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Fábio Nunes Da ,silva - One of the best experts on this subject based on the ideXlab platform.
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Superfícies regradas desenvolvíveis tipo tempo e tipo espaço no espaço de Minkowski
2013Co-Authors: Fábio Nunes Da ,silvaAbstract:Neste trabalho, baseado em [11], [13] e [7] estudamos superfícies regradas tipo espaço ou tipo tempo no espaço de Minkowski. Inicialmente, encontramos expressões para o triedro de Frenet de curvas tipo tempo, tipo espaço ou tipo luz. Mostramos que uma superfície regrada tipo tempo ou tipo espaço é desenvovível se, e somente se, o parâmetro de distribuição é nulo. Então, para o caso em que os vetores de Frenet da diretriz não são tipo luz, mostramos que a superfície regrada tipo espaço ou tipo tempo é desenvolvível se, e somente se, a diretriz é uma hélice. No caso em que algum dos vetores de Frenet da diretriz é tipo luz, mostramos que a superfície regrada tipo tempo ou tipo espaço é desenvolvível se, e somente se, a torção é constante. Estudamos casos especiais, nos quais a superfície regrada tipo tempo ou tipo espaço é gerada por retas que estão no plano osculador, ou no plano normal, ou no plano retificante, ou na direção de algum dos vetores do triedro de Frenet. _______________________________________________________________________________________ ABSTRACTIn thisWork, based in [11], [13] and [7] we study timelike and spacelike ruled surfaces in Minkowski space. Initially we find expressions for Frenet trihedron of lightlike, spacelike or timelike curves. We show that a timelike or spacelike ruled surface is developable if and only if the distribution parameter is null. Then for the case where some of the Frenet vectors of the directrix aren’t lightlike, we show that the timelike ou spacelike ruled surface is developable if and only if the directrix is helix. In the case where some of the Frenet vectors of the directrix is lightlike, we show that the timelike or spacelike ruled surface is developable if and only if the torsion is constant. We study special cases which the timelike or spacelike ruled surface is generated for straight line that are in Osculating Plane, or in normal Plane, or in rectifying Plane, or in the direction of some of the Frenet vectors
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Superfícies regradas desenvolvíveis tipo tempo e tipo espaço no espaço de Minkowski
2013Co-Authors: Fábio Nunes Da ,silvaAbstract:Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2013.Neste trabalho, baseado em [11], [13] e [7] estudamos superfícies regradas tipo espaço ou tipo tempo no espaço de Minkowski. Inicialmente, encontramos expressões para o triedro de Frenet de curvas tipo tempo, tipo espaço ou tipo luz. Mostramos que uma superfície regrada tipo tempo ou tipo espaço é desenvovível se, e somente se, o parâmetro de distribuição é nulo. Então, para o caso em que os vetores de Frenet da diretriz não são tipo luz, mostramos que a superfície regrada tipo espaço ou tipo tempo é desenvolvível se, e somente se, a diretriz é uma hélice. No caso em que algum dos vetores de Frenet da diretriz é tipo luz, mostramos que a superfície regrada tipo tempo ou tipo espaço é desenvolvível se, e somente se, a torção é constante. Estudamos casos especiais, nos quais a superfície regrada tipo tempo ou tipo espaço é gerada por retas que estão no plano osculador, ou no plano normal, ou no plano retificante, ou na direção de algum dos vetores do triedro de Frenet. _______________________________________________________________________________________ ABSTRACTIn thisWork, based in [11], [13] and [7] we study timelike and spacelike ruled surfaces in Minkowski space. Initially we find expressions for Frenet trihedron of lightlike, spacelike or timelike curves. We show that a timelike or spacelike ruled surface is developable if and only if the distribution parameter is null. Then for the case where some of the Frenet vectors of the directrix aren’t lightlike, we show that the timelike ou spacelike ruled surface is developable if and only if the directrix is helix. In the case where some of the Frenet vectors of the directrix is lightlike, we show that the timelike or spacelike ruled surface is developable if and only if the torsion is constant. We study special cases which the timelike or spacelike ruled surface is generated for straight line that are in Osculating Plane, or in normal Plane, or in rectifying Plane, or in the direction of some of the Frenet vectors
Ginoux Jean-marc - One of the best experts on this subject based on the ideXlab platform.
