Directrix Curve Research Articles

Geometric Characteristics of Surfaces with Curved Trapezoidal Plan

A method of forming a curved orthogonal coordinate system on a plane and a technique of constructing new surface shapes with curved trapezoidal plans are presented. Multiple examples of curved trapezoidal plans based on different directrix curves and surfaces with the given plans, including combinations of surfaces with different conjugate directrix curves, are illustrated. The proposed technique of surface forming may be used in architecture and construction for development of thin-walled space structures in both urban and industrial buildings. But for the analysis of thin shells, geometric characteristics of the middle surface of the shell are usually used. Vector equation of surfaces with curved trapezoidal plan was used to obtain the formulas for the fundamental form coefficients and surface curvatures. Examples of calculation of the fundamental form coefficients and curvatures of surfaces with particular directrix curves and vertical coordinate functions are presented.

On the curves lying on parallel-like surfaces of the ruled surface in $E^{3}$

In this paper, it has been researched curves lying on parallel-like surfaces $M^{f}$ of the ruled surface $M$ in $E^{3}$. Using the definition of parallel-like surfaces it has been found parametric expressions of parallel-like surface of the ruled surface and image curve of the directrix curve of the base surface. Moreover, obtaining Darboux frames of curves lying on surfaces $M$ and $M^{f}$, it has been compared the geodesic curvatures, the normal curvatures and the geodesic torsions of these curves.

ГЕОМЕТРИЯ ПРЯМЫХ КОНОИДОВ С ОРТОГОНАЛЬНОЙ СИСТЕМОЙ КООРДИНАТ.

The geometry and methods of forming right conoids are considered in the article. Right conoids are formed by the movement of a rectilinear generatrix perpendicular to a fixed straight line – the axis of the conoid, along which the rectilinear generatrix moves. When moving, the rectilinear generatrix rotates according to a given law around the directrix curve. Usually, the generatrix line is connected to some reference curve that the generatrix line touches. The coordinate grid along the generatrix lines is also connected with the reference curve. In this case, a non-orthogonal coordinate system of a right conoid is formed. Coefficients of quadratic forms of surfaces with a non-orthogonal coordinate system, usually are complex. Methods for analyzing shells with a non-orthogonal coordinate system of the middle surface of the shell are becoming more complicated. The article considers the possibility of forming right conoids with an orthogonal coordinate system, including with directrix reference curves. A vector equation is given for specifying right conoids with an orthogonal nonconjugate coordinate system. Formulas of coefficients of quadratic forms characterizing the internal geometry and curvature of right conoidal surfaces in space with the help of this vector equation are obtained. Based on the vector equations of the surface assignment, drawings of right conoids with various geometric parameters are constructed in the MathCAD program. The difference in the construction of right conoids with orthogonal and non-orthogonal curved coordinate systems is clearly demonstrated, so with different methods of specifying the angles of inclination of rectilinear generatrix of surfaces.

Determine When a Parametric Surface is a Surface of Revolution

A surface of revolution is a surface that can be generated by rotating a planar curve (the directrix) around a straight line (the axis) in the same plane. Using the mathematics of quaternions, we provide a parametric equation of a surface of revolution generated by rotating a directrix about an axis by quaternion multiplication of the parametric representations of the directrix curve and the line of axis. Then, we describe an algorithm to determine whether a parametric surface is a surface of revolution, and identify the axis and the directrix. Examples are provided to illustrate our algorithm.

Geometry of the normal ruled surfaces

The wide circle of the surfaces formed by the motion of the right line in the normal plain of some base directrix curve is regarded. The generate right line may rotate at some low at the normal plane of the base curve. The vector equation of the surface with any plane or space base curve is received. There are given the formulas of the geometry characteristics of the surfaces, on the base of them there is shown that the coordinate system of the normal ruled surfaces is orthogonal but there is not conjugated in common, that is that the normal ruled surfaces there are not developable surfaces in common way. The condition of the rotation of directrix plane line when the coordinate system of the normal ruled surfaces will be conjugated and the normal ruled surface will be developable is received. The condition that the normal ruled surface with space base curve will be the developable surface there is connected with its curvature of base curve. The developable normal ruled surface with plane base curve is formed by motion of right line at the normal plane of the base curve with the constant angle to the plane of the base curve; the received surface is a surface of constant slope. On the base of the vector equation of the surfaces there are made the figures of the normal ruled surfaces with the help of program complex MathCAD.

