RS51441B - ADJUSTMENT WITH AXLE OF VARIABLE CIRCUIT (KH) IN HORIZONTAL VARIABLE (Ψ) OVERVIEW - Google Patents

Abstract

A track comprising a track centre line of variable curvature on a horizontal section and having a variable superelevation angle. The curvature is determined on the basis of an assumed superelevation function such that the overall non-compensated lateral acceleration at a selected fixed aligning height, taking into account the amount of non-compensated lateral acceleration caused by the swaying movement, has a characteristic curve like that of said function and fulfils the following differential equation (formula I) wherein: KH(S) represents the curvature of the track centre line on a horizontal section, s represents the curve length along the track centre line, KC represents the constant reference curvature (in an arc), C represents the constant reference elevation angle (in an arc), (S) represents the elevation angle, h represents the aligning height, d represents the differential operator.

Description

The invention relates to a single track with a transition arch (crossing) of minimum force and to a ramp with a minimum force ascent as well as to the direction of the axis of such a single track.

The routing of railway tracks for railways, subways and other rail vehicles is usually performed as a sequence of elements with constant curvature in horizontal projection, such as straight and circular arcs, and elements of variable curvature. The transition from one straight track to another straight track that turns at an angle in relation to the first straight track can be performed either in two parts, that is, with two interconnected maximal transition arcs, or the transition can be performed in three parts, where one conventional transition arc is continued on one straight track, followed by a circular arc, followed by the next conventional transition arc, and then followed by the next straight track.

Vitoperenjc, that is, the transfer from one track to another parallel track at a certain distance, is performed this time with two circular arcs or with one series of one passing arc (crossing), one circular arc. one reversible passing arc, one circular arc and one passing arc.

In order to reduce the unbalanced lateral acceleration of the vehicle and the lateral traction force on curved track elements, the track plane is rotated around the track axis.

In railways, any slope measured on the upper side of the outer rail in relation to the upper side of the inner rail, for which the transverse direction of the track is jammed around the axis of the track, is referred to as overhang. To determine this overhang, an overhang proportional to the curvature of the track axis is usually prescribed. It does not give an overhang on straight sections, and on curved sections it gives an overhang on the outside of the track, which increases with the rotation of the track axis.

When tracing the tracks, the most common element is the clothoid (Cornu's spiral), which has a variable curve in the horizontal projection. It is one curve, along which the curve changes linearly from one value to another. The corresponding overhang, proportional to the curve, is a linear variable overhang, i.e. a flat ramp with an overhang. In doing so, bends are obtained at the points of connection with adjacent elements with a constant overhang. Then a linearly variable unbalanced lateral acceleration occurs in the track axis between constant values of adjacent elements.

At the same time, attention was not paid to the fact that due to the rolling of the vehicle along the track outside the axis of symmetry, speed jumps occur everywhere and that the acceleration will be endless because of this. Mathematically, in particular, the points of the vehicle on the axis of symmetry of the track are singular points where these infinite accelerations do not occur. Therefore, the existing kinematic description is not complete.

Infinite accelerations were practically avoided by planned or freely created rounding of the rails at the bend points. However, the proportionality between the curve and the overhang and thus the desired flow of unbalanced lateral accelerations in the axis of the track is lost in the areas of the bends with rounded edges.

The disadvantages of this type of tracing are known. In order to avoid them, instead of a linear process function, two connected parabolas of the second degree, or one polynomial of the third degree or one cosine half-wave are used to determine the curvature in the horizontal projection and the elevation. At the same time, the flow of speed change at every point of the vehicle is continuous. Accelerations remain finite but are unstable, and their changes in time - shocks - have infinity points as before. To remedy this, a linear function is used together with a sinusoid. Then the accelerations are continuous and the transition points have finite values.

The next possibility is to perform the track tracing in such a way that the desired kinematic properties should not be maintained along the central axis but at a certain point of the vehicle, for example in the center of gravity of the vehicle. In order to avoid geometric impact points in the guide elements, the function outside the axis of symmetry of the track must always be twice differentiable.

From AT 401 781B, a track is known with one real straight line with a continuous course of curves in the transition arch (crossing), in the course of which a fictitious straight line with one directrix is provided along which one point moving at a nominal speed has an unbalanced lateral acceleration equal to zero. From the directrix of a fictitious straight line, the directrix of a real straight line is created by moving each individual point of the directrix along the normal to the line by one constant distance.

