonormal frame of reference
E 1 , E 2 , E 3
O
, spanning a three-dimensional Euclidean space E
3 ,
attached to the origin O, which is the center of the sphere S
2
R of radius R.
C Λ , C Φ
Λ, Φ
is the local
frame of reference, i. e. Cartan’s moving frame (“rep´ ere mobile”), attached to a point {Λ, Φ} ∈ S
2
R . As
the reference curve, we have chosen the parallel circle Φ 0 = constant, namely the directrix x(U ), where
U = Λ is the parameter of the reference curve. The generator or the ruler of the surface is the vector
field Y (U ) := C Φ (U ), the unit vector which is normal to C Λ (Λ, Φ 0 ), directed towards North. The
linear manifold V Y (U ), also called the bundle of straight lines, is forming the circular cone C
2
R cos Φ 0
of radius R cos Φ 0 as soon as the ruler moves around the parallel circle. Finally, we have gained the
parameterized ruled surface X(U, V ). Its typical matrix of the metric, G, has been computed.
End of Example.
Additionally, let us more precisely define a ruled surface in Definition 4.1, which follows after Box 4.1
summarizing the vector definitions of Example 4.1.