Special Curves Research Articles

Rational conic fibrations containing special curves transverse to the fibers

Rational conic fibrations embedded in P N and containing an irreducible curve C transverse to the fibers are classified when C is either a plane curve or a rational normal curve of any degree d. This extends our previous results concerning lines, conics and twisted cubics. To emphasize the role of linear normality, the case in which C is a non-normal rational quartic is also discussed. Moreover, the rank of the d-th jet map of the surface at all points lying on such curves is evaluated.

Special homogeneous curves

We classify all special homogeneous curves. A special homogeneous curve $\mathcal {H}$ consists of connected components of the hyperbolic points in the level set $\{h=1\}$ of a homogeneous polynomial $h$ in two real variables of degree at least three, and admits a transitive group action of a subgroup $G\subset \mathrm {GL}(2)$ on $\mathcal {H}$ that acts via linear coordinate change.

БИОЛОГИЯЛЫҚ СҮТ ӨНІМДЕРІН СЕРПІМДІ ТОЛҚЫНДАРМЕН ЗЕРТТЕУ

The scientific article contains a pulsed ultrasonic installation in order to determine the temperature dependence of the density of biological fluids and the velocity of their elastic linear wave and the coefficient of adiabatic compressibility. With the help of experimental measurements and theoretical calculations, the temperature dependences of the density of the sample under study, the velocity of the linear wave, the coefficient of adiabatic compressibility (25-125)0C were studied for the first time, graphs of the dependences p=p(t), u=u(t), α5= α5 (t) were constructed. In order to study biological fluids, we used a thermos for domestic use. The source and receiver of the ultrasonic wave, a mirror reflecting the wave, a heater, and injection molding machines are placed in the biological liquid under study. We have positioned the piezoelectric element and the mirror at a certain distance. The required temperature and velocity of the elastic wave in the studied samples were measured using special curves a and B. Correct measurements of the installation.

On special curves in Lie groups with Myller configuration

In this study, we determine a new type comprehensive frame, which is called the generalized Frenet‐type frame in three‐dimensional Lie groups with Myller configurations, and it includes several special and classical type frames for Euclidean 3‐space and three‐dimensional Lie groups. After constructing this new comprehensive frame, we obtain derivative formulas with the help of the Lie curvature. In addition, we define some special type curves. The geometry of versor fields along a curve with Frenet‐type frame in three‐dimensional Lie groups with Myller configurations is a generalization of the usual theory of curves. Since this particular relationship, the osculating‐type and rectifying‐type curves with Frenet‐type frame in three‐dimensional Lie groups with Myller configurations include some special cases for osculating and rectifying curves in different spaces.

Partner Curves in Three Dimensional Degenerated Space Form

Using elementary and effective methods we study cone curves and their associated curves or partner curves in the three dimensional lightlike cone $\Q^3$, which is called three dimensional degenerated space form of the four dimensional Minkowski space $\E^4_1$. We define the associated curve of the cone curve and also partner curves of some special curves, such as a Bertrand curve and a Mannheim curve. We consider the properties and relations of a curve and its associated curve or partner curve. Some geometric characterizations of these curves are also given.

Peakon, Periodic Peakons, Compactons and Bifurcations of nonlinear Schrödinger’s Equation with Kudryashov’s Law of Refractive Index

In this paper, we consider the nonlinear Schrödinger’s equation with Kudryashov’s law of refractive index. By using the method of dynamical systems, we obtain bifurcations of the phase portraits of the traveling wave system under different parameter conditions. Corresponding to some special level curves, we derive possible exact explicit parametric representations of solutions (including peakon, periodic peakon, solitary wave solutions and compactons) under different parameter conditions.

Comments on Parallel Curves in 3-Dimensional Lie Group G

In this study, firstly, basic concepts in 3-dimensional Euclidean space and basic information about curves are given and some special curves are examined. Then, basic information about Lie algebra and Lie group basic concepts and curves are given and special curves such as helix, involute-evolute, Bertrand, Mannheim, Smarandache, are defined. Secondly, inspired by these special curves examined in the Lie group, the definitions of the parallel curve in the vector direction, the parallel curve in the direction of the vector B and the parallel curve in the direction of the linear combination of the vectors B and N of a curve according to the Frenet frame are given and characterized. Some theorems and results are obtained by finding the Frenet apparatus of these characterized curves. Finally, the findings are examined in more specific circumstances and new results are found.

On Characterization of Smarandache Curves Constructed by Modified Orthogonal Frame

In this study, we investigate Smarandache curves constructed by a space curve with a modified orthogonal frame. Firstly, the relations between the Frenet frame and the modified orthogonal frame are summarized. Later, the Smarandache curves based on the modified orthogonal frame are obtained. Finally, the tangent, normal, binormal vectors and the curvatures of the Smarandache curves are determined. A special curve known as the Gerono lemniscate curve whose curvature is not differentiable, the principal normal and binormal vectors are discontinuous at zero point is considered as an example, and the Smarandache curves of this curve are obtained by the aid of its modified orthogonal frame, and their graphics are given.

Null cartan geodesic isophote curves in Minkowski 3-space

In this paper, we investigate null Cartan geodesic isophote curves in the Minkowski [Formula: see text]-space, and give examples where such curves actually exist. By categorizing the types of light vectors, we characterize different types of null Cartan geodesic isophote curves. Moreover, we present the relationship between null Cartan geodesic isophote curves and other special curves.

