Closed Space Curve Research Articles

Construction of periodic adapted orthonormal frames on closed space curves

The construction of continuous adapted orthonormal frames along C1 closed–loop spatial curves is addressed. Such frames are important in the design of periodic spatial rigid–body motions along smooth closed paths. The construction is illustrated through the simplest non–trivial context — namely, C1 closed loops defined by a single Pythagorean–hodograph (PH) quintic space curve of a prescribed total arc length. It is shown that such curves comprise a two–parameter family, dependent on two angular variables, and they degenerate to planar curves when these parameters differ by an integer multiple of π. The desired frame is constructed through a rotation applied to the normal–plane vectors of the Euler–Rodrigues frame, so as to interpolate a given initial/final frame orientation. A general solution for periodic adapted frames of minimal twist on C1 closed–loop PH curves is possible, although this incurs transcendental terms. However, the C1 closed–loop PH quintics admit particularly simple rational periodic adapted frames.

Holditch’s Theorem in 3D Space

Holditch’s theorem is an old result on the area generated by a moving chord for closed planar curves. Some generalizations of this result have been given before, but none of these follows the same natural construction of the plane but done in the space. In this work, the notion of Holditch surface is defined, some properties of these surfaces are proved and they are used to generalize Holditch’s theorem for closed space curves naturally. Moreover, an approximation for the area of interest is given. Finally, it is showed that the only minimal non-planar Holditch surface is the helicoid.

From a Weightless Bent Wire Coat Hanger to Shell Structures Via the Beltrami Stress Tensor

A weightless wire coat hanger bent out of its plane is one of the simplest possible structures. All it does is transfer a force and moment around a closed space curve. Assembling a family of coat hangers enables us to build up trusses and frames and if we allow an infinite number of coat hangers which overlap we can assemble plates, shells and fully three dimensional structures. We can apply loads to these structures via a loading frame and wires, also made from coat hangers. A wire carrying an electric current produces a magnetic field in the space surrounding the wire. For the bent coat hanger we can imagine that there is a vector field surrounding the wire as a result of the force and moment in the wire. We can use this vector field to obtain expressions for the forces and moments in a shell in equilibrium with applied loads.

Boundary torsion and convex caps of locally convex surfaces

We prove that the torsion of any closed space curve which bounds a simply connected locally convex surface vanishes at least $4$ times. This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in Euclidean space. Furthermore, our result generalizes the $4$ vertex theorem of Sedykh for convex space curves, and thus constitutes a far reaching extension of the classical $4$ vertex theorem. The proof involves studying the arrangement of convex caps in a locally convex surface, and yields a Bose type formula for these objects.

Inference of power loss in spherical joints of the 6RSS parallel mechanism

This paper presents an analysis of the power losses due to the friction between the components of the spherical joints, belonging to a 6RSS parallel mechanism. The determination of power losses in spherical joints has been done with an application where the characteristic point follows a closed space curve. The trajectory of the point on the convex semicuple in contact with the concave semicuple it has been determined by using facilities of the CATIA software.

On the Bertrand offsets for ruled and developable surfaces

This work examines some classical results of Bertrand curves for ruled and developable surfaces using the E. Study’s dual line coordinates. In particular, a developable surface can have a developable Bertrand offset if a linear equation holds between the curvature and torsion of its edge of regression. From this result many of the classical results of closed space curve follow rather simply. Finally, an example of application is introduced and explained in detail.

On curves with nonnegative torsion

We provide new results and new proofs of results about the torsion of curves in \({\mathbb{R}^3}\). Let \({\gamma}\) be a smooth curve in \({\mathbb{R}^3}\) that is the graph over a simple closed curve in \({\mathbb{R}^2}\) with positive curvature. We give a new proof that if \({\gamma}\) has nonnegative (or nonpositive) torsion, then \({\gamma}\) has zero torsion and hence lies in a plane. Additionally, we prove the new result that a simple closed plane curve, without any assumption on its curvature, cannot be perturbed to a closed space curve of constant nonzero torsion. We also prove similar statements for curves in Lorentzian \({\mathbb{R}^{2,1}}\) which are related to important open questions about time flat surfaces in spacetimes and mass in general relativity.

Framed curves and knotted DNA

The present mini-review covers the local and global geometry of framed curves and the computation of twist and writhe in knotted DNA circles. Classical inequalities relating the total amount of bending of a closed space curve and associated knot parameters are also explained.

On Closed Space Curves in Minkowski Space–Time $${E_v^n}$$

A criterion for a 3-space curve to be closed has been presented in “Chung: Proc. Am. Math. Soc. 83, 357–361 (1981)”. In this paper, following same procedure, we give generalization of the criterion for a 3-space curve to an n-space curve to be closed in Minkowski space–time \({E_v^n}\) . Furthermore, we apply this criterion for a curve lying on an oriented surface in the Minkowski 3-space \({E_1^3}\) as an application.

The geometry of canal surfaces and the length of curves in de Sitter space

Abstract We find the minimal value of the length in de Sitter space of closed space-like curves with non-vanishing non-space-like geodesic curvature vector. These curves are in correspondence with closed almost-regular canal surfaces, and their length is a natural magnitude in conformal geometry. As an application, we get a lower bound for the total conformal torsion of closed space curves.

Closed polylines inscribed in a closed space curve

Let n be an odd positive integer. It is proved that if n + 2 is a power of a prime number and C is a regular closed non-self-intersecting curve in $ {\mathbb{R}^n} $ ,then C contains vertices of an equilateral (n + 2)-link polyline with n + 1 vertices lying in a hyperplane. It is also proved that if C is a rectifiable closed curve in $ {\mathbb{R}^n} $ ,then C contains n + 1 points that lie in a hyperplane and divide C into parts one of which is twice as long as each of the others. Bibliography: 6 titles.

