Explicit Parametric Equations Research Articles

Timelike zero mean curvature surfaces in ℝ1 4

Abstract We are interested in the solution of the Björling problem for timelike surfaces in R 1 4 \mathbb{R}_{1}^{4} . The main contribution of the paper is to present new and many examples of timelike zero mean curvature surfaces and give their explicit parametric equations. In particular cases, one observes that the parametric equations of these surfaces coincide with the timelike minimal surfaces in Lorentz–Minkowski 3-space.

Orthogonal analysis of space-time crystals.

This paper presents a space-time-wise orthogonal analysis of space-time crystals. This analysis provides a solution consisting of a pair of explicit parametric equations that result from a separate application of the Bloch-Floquet theorem in the (orthogonal) directions of space and time. Compared with previous approaches, this solution offers the benefits of greater simplicity, clearer emphasis on space-time duality, and deeper physical insight.

A Closed-Form Parametrization and an Alternative Computational Algorithm for Approximating Slices of Minkowski Sums of Ellipsoids in R3

The current work aims to develop an approximation of the slice of a Minkowski sum of finite number of ellipsoids, sliced up by an arbitrarily oriented plane in Euclidean space R3 that, to the best of the author’s knowledge, has not been addressed yet. This approximation of the actual slice is in a closed form of an explicit parametric equation in the case that the slice is not passing through the zones of the Minkowski surface with high curvatures, namely the “corners”. An alternative computational algorithm is introduced for the cases that the plane slices the corners, in which a family of ellipsoidal inner and outer bounds of the Minkowski sum is used to construct a “narrow strip” for the actual slice of Minkowski sum. This strip can narrow persistently for a few more number of constructing bounds to precisely coincide on the actual slice of Minkowski sum. This algorithm is also applicable to the cases with high aspect ratio of ellipsoids. In line with the main goal, some ellipsoidal inner and outer bounds and approximations are discussed, including the so-called “Kurzhanski’s” bounds, which can be used to formulate the approximation of the slice of Minkowski sum.

Biharmonic Curves in a Strict Walker 3-Manifold

In this paper, we study the geometry of biharmonic curves in a strict Walker 3-manifold and we obtain explicit parametric equations for biharmonic curves and time-like biharmonic curves, respectively. We discuss the conditions for a speed curve to be a slant helix in a Walker manifold. We give an example of biharmonic curve for illustrating the main result.

Construction of inverse curves of general helices in the Sol space Sol³

In this paper, we study inverse curves of general helices in the Sol³. Finally, we find out explicit parametric equations of inverse curves in the Sol³.

Partially reflected waves in water of finite depth

This paper presents a second-order asymptotic solution in the Lagrangian description for nonlinear partial standing wave in the finite water depth. The asymptotic solution that is uniformly valid satisfies the irrotationality condition and zero pressure at the free surface. In the Lagrangian approximation, the explicit nonlinear parametric equations for the particle trajectories are obtained. In particular, the Lagrangian mean level of a particle motion for the partial standing wave is found as a part of the solution which is different from that in an Eulerian system. This solution enables the description of wave profile and particle trajectory, which can be progressive, standing or partial standing waves. The dynamic properties of nonlinear partial standing waves, including mass transport velocity, radiation stress, wave setup and pressure due to reflection are also investigated.

Complete biconservative surfaces in the hyperbolic space [formula omitted

We construct simply connected, complete, non-CMC biconservative surfaces in the 3-dimensional hyperbolic space H3, in an intrinsic and extrinsic way. We obtain three families of such surfaces, and, for each surface, the set of points where the gradient of the mean curvature function does not vanish is dense and has two connected components. In the intrinsic approach, we first construct a simply connected, complete abstract surface and then prove that it admits a unique biconservative immersion in H3. Working extrinsically, we use the images of the explicit parametric equations and a gluing process to obtain our surfaces. They are made up of circles (or hyperbolas, or parabolas, respectively) which lie in 2-affine parallel planes and touch a certain curve lying in a totally geodesic hyperbolic surface H2 in H3.

Global dynamics of Chua Corsage Memristor circuit family: fixed-point loci, Hopf bifurcation, and coexisting dynamic attractors

This paper presents an in-depth and rigorous mathematical analysis of a family of nonlinear dynamical circuits whose only nonlinear component is a Chua Corsage Memristor (CCM) characterized by an explicit seven-segment piecewise-linear equation. When connected across an external circuit powered by a DC battery, or a sinusoidal voltage source, the resulting circuits are shown to exhibit four asymptotically stable equilibrium points, a unique stable limit cycle spawn from a supercritical Hopf bifurcation along with three static attractors, four coexisting dynamic attractors of an associated non-autonomous nonlinear differential equation, and four corresponding coexistingpinched hysteresis loops. Thebasin of attractions of the above static and dynamic attractors is derived numerically via global nonlinear analysis. When driven by a battery, the resulting CCM circuit exhibits a contiguous fixed-point loci, along with its DC V–I curve described analytically by two explicit parametric equations. We also proved the fundamental feature of theedge of chaos property; namely, it is possible to destabilize a stable circuit (i.e., without oscillation) and make it oscillate, by merely adding a passive circuit element, namely $$L >0$$. The CCM circuit family is one of the few known example of a strongly nonlinear dynamical system that is endowed with numerous coexisting static and dynamic attractors that can be studied both experimentally, and mathematically, via exact formulas.

