A new, simple and accurate transition curve type, for use in road and railway alignment design

Abstract

Purpose

This paper evaluates all the available transition curve types related to road and railway alignments and proposes a new, well verified, transition curve type that combines the accuracy of clothoid curve and the simplicity of cubic parabola curve.

Method

A methodology similar to clothoid’s curve formation is used to introduce a new transition curve type called “of clothoid Symmetrically Projected Transition Curve (SPTC). All three transition curve types are being compared to each other, for a variety of transition length value versus Radius value combinations. The cubic parabola is a simple function of the form of y=f(x). Clothoid is a transition curve in the form of x=f(l), y=f(l), having as main characteristic the linearity of curvature variation versus its length. A new transition curve will be defined in the form of y=f(x) having also as main characteristic the linearity of curvature variation versus its projection length on axis X. By using the same calculation procedure as the clothoid, the new transition curve will be fully defined. A relation similar to (1) was used as base, by defining a parameter Α similar to the one used in the clothoid. The new curve will be called Symmetrically Projected Transition Curve (SPTC).

Results

Some remarkable results that derived from transition curves comparison are: There are no significant differences between the 3 curves in the area of short transition lengths. For long transition lengths, cubic parabola is diverging from the other 2. The deviation of the cubic parabola from the other curves for large values of Χ, ratios Χ/Α > 0.7, as well as the affinity of the clothoid with the SPTC are obvious. The most remarkable observation than can be made in the table is the fact that ΔΧ always zero for the SPTC (10terms). Thus, the SPTC curve is symmetrically projected on its basic tangent. This property contributes to the simplicity of the alignment design. That is another reason to prefer the SPTC curve.

Conclusions

The use of cubic parabola in combination with approximate value of diversion can lead to design problems. The new transition curve can be used instead of cubic parabola especially when long transition lengths are required. The new transition curve can also be used successfully to join 2 homo-bending arcs. However, referring to cubic parabola calculations, for a ratio X/A ≥ 0.5 and taking in to account the approximate calculation procedure of ΔR it can lead to alignment design errors. Consequently, the usage limits for each transition curve should be well known. A new transition curve was also proposed in this work. The new curve is called Symmetrically Projected Transition Curve (SPTC). SPTC was found, in most cases, to have better performance than cubic parabola. Symmetry is an important characteristic of the SPTC and contributes to simplicity, accuracy and audit ability of the designed alignment. Finally SPTC can also be used as a transition curve between two adjacent circular arcs in the same direction.

1 Introduction

The selection of a suitable transition curve is of major importance towards a proper alignment design in road and railway projects. There are two very well known transition curves. The transition curve that is exclusively used in road alignments is the Clothoid [2]. The cubic parabola is used, for historical reasons only, in railway alignments. It is noted that currently there is no research underway related to each curve’s accuracy and usability.

A comparison of the cubic parabola with the clothoid reveals that for small lengths of transition curves in relation to the radius of curvature corresponding to the end of the transition curve there are no significant differences between the two transition curve types. Thus, the use of the cubic parabola has no advantage over the clothoid other than its simplicity which is its dominant characteristic. However, the selection of transition curve type based on simplicity cannot be justified nowadays that computer software is widely used to design road and railway alignments. Thus, the criterion of a suitable transition curve based on the simplicity of the calculations must be reconsidered.

The use of the cubic parabola is acceptable only for small values of transition curve length. There are cases, however, where the length of the transition curve exceeds a certain limit. One such case is to obtain a “flatter” alignment. Another case is when the transition curve is used to adjust two circular arcs in the same direction. In such cases the use of the cubic parabola has serious limitations. When the length of the transition curve has to be greater than the half of the radius of curvature, the differences between cubic parabola and clothoid become significant. This is due to approximations applied in order to link cubic parabola to an arc [1].

The detailed analysis of the differences between cubic parabola and clothoid led to the introduction of a new transition curve type. This curve is called Symmetrically Projected Transition Curve (SPTC). The new curve is a simple function of the form of y=f(x) like the cubic one but is not based on any approximation used in cubic parabola calculations. On the other hand it is equally accurate to the clothoid.

2 Transition curves

2.1 The clothoid

The new transition curve has certain similarities with the clothoid. Consequently, the clothoid is briefly described first in order to clarify and distinguish it from the proposed curve. The clothoid’s main characteristic is the linear variation in curvature versus the covered length on the curve [2] (Kasper et al. 1954). A transition curve with no linear variation in curvature has also been proposed [4]. It is known, that for every point of the clothoid the following relation which links the distance L (on the curve) from the starting point with the curvature radius R on that point is valid:

$$\mathit{RL}=A^2$$

(1)

Thus, the product of R and L is constant for every curve characterized by the specific parameter Α.