From AT 402 211 B, a track with a transitional arc is known, in which tracing for the elevation angle and for the curve of the directrix in the horizontal projection, functions that can be continuously differentiated twice were used. Further, special functions derived from hyperbolic tangent and squared sine functions are listed. The tangent hyperbolic function can be differentiated arbitrarily often, and all its derivatives fit on the edges of the arc with constant curve tracing elements connected to it. In addition, a non-linear dependence is given for large angles of inclination and cant for a leveled cant for a certain speed.

In the practical application of this routing, numerous problems arise: The actual track layout is not known in advance. It arises only after applying the transformation from a fictitious to a real track. It is contrary to the practice, which is used in all conventional layouts, to directly indicate the function of the track axis. There is a similar problem with the assessment of track position errors in terms of their influence on vehicle kinematics. Track position errors must be transformed from real track strips into fictitious track strips. Only then, for example, will the corresponding unbalanced lateral acceleration of the vehicle be determined.

When moving from one level of cantilever to another level of cantilever, some correspondingly drastic change must appear in the found extracts. In the case of a flat ramp, the first lead (corner) is admittedly minimal, and with it the twisting of the track and the angular speed of rolling, but all other leads are therefore unlimited on the edges. In the case of a ramp made up of two parabolas of the second degree, other derivatives in relation to the curve of the ramp have a minimum value. In this way, the angular accelerations of rolling in this version are minimal, while the ramp in the middle is steeper, and in the next version on the edges and in the middle, there is no impact when rolling. It is similar to that with third-degree polynomials and cosine half-waves. In the case of linear propagation with an oncoming sinusoid, there are still shocks at the edges so that the first and second derivatives have higher values than in other cantilever ramps.

The well-known requirement that there are still other derivatives of the process function is fulfilled for all known processes except the clothoid with a flat ramp. The infinitely common ability to differentiate a process function has its drawbacks. In the case of very flat transitions along the edges, the higher derivatives between them will be unnecessarily large and with them, for example, angular accelerations of rolling and impacts at an angle.

Functions that can be infinitely often continuously differentiated are transcendental, much like the hyperbolic tangent. They have a theoretically endless mathematical refinement at the connection points. Practically, their analytical ability to differentiate is no longer used after a small number of derivatives because the expressions are too long to handle. Analytical capability of integration, for example of curvature by position angle, which is in any case useful for practical work, has not been realized. In this way, only numerical differentiation and integration remain for the actual calculation of transcendental functions, where the continuity depends on the applied algorithm, but is limited in any case.

It is desirable to have a function flow that meets the necessary requirements regarding the possibility of differentiating at transition points and has the smallest possible values of all physical parameters for more convenient measurement.

Another aspect that has not been paid attention to is the limitations due to the flexibility of the continuously welded, initially straight rail. In the usual view of the rail as a continuously supported support, i.e. that the influence of the rail attachment is distributed, the overhang corresponds directly to the flow function of the bent rail. Its second derivative by place, according to the Clemens Bernoulli-Euler bending theory, is proportional to the bending moment in the rail, the third is proportional to the transverse force, while the fourth derivative corresponds to the distribution of forces in the base, with which the rail is brought into the desired ramp shape and in which it must remain.

It is the task of the invention to remove the mentioned shortcomings and to obtain a track that can be realized in reality and to achieve a smooth flow of unbalanced lateral accelerations.

As a solution, Pronalski proposes a single track with a central axis of the track with a variable curvature in the horizontal plane and a variable elevation angle. According to the invention, this track is characterized by the fact that the curve from a function accepted for overhang is determined in such a way that the total unbalanced lateral acceleration at a selected, fixed tracking height, taking into account the component of unbalanced lateral acceleration caused by swaying, has the same flow as the mentioned function and satisfies the following differential equation:

whereby

k,,(s) curve of the track axis in horizontal projection s arc length along the track axis

kcconstant base curve (in one circular arc)

\j/cugao of the basic overhang (in one circular arc)

vj/(s) angle of elevation

tracing height

d differential operator.

A further property of the invention provides that the function in its entire course, including the edges of the region, can be differentiated at least four times and that the fourth derivative of the function always has finite values.

One further feature of the invention provides that when determining the curvature (kh) of the track axis in the horizontal projection, the zero tracking height is selected as a fixed tracking height.