Defining developable ship surfaces considering material properties

This paper presents a practical method to define a quasi-developable surface that leans on two boundary curves or directrices. As a main difference with the rest of literature that works with quasi-developable surfaces assuming the fact that most of practical materials allow for a little bit of stretching, this paper presents how to quantify this developability in terms of a geometrical factor, the limit warp angle.The limit warp angle has been calculated numerically for steel and aluminum, applying a bending moment to plates of different geometries until a given percentage of the material's yield stress is surpassed. It has been demonstrated that developability depends on the material properties, thickness and aspect ratio of the plates that will be used to construct the real surface. The presented approach considers both physics and geometry and links them together.By considering the limit warp angle, a quasi-developable surface can be obtained without altering the directrices shape, unless the directrix curves present a very different shape between each other. This is very useful in engineering applications that produce designs based on curves and ease the use of parametric design techniques into the object definition. The constrained or parametric definition of ship hulls is beneficial during the initial phases of the ship design because it speeds up the creation process, provides more clarity about the influence of various parameters, and allows for the development of a range of concept hull surfaces.All the designs and calculations in this paper have been developed with a single, affordable and non specialized CAD software, indicating the capabilities of CAD products currently available for any designer.

The comparison of velaroidal shell structures of square plane load bearing properties

Velaroid surfaces is a class of analytical surfaces, formed by translating generatrix curves of variable curvature along the directrix curve of the same type, limited by plane contour lines. A new type of velaroidal shell –on the base of the catenary curve is introduced in this job, it’s surface equation obtained. The static structural analysis of the shells under equally distributed dead weight loading is conducted. Four types of structures of equal overall size and rising height were investigated: velaroidal shells on the base of catenary, parabola, ellipse, and sinusoid lines. The analysis is provided using the finite element method. The material has characteristics of reinforced concrete. The comparison revealed the peculiarities of each type of behavior under loading. The velaroidal shell with catenary generatrices demonstrates the most advantageous behavior under loading in the aspect of bending and membrane stresses ratio. This comparison is of interest for the potential introduction of such structures for roof elements.

Bisector Surfaces Through A Common Line of Curvatures and Its Classifications

In this paper, we study the Bisector surface, which defines the line of curvature on a surface play an importance role. Firstly, the Bisector surface constructed by a point and a space curve given in Euclidean 3-space. Then, it is investigated that the necessary and sufficient condition for directrix curve of this surface to satisfy line of curvature. After this, we classify the Bisector surfaces.

ОРТОГОНАЛЬНАЯ КРИВОЛИНЕЙНАЯ СИСТЕМА КООРДИНАТИ ПОСТРЕНИЕ ПОВЕРХНОСТЕЙ НА ТРАПЕЦИЕВИДНО-КРИВОЛИНЕЙНЫХ ПЛАНАХ

Thin space constructions on curved plans there are used at building of public structures, trade centers. Sport constructions. Working-out the methods of forming of the surfaces on the curved plans that is one of the modern task of the architecture and town building. The orthogonal coordinate system that is formed at the plane with some plane directrix curve and the system of the right lines orthogonal to the directrix curve there is regarded at the stat. The coordinate system forms some trapezium-curved segment. Taking some function of the vertical coordinate there is possible to receive different surfaces at the curved plans. Conjugating different directrix curves, it’s possible to receive the combined surfaces. The article presents a system of orthogonal coordinates of curvilinear-trapezoidal planes and methods for forming surfaces on these planes. Surfaces with a function of the vertical coordinate of a general view are considered, and surfaces on combined plans of segments of the same type are shown.

The exchange variations between bezier directrix curves of two developable ruled surfaces

Developable ruled surfaces and Bezier curves axe widely used in Computer Aided Geometric Deisgn (CAGD). A ruled surface is a surface generated by sweeping a line through space, or more specifically, sweeping a line along a directrix curve. A developable ruled surface is a surface where the surface normal at all points along a given ruling is constant. If the generators and directrix curve of a developable ruled surface are known, operations with these surfaces and designs of them are much simplified.In this paper, we present obtaining the Bezier directrix curve of a developable ruled surface from the Bezier directrix curve of the other developable ruled surface and the exchange variations of the arc lengths of these directrix curves. Also, we expressed the curvature exchanges between these Bezier directrix curves of two developable ruled surfaces. Also, based on this variation, we have discussed some ideas about the variations of developable ruled surfaces.

On the Parameterization of Rational Ringed Surfaces and Rational Canal Surfaces

Ringed surfaces and canal surfaces are surfaces that contain a one-parameter family of circles. Ringed surfaces can be described by a radius function, a directrix curve and vector field along the directrix curve, which specifies the normals of the planes that contain the circles. In particular, the class of ringed surfaces includes canal surfaces, which can be obtained as the envelopes of a one-parameter family of spheres. Consequently, canal surfaces can be described by a spine curve and a radius function. We present parameterization algorithms for rational ringed surfaces and rational canal surfaces. It is shown that these algorithms may generate any rational parameterization of a ringed (or canal) surface with the property that one family of parameter lines consists of circles. These algorithms are used to obtain rational parameterizations for Darboux cyclides and to construct blends between pairs of canal surfaces and pairs of ringed surfaces.

