Epihypocurves and epihypocyclic surfaces with arbitrary base curve

If a circle rolls around another motionless circle then a point bind with the rolling circle forms a curve. It is called epicycloid, if a circle is rolling outside the motionless circle; it is called hypocycloid if the circle is rolling inside the motionless circle. The point bind to the rolling circle forms a space curve if the rolling circle has the constant incline to the plane of the motionless circle. The cycloid curve is formed when the circle is rolling along a straight line. The geometry of the curves formed by the point bind to the circle rolling along some base curve is investigated at this study. The geometry of the surfaces formed when the circle there is rolling along some curve and rotates around the tangent to the curve is considered as well. Since when the circle rotates in the normal plane of the base curve, a point rigidly connected to the rotating circle arises the circle, then an epihypocycloidal cyclic surface is formed. The vector equations of the epihypocycloid curve and epihypocycloid cycle surfaces with any base curve are established. The figures of the epihypocycloids with base curves of ellipse and sinus are got on the base of the equations obtained. These figures demonstrate the opportunities of form finding of the surfaces arised by the cycle rolling along different base curves. Unlike epihypocycloidal curves and surfaces with a base circle, the shape of epihypocycloidal curves and surfaces with a base curve other than a circle depends on the initial rolling point of the circle on the base curve.

THE COMPARATIVE STUDY OF RAILWAY BRIDGE DESIGN LOAD BETWEEN PM 60/2012 AND EN 1991: 2-2003

In Referring to the government programs on the increasing speed of the Jakarta-Surabaya express train in 2017, problems arise in the field, namely the potential increase in freight transportation via the railway line where an increase in loading is required on the railway especially for the existing bridge. This Research was implemented by increasing of the loading of the standard PM 60/2012 bridge to be compared with the EN1991: 2/2003 standard. This research analyzed the increase in structural strength of the 30m span steel bridge of the BH 182 constructed in Operational Region 2 Bandung as a results of loading adjustment from PM 60/2012 to EN 1991:2/2003.This paper explained an effect caused by increasing load on railway bridges with similar span and materials on normal plane, shear plane, moment and deflection. Structural analysis and calculation was performed by means of SAP2000 software. Results of analysis showed that standard equalization of EN 1991:2/2003 caused increasing percentage of loading combination of Comb L, normal plane, shear plane, moment plane, and deflection are of 35%, 60%, 71%, and 31%, respectively. While for Comb R loading combination for normal plane, shear plane, moment plane, and deflection are of 30%, 64%, 71%, and 30%, respectively.Keywords : PM 60 of 2012; EN1991:2-2003; SAP2000; Normal Field; Shear Field; Moment Field; Deflection.

Constructing a New Frame of Spacelike Curves on Timelike Surfaces in Minkowski 3-Space

In this paper, we define a relativeMinkowski normal plane and relative tangent vector TM:We construct a new relative S-frame (Shnoda-Saad frame) of regular spacelike curves on timelike surfaces. It depends only on the curve lies on the surface, Euclidean and Minkowski unit normal vectors. Also, we define S-curve according to this frame with some related theorems.

Two Ways of Solving System of Nonlinear Structural Equations

By applying the inner product of vectors, two objective functions are found. These vectors are taken from the structural equilibrium path. Via minimizing these functions, with respect to the load incremental parameter and the angle between particular vectors, two new constraint equalities are achieved. Since the scheme of authors is general, three more constraints are also reached. These formulations are similar to the previous presented nonlinear solvers, which confirm the legitimacy of new procedure. Afterward, several numerical tests are performed to prove the ability of the proposed techniques. Findings show that the new algorithms are capable in passing the load and displacement limit points of the various benchmark problems with severe nonlinear behaviors. Based on the number of increments and iterations and also the total analysis duration, the suggested methods have the maximum rapid convergence rate, in comparison to the normal plane, the updated normal plane and the cylindrical arc length strategies.

A relation between the curvature ellipse and the curvature parabola

At each point in an immersed surface in ℝ4 there is a curvature ellipse in the normal plane which codifies all the local second order geometry of the surface. Recently, at the singular point of a corank 1 singular surface in ℝ3, a curvature parabola in the normal plane which codifies all the local second order geometry has been defined. When projecting a regular surface in ℝ4 to ℝ3 in a tangent direction, corank 1 singularities appear generically. The projection has a cross-cap singularity unless the direction of projection is asymptotic, where more degenerate singularities can appear. In this paper we relate the geometry of an immersed surface in ℝ4 at a certain point to the geometry of the projection of the surface to ℝ3 at the singular point. In particular we relate the curvature ellipse of the surface to the curvature parabola of its singular projection.

On Special Kinds of Involute and Evolute Curves in 4-Dimensional Minkowski Space

Recently, extensive research has been done on evolute curves in Minkowski space-time. However, the special characteristics of curves demand advanced level observations that are lacking in existing well-known literature. In this study, a special kind of generalized evolute and involute curve is considered in four-dimensional Minkowski space. We consider (1,3)-evolute curves with respect to the casual characteristics of the (1,3)-normal plane that are spanned by the principal normal and the second binormal of the vector fields and the (0,2)-evolute curve that is spanned by the tangent and first binormal of the given curve. We restrict our investigation of (1,3)-evolute curves to the (1,3)-normal plane in four-dimensional Minkowski space. This research contribution obtains a necessary and sufficient condition for the curve possessing the generalized evolute as well as the involute curve. Furthermore, the Cartan null curve is also discussed in detail.