One further property of the invention foresees that the seventh-degree polynomial specified in equation (2) is used as a normalized function, whereby the normalized function is used for the change of the cant angle and for the total unbalanced lateral acceleration, and from it the curve (kh) of the track axis in the horizontal projection according to equation (1) is derived:

Whereby:

s arc length along the track axis

1 length of transition arch and ramp with overhang.

A further feature of the invention provides that as a normalized function, the function specified in equation (3) with one polynomial of the third degree in combination with sine and cosine and one constant value (Z) is used, wherein that normalized function is used for the course of the cant angle and for the overall unbalanced lateral acceleration, and from it the curvature (kh) of the axis of the track in the horizontal projection according to equation (I) is derived:

Whereby:

s arc length along the track axis

1 length of transition arch and ramp with overhang.

A further feature of the invention provides that as a normalized function, a function with one polynomial of the third degree in combination with sine and cosine is used as a normalized function, and this normalized function is used for the course of the cant angle and for the overall unbalanced lateral acceleration, and from it the curvature (kh) of the axis of the track in the horizontal projection is derived according to equation (1):

Whereby:

s arc length along the track axis

1 length of transition arch and ramp with overhang.

A further feature of the invention provides that as a normalized function the function with one polynomial of the fifth degree in combination with only sine is used as the normalized function, and that normalized function is used for the course of the cant angle and for the overall unbalanced lateral acceleration, and from it the curve (kh) of the axis of the track in the horizontal projection according to the equation (1) is derived:

Whereby:

s arc length along the track axis

1 length of transition arch and ramp with overhang.

A further property of the invention provides that as a normalized function the function with one polynomial of the fifth degree in combination with cosine only is used as a normalized function, and that normalized function is used for the course of the cant angle and for the overall unbalanced lateral acceleration, and from it the curve (kh) of the axis of the track in the horizontal projection according to the equation (1) is derived:

Whereby:

s the length of the arc along the track axis 1 the length of the transition arc and ramp with an overhang.

A further property of the invention provides that as a normalized function the function specified in equation (7) with one polynomial of the ninth degree is used, wherein that normalized function is used for the course of the cant angle and for the overall unbalanced lateral acceleration, and from it the curve (kh) of the axis of the track in the horizontal projection is derived according to equation (1):

Whereby:

s arc length along the track axis

1 length of transition arch and ramp with overhang.

A further feature of the invention provides that an overhanging track element connecting one straight track to another straight track diverging from it by an angle is made in one piece.

A further feature of the invention provides that a four-times-differentiable finite-valued function is used for a one-piece cantilever routing element that connects one rectilinear track to another rectilinear track that deviates from it by some angle.

One further feature of the invention provides that a vitoperca track element with an overhang, which connects a straight track with a track parallel to it, is made in one piece.

A further feature of the invention provides that a four-times-differentiable finite-valued function is used for a one-piece cantilever routing element connecting a straight track to a parallel track.

Furthermore, the invention will be explained in more detail through examples of execution and with reference to the attached drawings, whereby

- Figure 1 schematically shows a vehicle located on a track with an overhang, - Figure 2 shows the normalized process function of a minimum force ramp according to the invention with its normalized derivatives, and - Figure 3 shows the normalized flow of a curve for a transition arc with an increased curve at the beginning and at the end.

Figure 1 shows the vehicle on its track, taking into account its height. The tracing height (h) means any height at which unbalanced lateral accelerations are observed and measured.

In order to build a track with a continuous base, they were used to extend the rails in the horizontal projection and according to the height of the process function where there are still fourth extensions. For accurate regulation of a predetermined geometry, its real feasibility and related requirements for limited fourth derivatives of the process function are of extreme importance. This will ensure that the impact distribution across the entire cross-section of the vehicle will be constant and the kinematics of the vehicle will meet all requirements.

The well-known requirement for the existence of other derivations does not exclude the existence of infinitely many derivations that lead to the described defects along the edges.

Figure 2 shows, as an example, the normalized process function of a minimum force ramp according to the invention with its also normalized derivatives, derived from formula 2.

The function itself, as the zero derivative, corresponds to the development of the overhang. Its first derivative is the slope angle of the ramp, which corresponds to the twisting of the track and the angular velocity of the vehicle around the longitudinal axis. The second derivative is still straight and proportional to the curve of the rail in the vertical projection, the angular acceleration around the longitudinal axis of the vehicle and the bending moment in the rails that form the ramp. The third derivative is still continuous and corresponds to the change in the curvature of the rail in the vertical projection, the impact at an angle around the longitudinal axis of the vehicle and the transverse force in the rails that form the ramp. There is still a fourth derivative. It has rebound points on the edges and is proportional to the distribution of forces per unit length acting on the rail fastening elements that form the ramp, and is necessary to maintain the ramp in that shape.

