A ruled surface is a shape swept out by moving a line in 3D space. Due to their simple geometric forms, ruled surfaces have applications in various domains such as architecture and engineering. In the past, various approaches have been proposed to approximate a target shape using developable surfaces, which are special ruled surfaces with zero Gaussian curvature. However, methods for shape approximation using general ruled surfaces remain limited and often require the target shape to be either represented as parametric surfaces or have non-positive Gaussian curvature. In this paper, we propose a method to compute a piecewise ruled surface that approximates an arbitrary freeform mesh surface. We first use a group-sparsity formulation to optimize the given mesh shape into an approximately piecewise ruled form, in conjunction with a tangent vector field that indicates the ruling directions. Afterward, we utilize the optimization result to extract seams that separate smooth families of rulings, and use the seams to construct the initial rulings. Finally, we further optimize the positions and orientations of the rulings to improve the alignment with the input target shape. We apply our
We consider ruled surfaces with finite multiplicity. We study behaviors of the striction curves and the singularities of the ruled surfaces. We also give geometric meanings of invariants related to the ruled surfaces.
Ruled surfaces play an important role in various types of design, architecture, manufacturing, art, and sculpture. They can be created in a variety of ways, which is a topic that has been the subject of much discussion in mathematics and engineering journals. In geometric modelling, ideas are successful if they are not too complex for engineers and practitioners to understand and not too difficult to implement, because these specialists put mathematical theories into practice by implementing them in CAD/CAM systems. Some of these popular systems such as AutoCAD, Solidworks, CATIA, Rhinoceros 3D, and others are based on simple polynomial or rational splines and many other beautiful mathematical theories that have not yet been implemented due to their complexity. Based on this philosophy, in the present work, we investigate a simple way to generate ruled surfaces whose generators are the curvature axes of curves. We show that this type of ruled surface is a developable surface and that there is at least one curve whose curvature axis is a line on the given developable surface. In addition, we discuss the classifications of developable surfaces corresponding to space curves with singu
In this work, ruled surfaces in 3-dimensional Riemannian manifolds are studied. We determine the expression for the extrinsic and sectional curvature of a parametrized ruled surface, where the former one is shown to be non-positive. We also quantify the set of ruling vector fields along a given base curve which allows to define a relevant reference frame that we refer to as Sannia frame. The fundamental theorem of existence and equivalence of Sannia-ruled surfaces in terms of a system of invariants is given. The second part of the article tackles the concept of the striction curve, which is proven to be the set of points where the so-called Jacobi evolution function vanishes on a ruled surface. This characterization of striction curves provides independent proof for their existence and uniqueness in space forms and disproves their existence or uniqueness in some other cases.
We study the extension of homologically trivial symplectic or Hamiltonian cyclic actions to Hamiltonian circle actions on irrational ruled symplectic $4$-manifolds. On one hand, we construct symplectic involutions on minimal irrational ruled $4$-manifolds that cannot extend to a symplectic circle action even with a possibly different symplectic form. Higher dimensional examples are also constructed. On the other hand, for homologically trivial symplectic cyclic actions of any other order, we show that such an extension always exists. We also classify finite groups of symplecticmorphisms that acts trivially on the first homology group, and prove the non-extendability of the Klein $4$-group action to the three dimensional rotation group action motivated by the classification of finite groups of symplectomorphisms.
The reduced density matrices (RDMs) of many-body quantum states form a convex set. The boundary of low dimensional projections of this convex set may exhibit nontrivial geometry such as ruled surfaces. In this paper, we study the physical origins of these ruled surfaces for bosonic systems. The emergence of ruled surfaces was recently proposed as signatures of symmetry-breaking phase. We show that, apart from being signatures of symmetry-breaking, ruled surfaces can also be the consequence of gapless quantum systems by demonstrating an explicit example in terms of a two-mode Ising model. Our analysis was largely simplified by the quantum de Finetti's theorem---in the limit of large system size, these RDMs are the convex set of all the symmetric separable states. To distinguish ruled surfaces originated from gapless systems from those caused by symmetry-breaking, we propose to use the finite size scaling method for the corresponding geometry. This method is then applied to the two-mode XY model, successfully identifying a ruled surface as the consequence of gapless systems.
This is the fourth in a series of papers math.DG/0008021, math.DG/0008155, math.DG/0010036 constructing explicit examples of special Lagrangian submanifolds (SL m-folds) in C^m. A submanifold of C^m is ruled if it is fibred by a family of real straight lines in C^m. This paper studies ruled special Lagrangian 3-folds in C^3, giving both general theory and families of examples. Our results are related to previous work of Harvey and Lawson, Borisenko and Bryant. An important class of ruled SL 3-folds is the special Lagrangian cones in C^3. Each ruled SL 3-fold is asymptotic to a unique SL cone. We study the family of ruled SL 3-folds N asymptotic to a fixed SL cone N_0. We find that this depends on solving a linear equation, so that the family of such N has the structure of a vector space. We also show that the intersection Sigma of N_0 with the unit sphere in C^3 is a Riemann surface, and construct a ruled SL 3-fold N asymptotic to N_0 for each holomorphic vector field w on Sigma. As corollaries of this we write down two large families of explicit SL 3-folds depending on a holomorphic function on C, which include many new examples of singularities of SL 3-folds. We also show that ea
In this study, we define some new types of ruled surfaces called slant ruled surfaces. We give some characterizations for a regular ruled surface to be a slant ruled surface in Euclidean 3- space. We show that if the slant ruled surface is developable then the striction curve is a general helix or a slant helix according to the kind of surface. Moreover, we give the relationships between slant ruled surfaces and some offset surfaces such as Bertrand offsets and Mannheim offsets.
