Smooth Space Curves Research Articles

A Surface Family with a Mutual Geodesic Curve in Galilean 3-Space G3

This article gives an approach for establishing a surface family with a mutual geodesic curve in Galilean 3-space G3. Given a smooth space curve, we derive the sufficient and necessary condition for the given curve to be geodesic on it. Furthermore, we resolve the conditions when the surface is a ruled surface. Consequently, its ability to develop with the mutual geodesic of the elements of the surface family is researched. Meanwhile, we explain this approach by submitting considerable examples.

Jacobian-free variational method for computing connecting orbits in nonlinear dynamical systems.

One approach for describing spatiotemporal chaos is to study the unstable invariant sets embedded in the chaotic attractor of the system. While equilibria, periodic orbits, and invariant tori can be computed using existing methods, the numerical identification of heteroclinic and homoclinic connections between them remains challenging. We propose a robust matrix-free variational method for computing connecting orbits between equilibrium solutions. Instead of a common shooting-based approach, we view the identification of a connecting orbit as a minimization problem in the space of smooth curves in the state space that connect the two equilibria. In this approach, the deviation of a connecting curve from an integral curve of the vector field is penalized by a non-negative cost function. Minimization of the cost function deforms a trial curve until, at a global minimum, a connecting orbit is obtained. The method has no limitation on the dimension of the unstable manifold at the origin equilibrium and does not suffer from exponential error amplification associated with time-marching a chaotic system. Owing to adjoint-based minimization techniques, no Jacobian matrices need to be constructed. Therefore, the memory requirement scales linearly with the size of the problem, allowing the method to be applied to high-dimensional dynamical systems. The robustness of the method is demonstrated for the one-dimensional Kuramoto-Sivashinsky equation.

Position Vectors of the Natural Mate and Conjugate of a Space Curve

The concept of the natural mate and the conjugate curves associated to a smooth curve in Euclidian 3-space were introduced initially by Dashmukh and others. In this paper, we give some extra results that add more properties of the natural mate and the conjugate curves associated with a smooth space curve in E 3 . The position vectors of the natural mate and the conjugate of a given smooth curve are investigated. Also, using the position vector of the natural mate, the necessary and sufficient condition for a smooth given curve to be a Bertrand curve is introduced. Moreover, a new characterization of a general helix is introduced.

MATLAB-Based Visualization of Helical Structures and Their Applications

A helix is a type of smooth space curve, i.e. a curve in three-dimensional space. This study aims to present a MATLAB-based approach to visualize helical structures and to demonstrate their applications in various fields. As a qualitative study, primary and secondary sources of data have been consulted. Primarily, observing the real-life activities of humans and secondly, the analysis of archival documents such as related research articles, analytical books, and related reports. In conclusion, most of the architectural work in the world is possible by different types of geometrical design through the helix. It also found that we can visualize the different types of helices by using MATLAB in mathematics teaching. The implication of helix is everywhere in different fields and areas such as constructing a basket, screw, etc. Finally, we used helical concepts related to daily life activities in the field of instruction in the mathematics classroom.

Certified numerical algorithm for isolating the singularities of the plane projection of generic smooth space curves

Isolating the singularities of a plane curve is the first step towards computing its topology. For this, numerical methods are efficient but not certified in general. We are interested in developing certified numerical algorithms for isolating the singularities. In order to do so, we restrict our attention to the special case of plane curves that are projections of smooth curves in higher dimensions. This type of curve appears naturally in robotics applications and scientific visualization. In this setting, we show that the singularities can be encoded by a regular square system whose solutions can be isolated with certified numerical methods. Our analysis is conditioned by assumptions that we prove to be generic using transversality theory. We also provide a semi-algorithm to check their validity. Finally, we present experiments, some of which are not reachable by other methods, and discuss the efficiency of our method.

A precise mathematical model for geometric modeling of wire rope strands structure

Based on the Frenet frame, this paper proposes a general mathematical spiral model with an arbitrary smooth space curve as the center path, which can accurately build complex skeleton lines of wire rope strands. From the aspect of geometry, all the wires are spatial cylinders and must meet the actual geometric requirements: 1. The center cylinder is tangent or separated from the spiral cylinder; 2. The adjacent spiral cylinders do not overlap each other. For requirement 1, Costello’ conclusion is referenced and extended universally to suit an arbitrary smooth space central curve case with rigorous proofs. For requirement 2, the overlapping problem is described as obtaining the minimum distance between the two adjacent spatial path curves, which is deduced by a novel cross section method (SCM) with rigorous proofs and solved by the General Particle Swarm Optimization (PSO) algorithm. Based on the above models, the geometric modeling of wire rope strands procedure is proposed and implemented on the platforms of MATLAB and SolidWorks. Validations are conducted through geometric graphical representations, compared with those from some previous researches. For the simple straight strand case, when the number of spiral cylinders and spiral radius are given, the critical relationship between the ratio of spiral wire radius to spiral radius and the spiral angle is firstly obtained, which can be a precise dimension design reference of simple straight strand for eliminating initial geometric overlap. Further, to show the advance, some precise graphical examples of complex wire rope strands like independent wire rope core (IWRC) and multilayered rope are presented. The wire rope strands geometric modeling method proposed in this paper is precise enough averting initial geometric overlap between the wires for the benefit of subsequent mechanical computation accuracy and efficiency.