These process functions can be applied on one fictitious track strip from which a real track strip can be obtained by projection. A special case is conventional routing, where both lanes are identical.

According to the invention, something different happens here: The cantilever ramp has been known for a long time. Direct tracking of the track is sought, so that the desired kinematic characteristics of the vehicle moving on the track are achieved.

Therefore, as always, unbalanced lateral acceleration will be observed. If it is introduced outside the plane of the track, a well-known expression is reached, which consists of the product of curves x speed of movement squared and another expression due to swaying, namely the angular acceleration of rolling around the longitudinal axis of the vehicle multiplied by the vertical distance from the axis of the track. If the curve in the horizontal projection is chosen so that one part of it in the first-mentioned expression is directly compensated while the other expression is proportional to the cant, then the unbalanced lateral acceleration will always be proportional to the cant. Therefore, the curve in the horizontal projection consists of two components, one conventional component corresponding to the cant flow and one component proportional to the second derivative of the cant flow. This produces the well-known damping of oscillations of the transitional arc, i.e., when transitioning from a straight line to a circular arc, they occur at the beginning of a curve with the opposite sign and one position on the other side of the reached circle. Figure 3 shows the corresponding normalized curve flow that is created by applying equations (1) and (12).

With this procedure, a description of the track axis is obtained completely independent of all kinematic quantities, whereby, as with conventional tracing, it can be conveniently done purely geometrically. Kinematic quantities are only required with control in terms of allowances in relation to certain regulations.

The described procedure can be applied as a whole to routing in areas of variable curvature and overhang and not only to transition arcs.

There are three characteristics that determine tracking properties: Geometric functions of camber and camber and kinematic functions of unbalanced lateral acceleration, preferably at tracking height.

In the case of the known routing, one moves from geometric functions to the track axis, and also the calculation of the kinematic function is always performed only for the track axis.

According to the invention, it is handled differently: Namely, one must start from one process function that can be differentiated at least three times, i.e. to fulfill the requirements of the Bending Theory, which arise from one process function that can be differentiated four times and from which there follows an overhang and an unbalanced lateral acceleration at the tracing height, taking into account the influence of the component caused by swaying on the unbalanced lateral acceleration, and from there the curve in the horizontal projection is determined.

With the missing tracking height selected (tracking height (h) equal to zero), a tracking flow is obtained where the track axis always follows that function, as is now common.

With the existing chosen tracing height, there is - due to the component of unbalanced lateral acceleration that needs to be compensated due to swaying - a change in the flow of the curve outside the process function, which in the case of a transitional arc leads from a straight line to a circle and then to damping of oscillations at the beginning.

As the formula is described, it reduces the sc unbalanced lateral acceleration to the angle (Froude number) which follows from:

Whereby:

Pq angle of unbalanced lateral acceleration

Oq unbalanced lateral acceleration

Earth's gravity

absence of redundancy in double-track railways

width of double-track railways

track axis curvature

about driving speed

tracing height

and the angular acceleration of rolling

\\ iugao cant.

The rolling angular acceleration is calculated from the second derivative in time of the pitch angle, which is replaced via the second derivative by road speed:

In doing so:

and the angular acceleration of rolling

d differential operator

\\ iugao cant.

time

at driving speed

arc length along the track axis.

According to the invention, such a track is provided that the curve is determined from a function accepted for the cant so that the entire unbalanced lateral acceleration has at one chosen, fixed tracking height (h), taking into account the component of the unbalanced lateral acceleration caused by the sway, the flow as this function has so that it satisfies the following differential equation:

whereby

kh(s) curve of the track axis in horizontal projection s arc length along the track axis

kcconstant base curve (in one circular arc)

v|/cugao of the basic overhang (in one circular arc)

v|/(s) angle of elevation

tracing height

d differential operator.

This differential equation can be directly calculated for a selected process function. For the tracing height h = 0, a conventional tracing is obtained. The basic curve and the basic overshoot must be selected on a circular arc or at the same point of the track.

