If we consider ruled surfaces of the projective 3-space as a one parameter family of lines, then they appear in the well-known KLEIN-model of lines in the projective 3-space as curves of a hyperquadric in the projective 5-space. The osculating spaces of such a curve are represented in the projective 3-space by spaces of linear complexes. Those points of a generator e of the ruled surface, in which the tangent bundles are in the same time complex line bundles in the accompanying osculating line complex of the ruled surface along e, are called the LIE-points of e. The LIE-points fulfil two (real or imaginary conjugate) curves on the ruled surface called the LIE-curves. The support of the osculating-3-space of the ruled surface along a regular non-torsal generator e are two, one or zero straight lines in the osculating regulus. If thes straight lines exist, one calls them the flecnode tangents of the ruled surface. On a hyperbolic ruled surface build the points of contact of the flecnode tangents two projective distinguished curves called the flecnode curves. In this work we present the different methods of treating these curves in the history, and we give a new explicit calculation of the flecnode points and the LIE-points depending on the basis of a PLÜCKER-coordinates representation of the ruled surface. In addition we study the questions that appears by considering the LIE-curves of a ruled surface to form a pair of BERTRAND curves for which this ruled surface is the surface of common main normals. For example, the question about ruled surfaces, whose LIE-curves are orthogonal to the generators will be answered here. / Regelflächen des projektiven 3-Raums erscheinen, als (eindimensionalen) Geradenmengen aufgefasst, im bekannten KLEINschen Punktmodell der Geradenmenge vom projektiven 3-Raum als Kurven einer Hyperquadrik in einem projektiven 5-Raum. Die Schmiegräume einer solchen Kurve werden im projektiven 3-Raum durch Räume linearer Komplexe repräsentiert. Diejenigen Punkte einer Erzeugende e der Regelfläche, in denen die Tangentenbüschel gleichzeitig auch Komplexgeradenbüschel im begleitenden Schmiegkomplex von e sind, heißen LIE-Punkte von e. Die LIE-Punkte erfüllen zwei (reelle oder konjugiert imaginäre) Kurvenzüge auf der Regelfläche, die LIE-Kurven. Die Träger des Schmieg-3-Raums der Regelfläche längs einer reguläre nichttorsalen Erzeugende e sind zwei, eine oder null Geraden im Schmiegregulus. Sofern diese Geraden existieren, nennt man sie die Fleknodaltangenten der Regelfläche. Auf hyperbolischen Regelflächen bilden die Berührpunkte der Fleknodaltangenten zwei projektiv ausgezeichnete Kurven, die Fleknodalkurven. In der vorliegenden Arbeit stellen wir die unterschiedlichen Behandelungen diesen ausgezeichneten Kurven in der Geschichte dar, und geben wir eine neue explizite Berechnung von den Fleknodal- bzw. LIE-Punkte auf der Basis einer PLÜCKER-Koordinaten-Darstellung der Regelfläche. Außerdem untersuchen wir die Fragestellungen, die man bekommt, wenn man versucht, dass das paarweise auftreten der LIE-Kurven irgendwie in Analogie zum klassischen euklidischen BERTRAND-Kurvenpaar zu stellen. Z.B. lässt sich die Frage nach Regelflächen, deren LIE-Kurven Orthogonaltrajektorien der Erzeugenden sind, hier beantwortet.