Scheme-theoretic Whitney conditions

We investigate a scheme-theoretic variant of Whitney condition (a). If X is a projective variety over the field of complex numbers and \(Y \subset X\) a subvariety, then X satisfies generically the scheme-theoretic Whitney condition (a) along Y provided that the projective dual of X is smooth. We give applications to tangency of projective varieties over \({\mathbb {C}}\) and to convex real algebraic geometry. In particular, we prove a Bertini-type theorem for osculating planes of smooth complex space curves and a generalization of a theorem of Ranestad and Sturmfels describing the algebraic boundary of an affine compact real variety.

Complete families of smooth space curves and strong semistability

We construct the first non-trivial examples of complete families of non-degenerate smooth space curves, and show that the base of such a family cannot be a rational curve. Both results rely on the study of the strong semistability of certain vector bundles.

Super space clothoids

Thin elastic filaments in real world such as vine tendrils, hair ringlets or curled ribbons often depict a very smooth, curved shape that low-order rod models --- e.g., segment-based rods --- fail to reproduce accurately and compactly. In this paper, we push forward the investigation of high-order models for thin, inextensible elastic rods by building the dynamics of a G 2 -continuous piecewise 3D clothoid: a smooth space curve with piecewise affine curvature. With the aim of precisely integrating the rod kinematic problem, for which no closed-form solution exists, we introduce a dedicated integration scheme based on power series expansions. It turns out that our algorithm reaches machine precision orders of magnitude faster compared to classical numerical integrators. This property, nicely preserved under simple algebraic and differential operations, allows us to compute all spatial terms of the rod kinematics and dynamics in both an efficient and accurate way. Combined with a semi-implicit time-stepping scheme, our method leads to the efficient and robust simulation of arbitrary curly filaments that exhibit rich, visually pleasing configurations and motion. Our approach was successfully applied to generate various scenarios such as the unwinding of a curled ribbon as well as the aesthetic animation of spiral-like hair or the fascinating growth of twining plants.

Characterization of the geometry and topology of DNA pictured as a discrete collection of atoms.

The structural and physical properties of DNA are closely related to its geometry and topology. The classical mathematical treatment of DNA geometry and topology in terms of ideal smooth space curves was not designed to characterize the spatial arrangements of atoms found in high-resolution and simulated double-helical structures. We present here new and rigorous numerical methods for the rapid and accurate assessment of the geometry and topology of double-helical DNA structures in terms of the constituent atoms. These methods are well designed for large DNA datasets obtained in detailed numerical simulations or determined experimentally at high-resolution. We illustrate the usefulness of our methodology by applying it to the analysis of three canonical double-helical DNA chains, a 65-bp minicircle obtained in recent molecular dynamics simulations, and a crystallographic array of protein-bound DNA duplexes. Although we focus on fully base-paired DNA structures, our methods can be extended to treat the geometry and topology of melted DNA structures as well as to characterize the folding of arbitrary molecules such as RNA and cyclic peptides.

On the generic plane section of smooth space curves in positive characteristic

Let C ⊂ P3 be a smooth and connected projective curve of genus g ≥ 2. Here in characteristic ≠ 2 we prove that its generic plane section is in uniform position and that its numerical character is connected.

Curves Having Schemes of Special Linear Systems with Multiple Components

In this article, we show that it is of frequent occurrence that smooth curves do have multiple components for some of their schemes of special divisors. For a smooth space curve C of degree d, we give sufficient conditions implying that has a multiple component, and we prove the existence of many space curves satisfying those conditions. As an example of a more general result for curves in ℙ r , we prove that general complete intersection curves of degree d in ℙ4 do have multiple components for the schemes and .

Projection genericity of space curves

If Γ is a smooth space curve, we consider the family of projections of Γ from a variable point not on Γ to a fixed plane. For a residual set of curves Γ, this family versally unfolds those singularities that occur in it. To obtain a family of curves which is open as well as dense in the space of smooth maps, we must compactify the parameter space, so we study curves in real projective space, and include projections from points of the curve itself. If Γ is smoothly embedded, the projection CP of Γ from P ∈ Γ is a well-defined smooth curve, and for generic Γ the family CP has generic singularities. However, when the point of projection moves off Γ, the projection varies discontinuously. We define a family of plane curves, parametrised by the blow-up X of P3 along Γ, such that for a point in the exceptional locus lying over P ∈ Γ , we have the union of the projection CP of Γ from P and a straight line L through the image of the tangent at P. A key result asserts that this is a flat family. We give an explicit list of restrictions on the family CP ∪ L (the key condition is that the total contact order of CP with L never exceeds 2), and show that these hold for a dense open set of curves Γ, and that if they do hold, there is a neighbourhood U of Γ, such that the family of projections from points of U \ Γ is generic. Combining this list of conditions with those obtained previously gives a natural definition of a dense set of space curves Γ, for which the complete family of projections has generic singularities, and we show that this set is also open.