In order to realize the accepted flow of a real continuously supported track, one convenient feature of the invention in the general case foresees the possibility of quadruple differentiation of the cantilever function. Then, from equation (1), the associated curvature of the track axis in the horizontal line is calculated.

For one transitional arc from one cantilevered circular arc to another cantilevered circular arc, the cantilever will be completely made using the normalized function (f(s/l)):

Next is:

vj/(s) angle of elevation

s arc length along the track axis

w(s) overhang in double-track railways

width of double-track railways

Hr*! constant elevation angle at the beginning of the overhanging ramp AY difference in elevation between the values in the circular arc ;/(-) ;y I axis function normalized between 0 and 1 ;1 length of the overhanging ramp ;<V>P2constant elevation angle at the end of the overhanging ramp. At the same time, the normalized function directly describes the course of the ramp after the overhang. ;As normed functions that can be differentiated at least four times, so as to meet the requirements of the Bernoulli-Euler Bending Theory for a cantilevered ramp for a transition arc from one circular arc with cantilever to another circular arc with cantilever, one polynomial of the seventh degree, one polynomial of the third degree combined with sine and cosine and one constant quantity (Z), one polynomial of the fifth degree combined with only sine, one polynomial of the fifth degree are used degree in combination with cosine only, as well as a polynomial of the ninth degree: ;Whereas: ;s the length of the arc along the axis of the track ;1 the length of the transition arc of the cantilever ramp ;/(-)basic function normalized to 1. ;\l) basic function normalized between 0 and 1. ;All these normalized functions, which are used for the transition arc with a cantilever ramp, which leads from one circular arc with cantilever to another circular arc with by excess, are either simple polynomials or simple combinations of trigonometric functions with short polynomials. They are not transcendental and in practice can be easily calculated, finally differentiated somewhat analytically to a physically significant degree and also integrated. ;The ability to differentiate can also be easily increased. Thus, for example, the increase by 1 of equation (7) for the normalized function is shown by a special polynomial of the ninth degree: With such a flow in the form of a cant function, the distribution of forces on the bed is not only limited, but also the continuous distribution of impacts is not only continuous but also flat. That is why the amplitudes are again slightly higher than those in the flow according to equation (2). On the basis of standardized functions, they form sc. ramps with an overhang according to equation (10). Double differentiation by the length of the arc along the track axis and insertion into equation (1) adapted for the transition arc in the following form gives the curve (kh) of the track axis in the horizontal projection; ;Furthermore: ;kh( s)curvature of the track axis in the horizontal projection s the length of the arc along the track axis ;Kjconstant curve of the circular arc at the beginning of the transition arc ;Akrazika in the curves between the values in the circular arc ;AL\basic function normalized between ;{I Basic function normalized between 0 and 1 ;h tracing height ;A<*>F difference in elevation between the values in the circular arc

d differential operator

k2 constant curvature of the circular arc at the end of the transition arc.

The following table shows, as an example, a numerical calculation with a normalized function according to equation (2). This numerical calculation is valid for one transition arc of length 200 m from one circle (index 1) of radius -2000 m and overhang of -64 mm to one circle (index 2) of radius +800 m and overhang of 160 mm for normal track width (track width 1435 mm; b = 1.5 m). All quantities are listed in order from the beginning of the transition arc: arc length, transition arc length with normalized arc length between 0 and 1, overhang angle, curvature in horizontal projection, local radius and twist important for measurement.

Tables for transitional arches and ramps with an overhang, which are formed from other normalized functions of equations (3) to (7), can be easily obtained in an analogous way, by numerically recalculating the formulas.

Other given functions can be handled analogously.

The procedure can be applied analogously for a one-piece transition with an overhang from one straight track to another straight track that continues at an angle to the first track. Choosing an appropriate overhang function and accepting the entire lateral acceleration, including the sway component with the same function, the track curve flow in the horizontal projection is obtained. For cantilever, one function is chosen that increases from zero to a maximum value and then drops to zero. In order to fulfill the requirements related to the bending of the rails, a function is chosen that can be differentiated at the edges of the area four times.

Likewise, wind-swept track elements with an overhang, which make the transition from one straight track to one parallel straight track, can be made in one piece. And there, one appropriate function will be taken, which can be differentiated four times in total, for overshooting and for the entire unbalanced-weighted acceleration, and from there the course of the curve of the track in the horizontal projection will be calculated.