Holditch theorem for the closed space curves in Lorentzian 3-space

In this article, we give the area formula of the closed projection curve of a closed space curve in Lorentzian 3-space L 3. For the 1-parameter closed Lorentzian space motion in L 3, we obtain a Holditch Theorem taking into account the Lorentzian matrix multiplication for the closed space curves by using their othogonal projections onto the Euclidean plane in the fixed Lorentzian space. Moreover, we generalize this Holditch Theorem for noncollinear three fixed points of the moving Lorentzian space and any other fixed point on the plane which is determined by these three fixed points.

Two perspectives on the twist of DNA.

Because of the double-helical structure of DNA, in which two strands of complementary nucleotides intertwine around each other, a covalently closed DNA molecule with no interruptions in either strand can be viewed as two interlocked single-stranded rings. Two closed space curves have long been known by mathematicians to exhibit a property called the linking number, a topologically invariant integer, expressible as the sum of two other quantities, the twist of one of the curves about the other, and the writhing number, or writhe, a measure of the chiral distortion from planarity of one of the two closed curves. We here derive expressions for the twist of supercoiled DNA and the writhe of a closed molecule consistent with the modern view of DNA as a sequence of base-pair steps. Structural biologists commonly characterize the spatial disposition of each step in terms of six rigid-body parameters, one of which, coincidentally, is also called the twist. Of interest is the difference in the mathematical properties between this step-parameter twist and the twist of supercoiling associated with a given base-pair step. For example, it turns out that the latter twist, unlike the former, is sensitive to certain translational shearing distortions of the molecule that are chiral in nature. Thus, by comparing the values for the two twists for each step of a high-resolution structure of a protein-DNA complex, we may be able to determine how the binding of various proteins contributes to chiral structural changes of the DNA.

Hamiltonian formulation of surfaces with constant Gaussian curvature

Dirac's method for constrained Hamiltonian systems is used to describe surfaces of constant Gaussian curvature. A geometrical free energy, for which these surfaces are equilibrium states, is introduced and interpreted as an action. An equilibrium surface can then be generated by the evolution of a closed space curve. Since the underlying action depends on second derivatives, the velocity of the curve and its conjugate momentum must be included in the set of phase-space variables. Furthermore, the action is linear in the acceleration of the curve and possesses a local symmetry—reparametrization invariance—which implies primary constraints in the canonical formalism. These constraints are incorporated into the Hamiltonian through Lagrange multiplier functions that are identified as the components of the acceleration of the curve. The formulation leads to four first-order partial differential equations, one for each canonical variable. With the appropriate choice of parametrization, only one of these equations has to be solved to obtain the surface which is swept out by the evolving space curve. To illustrate the formalism, several evolutions of pseudospherical surfaces are discussed.

An index formula for the self-linking number of a space curve

Given an embedded closed space curve with non-vanishing curvature, its self-linking number is defined as the linking number between the original curve and a curve pushed slightly off in the direction of its principal normals. We present an index formula for the self-linking number in terms of the writhe of a knot diagram of the curve and either (1) an index associated with the tangent indicatrix and its antipodal curve, (2) two indices associated with a stereographic projection of the tangent indicatrix, or (3) the rotation index (Whitney degree) of a stereographic projection of the tangent indicatrix minus the rotation index of the knot diagram.

ON MANIFOLD STRUCTURE OF CLOSED CURVES IN EUCLIDEAN SPACE

The space of all closed curves in Euclidean 3-space constitutes a Banach manifold. The existence of parallel mates to a given closed space curve depends on its total normal twist α. As a submanifold we consider the space of curves which has total normal twist an integer multiple of 2π. Also construction of special curves, like Bertrand, and helical curves with invariant total normal twist from a spherical curve will be described, and moreover, it constitutes a Banach submanifold.

One-Parameter Closed Dual Spherical Motions and Holditch’s Theorem

In this paper, for one-parameter closed dual spherical motions, we define the dual versions of the area vector of a given closed space curve, and the area projection of this curve in the direction of a given unit vector. The relationship between the above dual versions and the dual Steiner vector of the motion is used to give a generalization for Holditch’s theorem of planar kinematics into space kinematics. The geometry of ruled surfaces generated in a one-parameter spatial motion are treated in terms of their integral invariants. Finally, an example of application is investigated and explained in detail.

Some geometric properties of closed space curves and convex bodies

Some geometric properties of closed space curves and convex bodies

Geometry of Călugăreanu's theorem

A central result in the space geometry of closed twisted ribbons is Călugăreanu's theorem (also known as White's formula, or the Călugăreanu–White–Fuller theorem). This enables the integer linking number of the two edges of the ribbon to be written as the sum of the ribbon twist (the rate of rotation of the ribbon about its axis) and its writhe. We show that twice the twist is the average, over all projection directions, of the number of places where the ribbon appears edge-on (signed appropriately)—the ‘local’ crossing number of the ribbon edges. This complements the common interpretation of writhe as the average number of signed self-crossings of the ribbon axis curve. Using the formalism we develop, we also construct a geometrically natural ribbon on any closed space curve—the ‘writhe framing’ ribbon. By definition, the twist of this ribbon compensates its writhe, so its linking number is always zero.

Computing the Writhing Number of a Polygonal Knot

The writhing number measures the global geometry of a closed space curve or knot. We show that this measure is related to the average winding number of its Gauss map. Using this relationship, we give an algorithm for computing the writhing number for a polygonal knot with n edges in time roughly proportional to n1.6. We also implement a different, simple algorithm and provide experimental evidence for its practical efficiency.