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Stabilité de Systèmes Dynamiques Chaotiques et Variétés Singulières
HAL CCSD, 2005Co-Authors: Ginoux Jean-marcAbstract:This work aims to study the stability of chaotic dynamical systems starting from the geometrical structure of their attractors of which a part is based on a manifold called slow manifold. To this end, a new approach based on certain aspects of the formalism of Mechanics and Differential Geometry was developed and led to a geometrical and kinematics interpretation of the evolution of the trajectory curves, integrals of these dynamical systems in the vicinity of the slow manifold, and allowed to study their stability.Mechanics allowed, with the use of the velocity and instantaneous acceleration vectors, located on a point of the trajectory curve, to discriminate the slow domain from the fast domain and to locate the position of the slow manifold inside the phase space.Certain notions of Differential Geometry like the expressions of curvature, torsion and that of the Osculating Plane provided an analytical equation of the slow manifold independent of the slow eigenvectors of the tangent linear system, therefore defined on a greater domain of the phase space.The slow manifold was then considered as the location of the points where the curvature of the trajectory curves, integrals of these dynamical systems, is minimal (in dimension two this minimum becomes equal to zero). The sign of torsion allowed: to characterize its attractivity, to discriminate the attractive part from the repulsive part of the slow manifold and, to rule on the stability of these trajectory curves.Thus, the presence in the phase space of an attractive slow manifold compelling the trajectory curve, integrals of the dynamic system to visit its vicinity allowed analyzing the attractor structure.This approach based on certain aspects of the formalism of Mechanics andDifferential Geometry and which was accompanied by the development of numerical programs made it possible to constitute a new tool for investigation of chaotic dynamical systems.Its application to models of reference like that of B. Van der Pol., L.O. Chua or of E.N. Lorenz allowed obtaining more directly and with precision the analytical equation of their slow manifold. Moreover, a detailed study of the predator-prey models like that of Rosenzweig-MacArthur or Hastings-Powell, led on the one hand to the determination of their slow manifold and on the other hand to the design of a new three-dimensional model of predator-prey type: theVolterra-Gause model of which chaotic attractor has the shape of a snailshell (chaotic snail shell).Ce mémoire a pour objectif d'étudier la stabilité de systèmes dynamiques chaotiques à partir de la structure géométrique de leurs attracteurs dont une partie s'appuie sur une variété appelée variété lente. Dans ce but, une nouvelle approche basée sur certains aspects du formalisme de la Mécanique du Point et de la Géométrie Différentielle a été développée et a conduit à une interprétation géométrique et cinématique de l'évolution des courbes trajectoires, intégrales de ces systèmes dynamiques au voisinage de la variété lente.L'utilisation du formalisme de la Mécanique du Point a permis, grâce à l'emploi des vecteurs, vitesse et accélération instantanées attachées à un point courant de la courbe trajectoire, de discriminer le domaine lent du domaine rapide et de situer la position de la variété lente à l'intérieur de l'espace des phases. Certaines notions de Géométrie Différentielle, comme la courbure, la torsion et le plan osculateur, ont fourni une équation analytique de la variété lente indépendante des vecteurs propres lents du système linéaire tangent, donc définie sur un plus grand domaine de l'espace des phases. La variété lente a alors été envisagée comme le lieu des points où la courbure des courbes trajectoires, intégrales de ces systèmes dynamiques, est minimum (en dimension deux ce minimum devient égal à zéro). Le signe de la torsion a permis, de caractériser son attractivité et, de discriminer la partie attractive de la partie répulsive de la variété lente et de statuer sur la stabilité de ces courbes trajectoires.Ainsi, la présence dans l'espace des phases d'une variété lente attractive qui contraint les courbes trajectoires, intégrales du système dynamique à visiter son voisinage permet d'étudier la structure de l'attracteur.Cette approche basée sur certains aspects du formalisme de la Mécanique du Point et de la Géométrie Différentielle et qui s'est accompagnée de l'élaboration de programmes numériques a permis de constituer un nouvel outil d'investigation des systèmes dynamiques chaotiques.Son application à des modèles de référence comme celui de B. Van der Pol, de L.O. Chua ou d'E.N. Lorenz a permis d'obtenir plus directement et avec précision l'équation analytique de leur variété lente. De plus, une étude détaillée des modèles de type prédateur-proie comme celui de Rosenzweig-MacArthur ou d'Hastings-Powell, a conduit d'une part à la détermination de leur variété lente et d'autre part à la conception d'un nouveau modèle de type prédateur-proie à trois espèces appelé Volterra-Gause dont l'attracteur chaotique a la forme d'un escargot (chaotic snail shell)