Sobolev Active Contours

All previous geometric active contour models that have been formulated as gradient flows of various energies use the same L 2-type inner product to define the notion of gradient. Recent work has shown that this inner product induces a pathological Riemannian metric on the space of smooth curves. However, there are also undesirable features associated with the gradient flows that this inner product induces. In this paper, we reformulate the generic geometric active contour model by redefining the notion of gradient in accordance with Sobolev-type inner products. We call the resulting flows Sobolev active contours. Sobolev metrics induce favorable regularity properties in their gradient flows. In addition, Sobolev active contours favor global translations, but are not restricted to such motions; they are also less susceptible to certain types of local minima in contrast to traditional active contours. These properties are particularly useful in tracking applications. We demonstrate the general methodology by reformulating some standard edge-based and region-based active contour models as Sobolev active contours and show the substantial improvements gained in segmentation.

ON PRIME FANO THREEFOLDS OF GENUS 9

It is shown here that any prime Fano threefold of genus 9 is not exotic, that is, its scheme of lines is reduced at the generic point of every component. This extends a result of Prokhorov who showed that the only exotic Fano 3-fold of genus > 9 is the example of Mukai–Umemura. The problem is reduced to a result, obtained here, about the trisecants of a smooth space curve of genus 3 and degree 7.

On The Nodes of A Linear Projection of A Space Curve: A Uniqueness Problem

We study uniqueness of plane curves with prescribed set of nodes arising from a general linear projection of a given smooth irreducible space curve.

On certain loci of curves of genus g [ges ] 4 with Weierstrass points whose first non-gap is three

Let [Mfr ]g be the moduli space of smooth curves of genus g [ges ] 4 over the complex field [Copf ] and let [Tfr ]g ⊆ [Mfr ]g be the trigonal locus, i.e. the set of points [C] ∈ [Mfr ]g representing trigonal curves C of genus g [ges ] 4. Recall that each such curve C carries exactly one g13 (respectively at most two) if g [ges ] 5 (respectively g = 4). Let |D| be a g13 on C and suppose that it has a total ramification point at P (t.r. for short), i.e. that there is on C a point P such that 3P ∈ |D|. Such a P is a Weierstrass point whose first non-gap is three. In the present paper we study some sub-loci of [Tfr ]g related to curves possessing such points.

ON THE SLOPES OF THE MODULI SPACES OF CURVES

Let Mg be the moduli space of smooth curves of genus g and Mg the moduli space of stable curves of genus g. Then Mg = MgU A, where A = YJg£ ] A*> Ao is the closure of the locus of genus g — 1 curves with one double point, and A* (i ^ 0) is the the closure of the locus of the stable curves of type (i,g — i). Let A be the class of the Hodge line bundle on A4S, let Si be the class of Aj if i ^ 1, and let Si be half of the class of Ai . Let S = So H h 4 [11], and on the moduli functor, the canonical class is 13A — 2S, the only difference is the coefficient of Si (see [11, 9] for the details). This class is called effective if n(aX — bS) is effective for sufficiently large and sufficiently divisible n. Harris and Morrison [10] define the slope of the moduli space as

Smooth curves linked to thick curves

In this paper we construct smooth irreducible space curves which link geometrically by surfaces of minimal degree containing to curves of generic embedding dimension three. This produces interesting behavior with respect to both and . The curves link to smooth connected curves by surfaces of low degree but cannot link to smooth connected curves by surfaces of high degree. The curves give counterexamples to a conjecture of Martin-Deschamps and Perrin.

Plane projections of a smooth space curve

Let C be a smooth non-degenerate integral curve of degree d and genus g in P3 over an algebraically closed field of characteristic zero. For each point P in P3 let VP be the linear system on C induced by the hyperplanes through P . By VP one maps C onto a plane curve CP , such a map can be seen as a projection of C from P . If P is not the vertex of a cone of bisecant lines, then CP will have only finitely many singular points; or to put it slightly different: The secant scheme SP = (VP ) 1 2 parametrizing divisors in the second symmetric product C2 that fail to impose independent conditions on VP will be finite. Hence each such point P gives rise to a partition {a1 ≥ a2 ≥ . . . ≥ ak} of ∆(d, g) = 1 2 (d− 1)(d− 2)− g, where the ai are the local multiplicities of the scheme SP . If P is the vertex of a cone of bisecant lines (for example if P is a point of C), we set a1 = ∞. It is clear that the set of points P with a1 ≥ 2 is the surface F of stationary bisecant lines (including some tangent lines); a generic point P on F gives a tacnodial CP . We give two results valid for all curves C. The first one describes the set of points P with a1 ≥ 3. The second result describes the set of points with a1 ≥ 4.