Characterization of Rectifying Curves by Their Involutes and Evolutes

A rectifying curve is a twisted curve with the property that all of its rectifying planes pass through a fixed point. If this point is the origin of the Cartesian coordinate system, then the position vector of the rectifying curve always lies in the rectifying plane. A remarkable property of these curves is that the ratio between torsion and curvature is a nonconstant linear function of the arc-length parameter. In this paper, we give a new characterization of rectifying curves, namely, we prove that a curve is a rectifying curve if and only if it has a spherical involute. Consequently, rectifying curves can be constructed as evolutes of spherical twisted curves; we present an illustrative example of a rectifying curve obtained as the evolute of a spherical helix. We also express the curvature and the torsion of a rectifying spherical curve and give necessary and sufficient conditions for a curve and its involute to be both rectifying curves.

On f-rectifying curves in the Euclidean 4-space

A rectifying curve in the Euclidean 4-space 𝔼4 is defined as an arc length parametrized curve γ in 𝔼4 such that its position vector always lies in its rectifying space (i.e., the orthogonal complement Nγ ˔ of its principal normal vector field Nγ) in 𝔼4. In this paper, we introduce the notion of an f-rectifying curve in 𝔼4 as a curve γ in 𝔼4 parametrized by its arc length s such that its f-position vector γf, defined by γf (s) = ∫ f(s)dγ for all s, always lies in its rectifying space in 𝔼4, where f is a nowhere vanishing integrable function in parameter s of the curve γ. Also, we characterize and classify such curves in 𝔼4.

Representations of Rectifying Isotropic Curves and Their Centrodes in Complex 3-Space

In this work, the rectifying isotropic curves are investigated in three-dimensional complex space C3. The conclusion that an isotropic curve is a rectifying curve if and only if its pseudo curvature is a linear function of its pseudo arc-length is achieved. Meanwhile, the rectifying isotropic curves are expressed by the Bessel functions explicitly. Last but not least, the centrodes of rectifying isotropic curves are explored in detail.

Comparison of ion transport determinants between a TMEM16 chloride channel and phospholipid scramblase

Two TMEM16 family members, TMEM16A and TMEM16F, have different ion transport properties. Upon activation by intracellular Ca2+, TMEM16A—a Ca2+-activated Cl− channel—is more selective for anions than cations, whereas TMEM16F—a phospholipid scramblase—appears to transport both cations and anions. Under saturating Ca2+ conditions, the current–voltage (I-V) relationships of these two proteins also differ; the I-V curve of TMEM16A is linear, while that of TMEM16F is outwardly rectifying. We previously found that mutating a positively charged lysine residue (K584) in the ion transport pathway to glutamine converted the linear I-V curve of TMEM16A to an outwardly rectifying curve. Interestingly, the corresponding residue in the outwardly rectifying TMEM16F is also a glutamine (Q559). Here, we examine the ion transport functions of TMEM16 molecules and compare the roles of K584 of TMEM16A and Q559 of TMEM16F in controlling the rectification of their respective I-V curves. We find that rectification of TMEM16A is regulated electrostatically by the side-chain charge on the residue at position 584, whereas the charge on residue 559 in TMEM16F has little effect. Unexpectedly, mutation of Q559 to aromatic amino acid residues significantly alters outward rectification in TMEM16F. These same mutants show reduced Ca2+-induced current rundown (or desensitization) compared with wild-type TMEM16F. A mutant that removes the rundown of TMEM16F could facilitate the study of ion transport mechanisms in this phospholipid scramblase in the same way that a CLC-0 mutant in which inactivation (or closure of the slow gate) is suppressed was used in our previous studies.

A new aspect of rectifying curves and ruled surfaces in Galilean 3-space

A curve is named as rectifying curve if its position vector always lies in its rectifying plane. There are lots of papers about rectifying curves in Euclidean and Minkowski spaces. In this paper, we give some relations between extended rectifying curves and their modified Darboux vector fields in Galilean 3-Space. The other aim of the paper is to introduce the ruled surfaces whose base curve is rectifying curve. Further, we prove that the parameter curve of the surface is a geodesic.

Rectifying curves and geodesics on a cone in the Euclidean 3-space

A twisted curve in the Euclidean 3-space $\mathbb E^3$ is called a rectifying curve if its position vector field always lie in its rectifying plane. In this article we study geodesics on an arbitrary cone in $\mathbb E^3$, not necessary a circular one, via rectifying curves. Our main result states that a curve on a cone in $\mathbb E^3$ is a geodesic if and only if it is either a rectifying curve or an open portion of a ruling. As an application we show that the only planar geodesics in a cone in $\mathbb E^3$ are portions of rulings.

Characterization of Rectifying and Sphere Curves in ℝ3

Studies of curves in 3D-space have been developed by many geometers and it is known that any regular curve in 3D space is completely determined by its curvature and torsion, up to position. Many results have been found to characterize various types of space curves in terms of conditions on the ratio of torsion to curvature. Under an extra condition on the constant curvature, Y. L. Seo and Y. M. Oh found the series solution when the ratio of torsion to curvature is a linear function. Furthermore, this solution is known to be a rectifying curve by B. Y. Chen’s work. This project, uses a different approach to characterize these rectifying curves. This paper investigates two problems. The first problem relates to figuring out what we can say about a unit speed curve with nonzero curvature if every rectifying plane of the curve passes through a fixed point in ℝ3. Secondly, some formulas of curvature and torsion for sphere curves are identified. KEYWORDS: Space Curve; Rectifying Curve; Curvature; Torsion; Rectifying Plane; Tangent Vector; Normal Vector; Binormal Vector

A Curve Satisfying K/T = s with constant K>0

In the present paper, we investigate a space curve in which the curvature is constant and the torsion is a linear function. The aim of this paper is to find an explicit formula for this space curve when the ratioof the torsion to the curvature is a linear function when the curvature is constant. KEYWORDS: Space Curve, Curvature, Torsion, General Helix, Frenet Frame, Series Solution, Rectifying Curve

THE PHYSICAL PROPERTIES OF XeCl EXCIMER PULSED LASER DEPOSITED n-C:P/p-Si PHOTOVOLTAIC SOLAR CELLS

This paper reports on the successful deposition of phosphorus (P) -doped n-type (p-C:P) carbon (C) films, and fabrication of n-C:P/p-Si cells by pulsed laser deposition (PLD) using graphite target at room temperature. The cells performances have been given in the dark I–V rectifying curve and I–V working curve under illumination when exposed to AM 1.5 illumination condition (100mW/cm2, 25°C). The n-C:P/p-Si cell fabricated using a target with the amount of P by 7 weight percentages (Pwt%) shows the highest energy conversion efficiency η = 1.14% and fill factor FF = 41%. The quantum efficiency (QE) of the n-C:P/p-Si cells are observed to improve with Pwt%. The dependence of P content on the electrical and optical properties of the deposited films and the photovoltaic characteristics of the n-C:P/p-Si heterojunction solar cell are discussed.