Slant Curves in Contact Lorentzian Manifolds with CR Structures

In this paper, we first find the properties of the generalized Tanaka–Webster connection in a contact Lorentzian manifold. Next, we find that a necessary and sufficient condition for the ∇ ^ -geodesic is a magnetic curve (for ∇) along slant curves. Finally, we prove that when c ≤ 0 , there does not exist a non-geodesic slant Frenet curve satisfying the ∇ ^ -Jacobi equations for the ∇ ^ -geodesic vector fields in M. Thus, we construct the explicit parametric equations of pseudo-Hermitian pseudo-helices in Lorentzian space forms M 1 3 ( H ^ ) for H ^ = 2 c > 0 .

Null Cartan Helices in Lorentzian 3-Space: An Approximation

In this work, we investigate the null Cartan helices in Lorentzian 3-space. We derive the helices with the constant timelike, spacelike and lightlike Killing axis in Lorentzian 3-space. Then, we calculate the Bishop curvatures of the null Cartan helix and obtain the explicit parametric equations of these curves by using the Bishop curvatures. Finally, we present various related examples and draw their images using the Mathematica.

Eikonal Hypersurfaces in the Euclidean n-Space

We prove that a hypersurface in $$E^n$$ is of constant slope if and only if its normalized position vector provides a solution of the eikonal equation on the unit sphere. Explicit parametric equation of the constant slope hypersurface is given. We show that the constant slope hypersurfaces are based on the polar pair of hypersurfaces in the unit sphere.

Mannheim partner curves in Cartan-Vranceanu 3-space

In this paper, the Mannheim mate curves of the proper biharmonic curves in Cartan-Vranceanu 3-dimensional spaces (M,ds2l,m), with l2 ? 4m and m ? 0 are studied. We give the definition of the Mannheim mate of a proper biharmonic curve and give the explicit parametric equations of that Mannheim mate curve in Cartan-Vranceanu 3-dimensional space. Moreover, we show that the distance between corresponding points of the Mannheim pairs is constant in Cartan-Vranceanu 3-dimensional spaces.

Parametric equations of plane sextic curves with a maximal set of double points

We give explicit parametric equations for all irreducible plane projective sextic curves which have at most double points and whose total Milnor number is maximal (is equal to 19). In each case we find a parametrization over a number field of the minimal possible degree and try to choose coordinates so that the coefficients are as small as we can do.

Asymptotic curves on B-surfaces according to type-2 bishop frame in the sol space

In this paper, we study B— surfaces of biharmonic constant Π2— slope curves according to type-2 Bishop in the . We characterize asymptotic curves on B— surfaces of biharmonic constant Π2— slope curves in terms of their Bishop curvatures. Finally, we find out their explicit parametric equations in the .

<b>New characterization of b-m2 developable surfaces

In this paper, b-m2 developable surfaces of biharmonic b-slant helices in the special three- dimensional − φ Ricci symmetric para-Sasakian manifold P is studied. Explicit parametric equations of b-m2 developable surfaces of biharmonic b-slant helices in the special three-dimensional − φ Ricci

<b>One parameter family of S-tangent surfaces

In this paper, we study  S tangent surfaces according to Sabban frame in the Lorentzian Heisenberg group H. We obtained differential equations in terms of their geodesic curvatures in the Lorentzian Heisenberg group H. Finally, we found explicit parametric equations of one parameter family of  S tangent surfaces according to Sabban Frame.

On the Explicit Parametric Equation of a General Helix with First and Second Curvature in Nil 3-Space

Nil geometry is one of the eight geometries of Thurston's conjecture. In this paper we study in Nil 3-space and the Nil metric with respect to the standard coordinates (x,y,z) is gNil₃=(dx)²+(dy)²+(dz-xdy)² in IR³. In this paper, we find out the explicit parametric equation of a general helix. Further, we write the explicit equations Frenet vector fields, the first and the second curvatures of general helix in Nil 3-Space. The parametric equation the Normal and Binormal ruled surface of general helix in Nil 3-space in terms of their curvature and torsion has been already examined in [12], in Nil 3-Space.

Biharmonic constant Π₁-slope curves according to type-2 bishop frame in Heisenberg group Heis³

In this paper, we introduce constant Π₁- slope curves according to type-2 Bishop frame in the Heisenberg group Heis³. We characterize the biharmonic constant Π₁- slope curves in terms of their Bishop curvatures. Finally, we find out their explicit parametric equations in the Heisenberg group Heis³. Additionally, we illustrate our main theorem.

B-Smarandache tm₂ curves of biharmonic new type b-slant helices according to Bishop frame in the Sol space Sol³

In this paper, we study b-Smarandache tm₂ curves of biharmonic new type b-slant helix in the Sol³. We characterize the b-Smarandache tm₂ curves in terms of their Bishop curvatures. Finally, we find out their explicit parametric equations in the Sol³.

Biharmonic B-slant helices according to bishop frame in the SL₂(R)

In this paper, we study biharmonic B-slant helices in the SL₂(R). We characterize the biharmonic B-slant helices in terms of their curvature and torsion. Finally, we find out their explicit parametric equations.