Superminimal surfaces in the 6-sphere

In this article, we use the harmonic sequence associated to a weakly conformal harmonic map f: S → S 6 in order to determine explicit examples of linearly full almost complex 2-spheres of S 6 with at most two singularities. We prove that the singularity type of these almost complex 2-spheres has an extra symmetry and this allows us to determine the moduli space of such curves with suitably small area. We also characterize projectively equivalent almost complex curves of S 6 in terms of G 2 ℂ -equivalence of their directrix curves.

Using μ-bases to implicitize rational surfaces with a pair of orthogonal directrices

A rational surfaceS(s,t)=(a(t)a⁎(s),a(t)b⁎(s),b(t)c⁎(s),c(t)c⁎(s)) can be generated from two orthogonal rational planar directrices: P(t)=(a(t),b(t),c(t)) in the xz-plane and P⁎(s)=(a⁎(s),b⁎(s),c⁎(s)) in the xy-plane. Moving a scaled copy of the curve P⁎(s) up and down along the z-axis with the size controlled by the curve P(t), we get the surface S(s,t). For example, when P⁎(s) is a circle with center at the origin, the surface S(s,t) is a surface of revolution. Many other useful and interesting surfaces whose cross sections are not circles can also be generated in this manner. We provide a new technique to implicitize this kind of rational surface using μ-bases. Let P(t) be a rational planar curve of degree n with a μ-basis consisting of two moving lines of degree μ and n−μ, and let P⁎(s) be a rational planar curve of degree m with a μ-basis consisting of two moving lines of degree μ⁎ and m−μ⁎. From the μ-bases for these two directrix curves P(t), P⁎(s), we can easily generate a μ-basis for the surface S(s,t) consisting of three moving planes that follow the surface with generic bidegrees (m−1,μ), (m−1,n−μ), (m,0). To implicitize the surface S(s,t), we construct a (2m−1)n×(2m−1)n sparse resultant matrix Rs,t for these three polynomials. We show that det(Rs,t)=0 is the implicit equation of the surface S(s,t) with a known extraneous factor of degree (m−1)n. To decrease the size of this matrix and to eliminate entirely the extraneous factor, we construct a new mn×mn Sylvester style sparse matrix Ss,t from four moving planes that follow the surface S(s,t). We prove that det(Ss,t)=0 is the exact implicit equation of the surface S(s,t) without any extraneous factors. Examples are presented to illustrate our methods.

Implicitizing rational surfaces of revolution using μ-bases

We provide a new technique for implicitizing rational surfaces of revolution using μ-bases. A degree n rational plane curve rotating around an axis generates a degree 2n rational surface. From a μ-basis p,q of this directrix curve, where μ=deg(p)⩽deg(q)=n−μ, and a rational parametrization of the circle r(s)=(2s,1−s2,1+s2), we can easily generate three moving planes p⁎,q⁎,r⁎ with generic bidegrees (1,μ),(1,n−μ),(2,0) that form a μ-basis for the corresponding surface of revolution. We show that this μ-basis is a powerful bridge connecting the parametric representation and the implicit representation of the surface of revolution. To implicitize the surface, we construct a 3n×3n Sylvester style sparse resultant matrix Rs,t for the three bidegree polynomials p⁎,q⁎,r⁎. Applying Gaussian elimination, we derive a 2n×2n sparse matrix Ss,t, and we prove that det(Ss,t)=0 is the implicit equation of the surface of revolution. Using Bezoutians, we also construct a 2(n−μ)×2(n−μ) matrix Bs,t, and we show that det(Bs,t)=0 is also the implicit equation of the surface of revolution. Examples are presented to illustrate our methods.

Minimal two-spheres in G(2, 4)

In this paper, we mainly study the geometry of conformal minimal immersions of two-spheres in a complex Grassmann manifold G(2,4). At first, we give a precise description of any non-±holomorphic harmonic 2-sphere in G(2,4) with the linearly full holomorphic maps ψ0: S2 → ℂP3 (called its directrix curve) and then, it is proved that such a conformal minimal immersion ϕ: S2 → G(2, 4) with constant curvature has constant Kaahler angle. Furthermore, ϕ is either V1(3) + V3(3), which is totally geodesic, with constant Gauss curvature 2/5 and constant Kaahler angle given by t = 3/2 or V3(3) + V2(3), which is totally real, but it is not totally geodesic, with constant Gauss curvature 2/3, where V0(3), V1(3), V2(3), V3(3): S2 → ℂP3 is the Veronese sequence.