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Stabilité des systèmes dynamiques chaotiques et variétés singulières
2005Co-Authors: Ginoux Jean-marc, Rossetto Bruno, Jamet Jean-louisAbstract:Ce mémoire a pour objectif d'étudier la stabilité de systèmes dynamiques chaotiques à partir de la structure géométrique de leurs attracteurs dont une partie s'appuie sur une variété appelée variété lente. Dans ce but, une nouvelle approche basée sur certains aspects du formalisme de la Mécanique du Point et de la Géométrie Différentielle a été développée et a conduit à une interprétation géométrique et cinématique de l'évolution des courbes trajectoires, intégrales de ces systèmes dynamiques au voisinage de la variété lente. L'utilisation du formalisme de la Mécanique du Point a permis, grâce à l'emploi des vecteurs, vitesse et accélération instantanées attachées à un point courant de la courbe trajectoire, de discriminer le domaine lent du domaine rapide et de situer la position de la variété lente à l'intérieur de l'espace des phases. Certaines notions de Géométrie Différentielle, comme la courbure, la torsion et le plan osculateur, ont fourni une équation analytique de la variété lente indépendante des vecteurs propres lents du système linéaire tangent, donc définie sur un plus grand domaine de l'espace des phases. La variété lente a alors été envisagée comme le lieu des points où la courbure des courbes trajectoires, intégrales de ces systèmes dynamiques, est minimum (en dimension deux ce minimum devient égal à zéro). Le signe de la torsion a permis, de caractériser son attractivité et, de discriminer la partie attractive de la partie répulsive de la variété lente et de statuer sur la stabilité de ces courbes trajectoires. Ainsi, la présence dans l'espace des phases d'une variété lente attractive qui contraint les courbes trajectoires, intégrales du système dynamique à visiter son voisinage permet d'étudier la structure de l'attracteur. Cette approche basée sur certains aspects du formalisme de la Mécanique du Point et de la Géométrie Différentielle et qui s'est accompagnée de l'élaboration de programmes numériques a permis de constituer un nouvel outil d'investigation des systèmes dynamiques chaotiques. Son application à des modèles de référence comme celui de B. Van der Pol, de L.O. Chua ou d'E.N. Lorenz a permis d'obtenir plus directement et avec précision l'équation analytique de leur variété lente. De plus, une étude détaillée des modèles de type prédateur-proie comme celui de Rosenzweig-MacArthur ou d'Hastings-Powell, a conduit d'une part à la détermination de leur variété lente et d'autre part à la conception d'un nouveau modèle de type prédateur-proie à trois espèces appelé Volterra-Gause dont l'attracteur chaotique a la forme d'un escargot (chaotic snail shell).This work aims to study the stability of chaotic dynamical systems starting from the geometrical structure of their attractors of which a part is based on a manifold called slow manifold. To this end, a new approach based on certain aspects of the formalism of Mechanics and Differential Geometry was developed and led to a geometrical and kinematics interpretation of the evolution of the trajectory curves, integrals of these dynamical systems in the vicinity of the slow manifold, and allowed to study their stability. Mechanics allowed, with the use of the velocity and instantaneous acceleration vectors, located on a point of the trajectory curve, to discriminate the slow domain from the fast domain and to locate the position of the slow manifold inside the phase space. Certain notions of Differential Geometry like the expressions of curvature, torsion and that of the Osculating Plane provided an analytical equation of the slow manifold independent of the slow eigenvectors of the tangent linear system, therefore defined on a greater domain of the phase space. The slow manifold was then considered as the location of the points where the curvature of the trajectory curves, integrals of these dynamical systems, is minimal (in dimension two this minimum becomes equal to zero). The sign of torsion allowed: to characterize its attractivity, to discriminate the attractive part from the repulsive part of the slow manifold and, to rule on the stability of these trajectory curves. Thus, the presence in the phase space of an attractive slow manifold compelling the trajectory curve, integrals of the dynamic system to visit its vicinity allowed analyzing the attractor structure. This approach based on certain aspects of the formalism of Mechanics and Differential Geometry and which was accompanied by the development of numerical programs made it possible to constitute a new tool for investigation of chaotic dynamical systems. Its application to models of reference like that of B. Van der Pol., L.O. Chua or of E.N. Lorenz allowed obtaining more directly and with precision the analytical equation of their slow manifold. Moreover, a detailed study of the predator-prey models like that of Rosenzweig-MacArthur or Hastings-Powell, led on the one hand to the determination of their slow manifold and on the other hand to the design of a new three-dimensional model of predator-prey type: theVolterra-Gause model of which chaotic attractor has the shape of a snailshell (chaotic snail shell).TOULON-BU Centrale (830622101) / SudocSudocFranceF