If we consider ruled surfaces of the projective 3-space as a one parameter family of lines, then they appear in the well-known KLEIN-model of lines in the projective 3-space as curves of a hyperquadric in the projective 5-space. The osculating spaces of such a curve are represented in the projective 3-space by spaces of linear complexes. Those points of a generator e of the ruled surface, in which the tangent bundles are in the same time complex line bundles in the accompanying osculating line complex of the ruled surface along e, are called the LIE-points of e. The LIE-points fulfil two (real or imaginary conjugate) curves on the ruled surface called the LIE-curves. The support of the osculating-3-space of the ruled surface along a regular non-torsal generator e are two, one or zero straight lines in the osculating regulus. If thes straight lines exist, one calls them the flecnode tangents of the ruled surface. On a hyperbolic ruled surface build the points of contact of the flecnode tangents two projective distinguished curves called the flecnode curves. In this work we present the different methods of treating these curves in the history, and we give a new explicit calculation of the flecnode points and the LIE-points depending on the basis of a PLÜCKER-coordinates representation of the ruled surface. In addition we study the questions that appears by considering the LIE-curves of a ruled surface to form a pair of BERTRAND curves for which this ruled surface is the surface of common main normals. For example, the question about ruled surfaces, whose LIE-curves are orthogonal to the generators will be answered here. / Regelflächen des projektiven 3-Raums erscheinen, als (eindimensionalen) Geradenmengen aufgefasst, im bekannten KLEINschen Punktmodell der Geradenmenge vom projektiven 3-Raum als Kurven einer Hyperquadrik in einem projektiven 5-Raum. Die Schmiegräume einer solchen Kurve werden im projektiven 3-Raum durch Räume linearer Komplexe repräsentiert. Diejenigen Punkte einer Erzeugende e der Regelfläche, in denen die Tangentenbüschel gleichzeitig auch Komplexgeradenbüschel im begleitenden Schmiegkomplex von e sind, heißen LIE-Punkte von e. Die LIE-Punkte erfüllen zwei (reelle oder konjugiert imaginäre) Kurvenzüge auf der Regelfläche, die LIE-Kurven. Die Träger des Schmieg-3-Raums der Regelfläche längs einer reguläre nichttorsalen Erzeugende e sind zwei, eine oder null Geraden im Schmiegregulus. Sofern diese Geraden existieren, nennt man sie die Fleknodaltangenten der Regelfläche. Auf hyperbolischen Regelflächen bilden die Berührpunkte der Fleknodaltangenten zwei projektiv ausgezeichnete Kurven, die Fleknodalkurven. In der vorliegenden Arbeit stellen wir die unterschiedlichen Behandelungen diesen ausgezeichneten Kurven in der Geschichte dar, und geben wir eine neue explizite Berechnung von den Fleknodal- bzw. LIE-Punkte auf der Basis einer PLÜCKER-Koordinaten-Darstellung der Regelfläche. Außerdem untersuchen wir die Fragestellungen, die man bekommt, wenn man versucht, dass das paarweise auftreten der LIE-Kurven irgendwie in Analogie zum klassischen euklidischen BERTRAND-Kurvenpaar zu stellen. Z.B. lässt sich die Frage nach Regelflächen, deren LIE-Kurven Orthogonaltrajektorien der Erzeugenden sind, hier beantwortet.

Regelflächen, das sind durch die Bewegung von Geraden erzeugte Flächen, haben für den Betonleichtbau unter den Gesichtspunkten Statik und Herstellung herausragende Eigenschaften: Auch wenn sie doppelt gekrümmt sind, können sie geradlinig bewehrt oder vorgespannt werden. Außerdem kann die Schalung beispielsweise durch Heißdrahtschneiden aus Polystyrol-Hartschaum gewonnen werden. In gängigen CAD-Systemen ist die Klasse der Regelflächen bislang nicht angemessen repräsentiert und steht deshalb für die Bauteilgestaltung nur eingeschränkt zur Verfügung. Liniengeometrie für den Leichtbau liefert nun ein mathematisches Modell, das Regelflächen und auf sie wirkende Kräfte abbildet, und entwickelt daraus Formfindungswerkzeuge, die in einer vertrauten Entwurfsumgebung das Prinzip form follows force unterstützen. [Aus. Einführung] / Ruled surfaces, which are surfaces created by the movement of straight lines, have outstanding properties for lightweight concrete construction from the viewpoints of statics and production: even if they are double-curved, they can be reinforced or prestressed in a rectilinear fashion. In addition, the formwork can be obtained ef ciently from rigid polystyrene foam by hot wire cutting, for example. In current CAD systems, the class of ruled surfaces has not yet been adequately implemented and is therefore only available to a limited extent for component design. This project Line Geometry for Lightweight Structures provides a mathematical model that represents ruled surfaces and the forces acting on them, and uses this to develop form finding tools that support the principle of form follows force in a familiar design environment. [Off: Introduction]