Three-dimensional vibrations of cylindrical elastic solids with V-notches and sharp radial cracks

This paper provides free vibration data for cylindrical elastic solids, specifically thick circular plates and cylinders with V-notches and sharp radial cracks, for which no extensive previously published database is known to exist. Bending moment and shear force singularities are known to exist at the sharp reentrant corner of a thick V-notched plate under transverse vibratory motion, and three-dimensional (3-D) normal and transverse shear stresses are known to exist at the sharp reentrant terminus edge of a V-notched cylindrical elastic solid under 3-D free vibration. A theoretical analysis is done in this work utilizing a variational Ritz procedure including these essential singularity effects. The procedure incorporates a complete set of admissible algebraic–trigonometric polynomials in conjunction with an admissible set of “ edge functions” that explicitly model the 3-D stress singularities which exist along a reentrant terminus edge (i.e., α>180°) of the V-notch. The first set of polynomials guarantees convergence to exact frequencies, as sufficient terms are retained. The second set of edge functions—in addition to representing the corner stress singularities—substantially accelerates the convergence of frequency solutions. This is demonstrated through extensive convergence studies that have been carried out by the investigators. Numerical analysis has been carried out and the results have been given for cylindrical elastic solids with various V-notch angles and depths. The relative depth of the V-notch is defined as (1− c/ a), and the notch angle is defined as (360°− α). For a very small notch angle (1° or less), the notch may be regarded as a “sharp radial crack.” Accurate (four significant figure) frequencies are presented for a wide spectrum of notch angles (360°− α), depths (1− c/ a), and thickness ratios ( a/ h for plates and h/ a for cylinders). An extended database of frequencies for completely free thick sectorial, semi-circular, and segmented plates and cylinders are also reported herein as interesting special cases. A generalization of the elasticity-based Ritz analysis and findings applicable here is an arbitrarily shaped V-notched cylindrical solid, being a surface traced out by a family of generatrix, which pass through the circumference of an arbitrarily shaped V-notched directrix curve, r( θ), several of which are described for future investigations and close extensions of this work.

Projective Normality of Special Scrolls

We study the projective normality of a linearly normal special scroll R of degree d and speciality i over a smooth curve X of genus g. We relate it with the Clifford index of the base curve X. If d ≥ 4g − 2i − Cliff(X) + 1, i ≥ 3 and R is smooth, we prove that the projective normality of the scroll is equivalent to the projective normality of its directrix curve of minimum degree.

The pitch and the angle of pitch of a closed piece of ruled surface in R_1^3

Abstract It is well known that in the Euclidean space R3, the pitch and the angle of pitch on a ruled surface whose directrix curve is periodic can be given [6]. On the other hand, since there is no periodic timelike curve in the Minkowsky space R3 1, we give similar formulas on a closed piece of a ruled surface which is obtained by restricting the directrix curve of a ruled surface to a closed interval [a,b] contained in the domain of the directrix curve. Since the directrix curve is periodic spacelike, the formulas are the same in the Euclidean case when the length of the closed interval is equal to the period of the curve.

Another classification of incidence scrolls

The aim of this paper is the computation of the degree and genus of all incidence scrolls in $ \mathbb{P}^n $ . For this, we fix the dimension of a linear space which must be contained in the base of an incidence scroll. Then we will find all the incidence scrolls which have a base space of this fixed dimension. In this way, we can obtain all the incidence scrolls with a line as directrix curve, those whose base contains a plane, and so on.

The Lund fragmentation process for a multi-gluon string according to the area law

The Lund Area Law describes the probability for the production of a set of colourless hadrons from an initial set of partons, in the Lund string fragmentation model. In this paper we will present a general method to implement the Area Law for a multi-gluon string state. The partonic states are in general given by a perturbative QCD cascade and are consequently defined only down to a cutoff in the energy momentum fluctuations. We will show that our method defines the states down to the hadronic mass scale inside an analytically calculable scenario. We will then show that there is a differential version of our process which is closely related to the generalised rapidity range \lambda, which has been used as a measure on the partonic states. We identify \lambda as the area spanned between the directrix curve (the curve given by the parton energy momentum vectors laid out in colour order, which determines the string surface) and the average curve (to be called the P-curve) of the stochastic X-curves (curves obtained when the hadronic energy-momentum vectors are laid out in rank order). Finally we show that from the X-curve corresponding to a particular stochastic fragmentation situation it is possible to reproduce the directrix curve (up to one starting vector and a set of sign choices, one for each hadron).