In an analogous way, a detour around an obstacle can be made in one piece, that is, a route that starts from a straight track, goes around an obstacle on one side, then returns to the aforementioned extension of the straight track and crosses it, then extends further on the other side and turns into a straight track that continues further.

According to the method according to the invention, the track flows and the ramp shapes derived from the rail bending stress can be matched with perfect dynamic properties for all imaginable uses.

Claims (13)

1. A track with a track axis of variable curvature (kh) in the horizontal projection and with a variable angle (v|/) of elevation, indicated by the fact that the curve (kh) from a function accepted for cantilever is determined in such a way that the total unbalanced lateral acceleration at one chosen, fixed height (h) of the route, taking into account the component of unbalanced lateral acceleration caused by swaying, has the same flow as the aforementioned function and satisfies the following differential equation: where is it kh(s) curve of the track axis in horizontal projection s arc length along the track axis kcconstant base curve (in one circular arc) v|/base overhang angle (in one circular arc)\|/(s) overhang angle whose tracing d differential operator. 2. A line according to claim 1, characterized by the fact that the function in its entire flow, including the edges of the area, can be differentiated at least four times and that the fourth derivative of the function always has finite values. 3. The track according to claim 1 or 2, characterized by the fact that when determining the curvature (kh) of the track axis in the horizontal projection, the zero height of the routing is chosen as a fixed height (h) of the routing. 4. A line according to one of claims 1 to 3, characterized in that the following polynomial of the seventh degree is used as a normalized function: where: s is the arc length along the track axis 1 length of transition arch and ramp with overhang, where the normalized function is used for the change of the cant angle and for the total unbalanced lateral acceleration, and from it the curve (kh) of the track axis in the horizontal projection is derived according to equation (1). 5. A line according to one of claims 1 to 3, characterized in that the following third-degree polynomial in combination with sine and cosine and one constant value (Z) is used as a normalized function: where: s is the arc length along the track axis 1 length of transition arch and ramp with overhang, where the normalized function is used for the change of the cant angle and for the total unbalanced lateral acceleration, and from it the curve (kh) of the track axis in the horizontal projection is derived according to equation (1). 6. A line according to one of claims 1 to 3, characterized in that the following third degree polynomial in combination with sine and cosine is used as a normalized function: where: s is the arc length along the track axis 1 length of transition ramp and ramp with overhang. where the normalized function is used for the change of the cant angle and for the total unbalanced lateral acceleration, and from it the curve (kh) of the track axis in the horizontal projection is derived according to equation (1). 7. A line according to one of claims 1 to 3, characterized in that the following polynomial of the fifth degree is used as a normalized function in combination with only a sine: where: s is the length of the arc along the axis of the track 1 is the length of the transition arc and ramp with an overhang. where the normalized function is used for the change of the cant angle and for the total unbalanced lateral acceleration, and from it the curve (kh) of the track axis in the horizontal projection is derived according to equation (1). 8. A strip according to one of claims 1 to 3, characterized in that the following polynomial of the fifth degree is used as a normalized function in combination with only cosine: where: s is the arc length along the track axis 1 length of transition arch and ramp with overhang. where the normalized function is used for the change of the cant angle and for the total unbalanced lateral acceleration, and from it the curve (kh) of the track axis in the horizontal projection is derived according to equation (1). 9. A line according to one of claims 1 to 3, characterized in that the following ninth-degree polynomial is used as a normalized function whereby: s the length of the arc along the axis of the track 1 the length of the transition arc and ramp with overhang, whereby the normalized function is used for the change of the angle of overhang and for the total unbalanced lateral acceleration, and from it the curvature (kh) of the axis of the track in the horizontal projection is derived according to equation (1). 10. The track according to one of the claims 1 to 3, indicated by the fact that one element for routing with or without an overhang, which connects one straight track with another straight track that deviates from it by some angle, is formed by one single function. 11. The track according to claim 10, characterized by the fact that the one-part function, which forms the basis of one element for routing, performed with or without an overhang, and which connects one straight track with another straight track that deviates from it by some angle, can be differentiated four times with finite values. 12. The railway according to one of the claims 1 to 3, indicated by the fact that one wing element of the track with or without an overhang, which connects one straight track with one straight track parallel to it, is formed by a one-part function. 13. The track according to claim 12, characterized by the fact that the one-part function, which forms the basis of one weathered track element, performed with or without an overhang, and which connects one straight track with another straight track that deviates from it by some angle, can be differentiated four times with finite values.