In this study, we give the relationships between the conical curvatures of ruled surfaces drawn by the unit vectors of the ruling, central normal and central tangent of a regular ruled surface in the Euclidean -space. We obtain the differential equations characterizing slant ruled surfaces and if the reference ruled surface is a slant ruled surface, we give some conditions for the surfaces drawn by the central normal and the central tangent vectors to be slant ruled surfaces.
In this study, we define some new types of non-null ruled surfaces called slant ruled surfaces in the Minkowski 3-space E_1^3. We introduce some characterizations for a non-null ruled surface to be a slant ruled surface in E_1^3. Moreover, we obtain some corollaries which give the relationships between a non-null slant ruled surface and its striction line in E_1^3.
We study ruled orders. These arise naturally in the Mori program for orders on projective surfaces and morally speaking are orders on a ruled surface ramified on a bisection and possibly some fibres. We describe fibres of a ruled order and show they are in some sense rational. We also determine the Hilbert scheme of rational curves and hence the corresponding non-commutative Mori contraction. This gives strong evidence that ruled orders are examples of the non-commutative ruled surfaces introduced by Van den Bergh.
In this study, we introduce Darboux slant ruled surfaces in the Euclidean 3-space which is defined by the property that the Darboux vector of orthonormel frame of ruled surface makes a constant angle with a fixed, non-zero direction. We obtain the characterizations of Darboux slant ruled surfaces regarding the conical curvature kappa and give the relations between Darboux slant ruled surfaces and some other slant ruled surfaces.
We classify 4-dimensional austere submanifolds in Euclidean space ruled by 2-planes. The algebraic possibilities for second fundamental forms of an austere 4-fold M were classified by Bryant, falling into three types which we label A, B, and C. We show that if M is 2-ruled of Type A, then the ruling map from M into the Grassmannian of 2-planes in R^n is holomorphic, and we give a construction for M starting with a holomorphic curve in an appropriate twistor space. If M is 2-ruled of Type B, then M is either a generalized helicoid in R^6 or the product of two classical helicoids in R^3. If M is 2-ruled of Type C, then M is either a one of the above, or a generalized helicoid in R^7. We also construct examples of 2-ruled austere hypersurfaces in R^5 with degenerate Gauss map.
In this paper, using the classifications of timelike and spacelike ruled surfaces, we study the Mannheim offsets of timelike ruled surfaces in Minkowski 3-space. Firstly, we define the Mannheim offsets of a timelike ruled surface by considering the Lorentzian casual character of the offset surface. We obtain that the Mannheim offsets of a timelike ruled surface may be timelike or spacelike. Furthermore, we characterize the developable of Mannheim offset of a timelike ruled surface by the derivative of the conical curvature of the directing cone.
In this study, we consider the notion of similar ruled surface for timelike and spacelike ruled surfaces in Minkowski 3-space. We obtain some properties of these special surfaces in E_1^3 and we show that developable ruled surfaces in E_1^3 form a family of similar ruled surfaces if and only if the striction curves of the surfaces are similar curves with variable transformation. Moreover, we obtain that cylindrical surfaces and conoids form two families of similar ruled surfaces in E_1^3.
In this paper, we define dual geodesic trihedron(dual Darboux frame) of a spacelike ruled surface. Then, we study Mannheim offsets of spacelike ruled surfaces in dual Lorentzian space by considering the E. Study Mapping. We represent spacelike ruled surfaces by dual Lorentzian unit spherical curves and define Mannheim offsets of the spacelike ruled surfaces by means of dual Darboux frame. We obtain relationships between the invariants of Mannheim spacelike offset surfaces and offset angle, offset distance. Furthermore, we give conditions for these surface offsets to be developable.
We classify ruled minimal surfaces in $\Bbb R^3$ with density $e^z.$ It is showed that there is no noncylindrical ruled minimal surface and there is a family of cylindrical ruled minimal surfaces in $\Bbb R^3$ with density $e^z.$ It is also proved that all translation minimal surfaces are ruled.
In this study, we define a family of ruled surfaces in the Euclidean 3-space E^3 and called similar ruled surfaces. We obtain some properties of these special surfaces and we show that developable ruled surfaces form a family of similar ruled surfaces if and only if the striction curves of the surfaces are similar curves with variable transformation.
The present paper concerns the invariants of generically nef vector bundles on ruled surfaces. By Mehta - Ramanathan Restriction Theorem and by Miyaoka characterization of semistable vector bundles on a curve, the generic nefness can be considered as a weak form of semistability. We establish a Bogomolov type inequality for generically nef vector bundles with nef general fiber restriction on ruled surfaces with no negative section. This gives an affermative answer in this case to a problem posed by Th. Peternell. Concerning ruled surfaces with a negative section, we prove a a similar result for generically nef vector bundles, with nef and balanced general fiber restriction and with a numerical condition on first Chern class, which is satisfied, for instance, if in its class there is a reduced divisor. Finally, we use such results to bound the invariants of curve fibrations, which factorize through finite covers of ruled surfaces.
This paper deals with skew ruled surfaces $\varPhi$ in the Euclidean space $\mathbb{E}^{3}$ which are right normalized, that is they are equipped with relative normalizations, whose support function is of the form $q(u,v) = \frac{f(u) + g(u)\, v}{w(u,v)}$, where $w^2(u,v)$ is the discriminant of the first fundamental form of $\varPhi$. This class of relatively normalized ruled surfaces contains surfaces such that their relative image $\varPhi^{*}$ is either a curve or it is as well as $\varPhi$ a ruled surface whose generators are, additionally, parallel to those of $\varPhi$. Moreover we investigate various properties concerning the Tchebychev vector field and the support vector field of such ruled surfaces.