Geography Reference
In-Depth Information
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
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