In a Minkowski three dimensional space, whose metric is based on a strictly convex and centrally symmetric unit ball , we deal with ruled surfaces Φ in the sense of E. Kruppa. This means that we have to look for Minkowski analogues of the classical differential invariants of ruled surfaces in a Euclidean space. Here, at first – after an introduction to concepts of a Minkowski space, like semi-orthogonalities and a semi-inner-product based on the so-called cosine-Minkowski function - we construct an orthogonal 3D moving frame using Birkhoff’s left-orthogonality. This moving frame is canonically connected to ruled surfaces: beginning with the generator direction and the asymptotic plane of this generator g we complete this flag to a frame using the left-orthogonality defined by ; ( is described either by its supporting function or a parameter representation). The plane left-orthogonal to the asymptotic plane through generator g(t) is called Minkowski central plane and touches Φ in the striction point s(t) of g(t). Thus the moving frame defines the Minkowski striction curve S of the considered ruled surface Φ similar to the Euclidean case. The coefficients occurring in the Minkowski analogues to Frenet-Serret formulae of the moving frame of Φ in a Minkowski space are called “M-curvatures” and “M-torsions”. Here we essentially make use of the semi-inner product and the sine-Minkowski and cosine-Minkowski functions. Furthermore we define a covariant differentiation in a Minkowski 3-space using a new vector called “deformation vector” and locally measuring the deviation of the Minkowski space from a Euclidean space. With this covariant differentiation it is possible to declare an “M-geodesicc parallelity” and to show that the vector field of the generators of a skew ruled surface Φ is an M-geodesic parallel field along its Minkowski striction curve s. Finally we also define the Pirondini set of ruled surfaces to a given surface Φ. The surfaces of such a set have the M-striction curve and the strip of M-central planes in common

In a Minkowski three dimensional space, whose metric is based on a strictly convex and centrally symmetric unit ball , we deal with ruled surfaces Φ in the sense of E. Kruppa. This means that we have to look for Minkowski analogues of the classical differential invariants of ruled surfaces in a Euclidean space. Here, at first – after an introduction to concepts of a Minkowski space, like semi-orthogonalities and a semi-inner-product based on the so-called cosine-Minkowski function - we construct an orthogonal 3D moving frame using Birkhoff’s left-orthogonality. This moving frame is canonically connected to ruled surfaces: beginning with the generator direction and the asymptotic plane of this generator g we complete this flag to a frame using the left-orthogonality defined by ; ( is described either by its supporting function or a parameter representation). The plane left-orthogonal to the asymptotic plane through generator g(t) is called Minkowski central plane and touches Φ in the striction point s(t) of g(t). Thus the moving frame defines the Minkowski striction curve S of the considered ruled surface Φ similar to the Euclidean case. The coefficients occurring in the Minkowski analogues to Frenet-Serret formulae of the moving frame of Φ in a Minkowski space are called “M-curvatures” and “M-torsions”. Here we essentially make use of the semi-inner product and the sine-Minkowski and cosine-Minkowski functions. Furthermore we define a covariant differentiation in a Minkowski 3-space using a new vector called “deformation vector” and locally measuring the deviation of the Minkowski space from a Euclidean space. With this covariant differentiation it is possible to declare an “M-geodesicc parallelity” and to show that the vector field of the generators of a skew ruled surface Φ is an M-geodesic parallel field along its Minkowski striction curve s. Finally we also define the Pirondini set of ruled surfaces to a given surface Φ. The surfaces of such a set have the M-striction curve and the strip of M-central planes in common