A new approach to special curved surface families according to modified orthogonal frame

The main purpose of this paper was to investigate the problem of finding the surface family with respect to two different types of modified orthogonal frames defined for curves with curvature and torsion different from zero, respectively. For this purpose, conditions were given for the parametric curve with the modified orthogonal frame in three-dimensional Euclidean space to be a geodesic, asymptotic or line of curvature on the surface, respectively. It has been shown that a member of the surface family with the same special curve such as geodesic, asymptotic, or line of curvature can be obtained by choosing different deviation functions in the parametric writing of the surface to satisfy the conditions. Finally, several examples were given to support the study.

Tubular Surfaces According to a Focal Curve in E^3

A spine curve moves through the middle of a canal or a tubular surface. It might be asked whether it is possible to carry a spine curve over a tubular surface. For a tubular surface, we have seen that it can be done. In this study, we have given the general equations of a canal surface and a tubular surface according to a focal curve. In this case, we found the fundamental curvatures of a tubular surface. We gave theorems and proofs about the focal curve being a special curve.

Bifurcations of robust features on surfaces in the Minkowski 3-space

We obtain the bifurcation of some special curves on generic 1-parameter families of surfaces in the Minkowski 3-space. The curves treated here are the locus of points where the induced pseudo metric is degenerate, the discriminant of the lines principal curvature, the parabolic curve and the locus of points where the mean curvature vanishes.

Developable Quad Meshes and Contact Element Nets

The property of a surface being developable can be expressed in different equivalent ways, by vanishing Gauss curvature, or by the existence of isometric mappings to planar domains. Computational contributions to this topic range from special parametrizations to discrete-isometric mappings. However, so far a local criterion expressing developability of general quad meshes has been lacking. In this paper, we propose a new and efficient discrete developability criterion that is applied to quad meshes equipped with vertex weights, and which is motivated by a well-known characterization in differential geometry, namely a rank-deficient second fundamental form. We assign contact elements to the faces of meshes and ruling vectors to the edges, which in combination yield a developability condition per face. Using standard optimization procedures, we are able to perform interactive design and developable lofting. The meshes we employ are combinatorially regular quad meshes with isolated singularities but are otherwise not required to follow any special curves on a developable surface. They are thus easily embedded into a design workflow involving standard operations like remeshing, trimming, and merging operations. An important feature is that we can directly derive a watertight, rational bi-quadratic spline surface from our meshes. Remarkably, it occurs as the limit of weighted Doo-Sabin subdivision, which acts in an interpolatory manner on contact elements.

Osculating mate of a Frenet curve in the Euclidean 3-space

A new kind of partner curves called osculating mate of a Frenet curve is introduced. Some characterizations for osculating mate are obtained and using the obtained results some special curves such as slant helix, spherical helix, C-slant helix and rectifying curve are constructed.

Canal surfaces generated by special curves and quaternions

Canal surfaces generated by special curves and quaternions

Ruled surface family with common special curve

In this paper, we investigate the problem of the construction of a ruled surface family passing through the striction curve (c) of a ruled surface φ. We first define ruled surface family with common striction curve (c). Using the geodesic Frenet frame {e,t,g} of the ruled surface φ, we present sufficient conditions for that striction curve to be geodesic, line of curvature and asymptotic curve. Finally, we give some examples to show ruled surfaces with common geodesic, line of curvature and asymptotic curve.

Translation surfaces generating with some partner curve

In this article, generating curves of translation surfaces are paired with some special curve pairs. With the results obtained from these pairings, the developable and minimal translation surfaces are characterized. In addition, the surface curvatures of the translation surface are obtained. For a better understanding of the results, examples are given and their drawings are made with the help of Mathematica.

A necessary and sufficient condition for the global existence of solutions to nonlinear reaction‐diffusion equations on the half‐spaces in ℝN

In this paper, we study the existence and nonexistence of the global solutions to nonlinear reaction‐diffusion equations where is the half‐space , is a nonnegative continuous function, and is a locally Lipschitz function with some additional properties. The purpose of this paper is to give a necessary and sufficient condition for the existence of global solutions as follows: There is no global solution for any nonnegative and nontrivial initial data if and only if for every . In fact, we introduce a very special curve in to obtain the lower bound of decay of the heat semigroup, which is essential to prove the main result.

Calculation of mutual inductance between circular and arbitrarily shaped filaments via segmentation method

In this article, two analytical formulas for the calculation of mutual inductance between a circular filament and line segment arbitrarily positioning in the space are derived by using Mutual Inductance Method (MIM) and Babic’s Method (BM), respectively. Using the fact that any curve can be interpolated by a set of line segments, a method for calculation of mutual inductance between a circular filament and filament having an arbitrary shape in the space is proposed based on the derived analytical formulas. The derived two formulas and the proposed method (Segmentation Method) were numerically validated by using FastHenry software and reference examples from the literature. In particular, the proposed method was successfully applied to the calculation of mutual inductance between the circular filament and the following special curves such as circle, circular arc, elliptic arc, ellipse, spiral, helices and conical helices. All results of the calculation are in good agreement with the reference examples.

Geometry of Admissible Curves of Constant-Ratio in Pseudo-Galilean Space

An admissible curve of a pseudo-Galilean space is said to be of constant-ratio if the ratio of the length of the tangent and normal components of its position vector function is a constant. In this paper, we investigate and characterize a spacelike admissible curve of constant-ratio in terms of its curvature functions in the pseudo-Galilean space G13. Also, we study some special curves of constantratio such as T-constant and N-constant types of these curves. Finally, we give some computational examples for constructing the meant curves to demonstrate our theoretical results.