Rida T Farouki - One of the best experts on this subject based on the ideXlab platform.
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Rotation–minimizing Osculating frames
'Elsevier BV', 2014Co-Authors: Rida T Farouki, C. Giannelli, M. L. Sampoli, A. SestiniAbstract:An orthonormal frame (f1,f2,f3) is rotation–minimizing with respect to fi if its angular velocity ω satisfies ω⋅fi≡0 — or, equivalently, the derivatives of fj and fk are both parallel to fi. The Frenet frame (t,p,b) along a space curve is rotation–minimizing with respect to the principal normal p, and in recent years adapted frames that are rotation–minimizing with respect to the tangent t have attracted much interest. This study is concerned with rotation–minimizing Osculating frames (f,g,b) incorporating the binormal b, and Osculating–Plane vectors f,g that have no rotation about b. These frame vectors may be defined through a rotation of t,p by an angle equal to minus the integral of curvature with respect to arc length. In aeronautical terms, the rotation–minimizing Osculating frame (RMOF) specifies yaw–free rigid–body motion along a curved path. For polynomial space curves possessing rational Frenet frames, the existence of rational RMOFs is investigated, and it is found that they must be of degree 7 at least. The RMOF is also employed to construct a novel type of ruled surface, with the property that its tangent Planes coincide with the Osculating Planes of a given space curve, and its rulings exhibit the least possible rate of rotation consistent with this constraint
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Rotation–minimizing conformal frames
2013Co-Authors: Rida T Farouki, C. Giannelli, M. L. Sampoli, A. SestiniAbstract:An orthonormal frame (f1,f2,f3) is rotation–minimizing with respect to fi if its angular velocity ω satisfies ω · fi ≡ 0 or, equivalently, the derivatives of fj, fk are both parallel to fi. The Frenet frame (t,p,b) is rotation–minimizing with respect to the principal normal p, and in recent years adapted frames that are rotation–minimizing with respect to the tangent t have attracted much interest. This study is concerned with conformal frames, that are rotation–minimizing with respect to the binormal b along a space curve. Such a frame (f,g,b) incorporates Osculating–Plane vectors f,g that have no rotation about b, and may be defined through a rotation of t,p by an amount equal to minus the integral of curvature with respect to arc length. In aeronautical terms, a rotation–minimizing conformal frame (RMCF) specifies “yaw–free ” rigid–body motion on a curved path. The existence of rational RMCFs on polynomial space curves with rational Frenet frames is investigated, and it is shown that they must be degree 7 at least. The RMCF is also employed to construct a novel type of ruled surface, characterized by tangent Planes that coincide with the Osculating Planes along a given space curve, and rulings that exhibit the least possible rate of rotation consistent with this constraint
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rational pythagorean hodograph space curves
Computer Aided Geometric Design, 2011Co-Authors: Rida T FaroukiAbstract:A method for constructing rational Pythagorean-hodograph (PH) curves in R^3 is proposed, based on prescribing a field of rational unit tangent vectors. This tangent field, together with its first derivative, defines the orientation of the curve Osculating Planes. Augmenting this orientation information with a rational support function, that specifies the distance of each Osculating Plane from the origin, then completely defines a one-parameter family of Osculating Planes, whose envelope is a developable ruled surface. The rational PH space curve is identified as the edge of regression (or cuspidal edge) of this developable surface. Such curves have rational parametric speed, and also rational adapted frames that satisfy the same conditions as polynomial PH curves in order to be rotation-minimizing with respect to the tangent. The key properties of such rational PH space curves are derived and illustrated by examples, and simple algorithms for their practical construction by geometric Hermite interpolation are also proposed.
Jamet Jean-louis - One of the best experts on this subject based on the ideXlab platform.
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Stabilité des systèmes dynamiques chaotiques et variétés singulières
2005Co-Authors: Ginoux Jean-marc, Rossetto Bruno, Jamet Jean-louisAbstract:Ce mémoire a pour objectif d'étudier la stabilité de systèmes dynamiques chaotiques à partir de la structure géométrique de leurs attracteurs dont une partie s'appuie sur une variété appelée variété lente. Dans ce but, une nouvelle approche basée sur certains aspects du formalisme de la Mécanique du Point et de la Géométrie Différentielle a été développée et a conduit à une interprétation géométrique et cinématique de l'évolution des courbes trajectoires, intégrales de ces systèmes dynamiques au voisinage de la variété lente. L'utilisation du formalisme de la Mécanique du Point a permis, grâce à l'emploi des vecteurs, vitesse et accélération instantanées attachées à un point courant de la courbe trajectoire, de discriminer le domaine lent du domaine rapide et de situer la position de la variété lente à l'intérieur de l'espace des phases. Certaines notions de Géométrie Différentielle, comme la courbure, la torsion et le plan osculateur, ont fourni une équation analytique de la variété lente indépendante des vecteurs propres lents du système linéaire tangent, donc définie sur un plus grand domaine de l'espace des phases. La variété lente a alors été envisagée comme le lieu des points où la courbure des courbes trajectoires, intégrales de ces systèmes dynamiques, est minimum (en dimension deux ce minimum devient égal à zéro). Le signe de la torsion a permis, de caractériser son attractivité et, de discriminer la partie attractive de la partie répulsive de la variété lente et de statuer sur la stabilité de ces courbes trajectoires. Ainsi, la présence dans l'espace des phases d'une variété lente attractive qui contraint les courbes trajectoires, intégrales du système dynamique à visiter son voisinage permet d'étudier la structure de l'attracteur. Cette approche basée sur certains aspects du formalisme de la Mécanique du Point et de la Géométrie Différentielle et qui s'est accompagnée de l'élaboration de programmes numériques a permis de constituer un nouvel outil d'investigation des systèmes dynamiques chaotiques. Son application à des modèles de référence comme celui de B. Van der Pol, de L.O. Chua ou d'E.N. Lorenz a permis d'obtenir plus directement et avec précision l'équation analytique de leur variété lente. De plus, une étude détaillée des modèles de type prédateur-proie comme celui de Rosenzweig-MacArthur ou d'Hastings-Powell, a conduit d'une part à la détermination de leur variété lente et d'autre part à la conception d'un nouveau modèle de type prédateur-proie à trois espèces appelé Volterra-Gause dont l'attracteur chaotique a la forme d'un escargot (chaotic snail shell).This work aims to study the stability of chaotic dynamical systems starting from the geometrical structure of their attractors of which a part is based on a manifold called slow manifold. To this end, a new approach based on certain aspects of the formalism of Mechanics and Differential Geometry was developed and led to a geometrical and kinematics interpretation of the evolution of the trajectory curves, integrals of these dynamical systems in the vicinity of the slow manifold, and allowed to study their stability. Mechanics allowed, with the use of the velocity and instantaneous acceleration vectors, located on a point of the trajectory curve, to discriminate the slow domain from the fast domain and to locate the position of the slow manifold inside the phase space. Certain notions of Differential Geometry like the expressions of curvature, torsion and that of the Osculating Plane provided an analytical equation of the slow manifold independent of the slow eigenvectors of the tangent linear system, therefore defined on a greater domain of the phase space. The slow manifold was then considered as the location of the points where the curvature of the trajectory curves, integrals of these dynamical systems, is minimal (in dimension two this minimum becomes equal to zero). The sign of torsion allowed: to characterize its attractivity, to discriminate the attractive part from the repulsive part of the slow manifold and, to rule on the stability of these trajectory curves. Thus, the presence in the phase space of an attractive slow manifold compelling the trajectory curve, integrals of the dynamic system to visit its vicinity allowed analyzing the attractor structure. This approach based on certain aspects of the formalism of Mechanics and Differential Geometry and which was accompanied by the development of numerical programs made it possible to constitute a new tool for investigation of chaotic dynamical systems. Its application to models of reference like that of B. Van der Pol., L.O. Chua or of E.N. Lorenz allowed obtaining more directly and with precision the analytical equation of their slow manifold. Moreover, a detailed study of the predator-prey models like that of Rosenzweig-MacArthur or Hastings-Powell, led on the one hand to the determination of their slow manifold and on the other hand to the design of a new three-dimensional model of predator-prey type: theVolterra-Gause model of which chaotic attractor has the shape of a snailshell (chaotic snail shell).TOULON-BU Centrale (830622101) / SudocSudocFranceF
A. Sestini - One of the best experts on this subject based on the ideXlab platform.
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Rotation–minimizing Osculating frames
'Elsevier BV', 2014Co-Authors: Rida T Farouki, C. Giannelli, M. L. Sampoli, A. SestiniAbstract:An orthonormal frame (f1,f2,f3) is rotation–minimizing with respect to fi if its angular velocity ω satisfies ω⋅fi≡0 — or, equivalently, the derivatives of fj and fk are both parallel to fi. The Frenet frame (t,p,b) along a space curve is rotation–minimizing with respect to the principal normal p, and in recent years adapted frames that are rotation–minimizing with respect to the tangent t have attracted much interest. This study is concerned with rotation–minimizing Osculating frames (f,g,b) incorporating the binormal b, and Osculating–Plane vectors f,g that have no rotation about b. These frame vectors may be defined through a rotation of t,p by an angle equal to minus the integral of curvature with respect to arc length. In aeronautical terms, the rotation–minimizing Osculating frame (RMOF) specifies yaw–free rigid–body motion along a curved path. For polynomial space curves possessing rational Frenet frames, the existence of rational RMOFs is investigated, and it is found that they must be of degree 7 at least. The RMOF is also employed to construct a novel type of ruled surface, with the property that its tangent Planes coincide with the Osculating Planes of a given space curve, and its rulings exhibit the least possible rate of rotation consistent with this constraint
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Rotation–minimizing conformal frames
2013Co-Authors: Rida T Farouki, C. Giannelli, M. L. Sampoli, A. SestiniAbstract:An orthonormal frame (f1,f2,f3) is rotation–minimizing with respect to fi if its angular velocity ω satisfies ω · fi ≡ 0 or, equivalently, the derivatives of fj, fk are both parallel to fi. The Frenet frame (t,p,b) is rotation–minimizing with respect to the principal normal p, and in recent years adapted frames that are rotation–minimizing with respect to the tangent t have attracted much interest. This study is concerned with conformal frames, that are rotation–minimizing with respect to the binormal b along a space curve. Such a frame (f,g,b) incorporates Osculating–Plane vectors f,g that have no rotation about b, and may be defined through a rotation of t,p by an amount equal to minus the integral of curvature with respect to arc length. In aeronautical terms, a rotation–minimizing conformal frame (RMCF) specifies “yaw–free ” rigid–body motion on a curved path. The existence of rational RMCFs on polynomial space curves with rational Frenet frames is investigated, and it is shown that they must be degree 7 at least. The RMCF is also employed to construct a novel type of ruled surface, characterized by tangent Planes that coincide with the Osculating Planes along a given space curve, and rulings that exhibit the least possible rate of rotation consistent with this constraint