With reference to Fig. 1 and using the above relation as well as the relation:

Fig. 1

Typical transition curve graph

$$\mathit{dL}=\mathit{Rd\tau }$$

(2)

the following relation is obtained:

$$\mathit{LdL}=A^2\mathit{d\tau }$$

(3)

where, τ is the tangent angle on the particular point of the clothoid in relation to the axis x and dτ the elementary angle corresponding to length dL.

Integration of Eq. (3) yields:

$$L^2/2=A^2\tau +C$$

(4)

Given that L = 0 for τ = 0 the value of C is equal to 0 and the deflection angle τ (in rad) will be:

$$\tau =\frac{L^2}{2A^2}$$

(5)

Based on Fig. 1 the following relations are obtained:

$$\mathit{dX}=\mathit{dL}cos\tau ,\mathit{dY}=\mathit{dL}sin\tau$$

(6)

After taking into account Eq. (5) dX and dY are as follows:

$$\mathit{dX}=\mathit{dL}cos\left(\frac{L^2}{2A^2}\right),\mathit{dY}=\mathit{dL}sin\left(\frac{L^2}{2A^2}\right)$$

(7)

Integration of the above equation yields the following relations used to calculate the coordinates of any point of the clothoid at a distance ℓ away from its starting point [3]

$$x=\ell -\frac{{\ell }^5}{40A^4}+\frac{{\ell }^9}{3456A^8}-\frac{{\ell }^{13}}{599040A^{12}}\cdots$$

(8)

$$y=\frac{{\ell }^3}{6A^2}-\frac{{\ell }^7}{336A^6}+\frac{{\ell }^{11}}{42240A^{10}}-\frac{{\ell }^{15}}{9676800A^{14}}\cdots$$

(9)

2.2 The cubic parabola

Using only the first terms of the relations (8) and (9) the following relation is obtained:

$$y=\frac{x^3}{6A^2}$$

(10)

which is known as the cubic parabola but is normally used as follows:

$$y=\frac{x^3}{6\mathit{RX}}$$

(11)

where R is the radius of the circle, which links to the end of the cubic parabola. L is the actual length of cubic parabola and X is its respective projection’s length on axis x. The length L of the cubic parabola is considered to be equal with the projection of Χ on axis x. As noted above relation (11) is commonly used instead of (10). This is because in the cubic parabola parameter Α has not been defined.

According to Esveld [1] the same relation with (11) can be derived starting from the general equation of the cubic parabola:

$$y=k{x}^3$$

(12)

In order for it to link with a circle of a radius R, its curvature, its second derivation with respect to x, at a distance X from its starting point, has to be the same with that of the circle, as shown in the following relation:

$${\left.\frac{d^{2}y}{d{x}^2}\right|}_{x=X}=6\mathit{kX}=\frac{1}{R}$$

(13)

from which the value of k is derived which, if set in (12) then we get (11). This relation gives the y coordinate of any point of the cubic parabola versus its projection on axis x.

From the relations (10)–(11), the deflection angle τ for the cubic parabola can be calculated, using the relation:

$$tan\tau =\frac{\mathit{dy}}{\mathit{dx}}=\frac{x^2}{2A^2}=\frac{x^2}{2\mathit{RX}}$$

(14)

As it is already mentioned, the length L of the cubic parabola is considered to be equal with the projection of Χ on axis x. This is an approximation, which is usually satisfactory. Relation (13) is based on this approximation that is valid for minor values of x in relation to radius R.

To calculate the real length L on the cubic parabola the following relation is used:

$$L=X\left[1+\frac{1}{10}{\left(\frac{X}{2R}\right)}^2\right]$$

(15)

This relation is derived from the following procedure. From Fig. 1 we get:

$$\mathit{dL}=\mathit{dX}/cos\tau =\mathit{dX}\sqrt{1+{tan}^2\tau }$$

(16)

which after the use of (14) becomes:

$$\mathit{dL}=\sqrt{1+\frac{X^4}{4A^4}}\mathit{dX}$$

(17)

Analyzing it in Tailor series we get:

$$\mathit{dL}=\left(1+\frac{X^4}{8A^4}-\frac{X^8}{128A^8}+..\right)\mathit{dX}$$

(18)

Its integration results in the following relation for the final length of the cubic parabola (x=X):

$$L=X\left[1+\frac{1}{10}{\left(\frac{X}{2R}\right)}^2-\frac{1}{72}{\left(\frac{X}{2R}\right)}^4+..\right]$$

(19)

which, if only the first two terms are considered as important, coincides with relation (15).

2.3 The new transition curve

The cubic parabola is a simple function of the form of y = f(x) and is based on the acknowledgment that its length is equal to its projection on axis X. Clothoid is a transition curve in the form of x = f(l), y = f(l), having as main characteristic the linearity of curvature variation versus its length. A new transition curve will be defined in the form of y = f(x) having also as main characteristic the linearity of curvature variation versus its projection length on axis X. By using the same calculation procedure as the clothoid, the new transition curve will be fully defined without having to acknowledge that its length L is equal to its projection on axis X. A relation similar to (1) was used as base, by defining a parameter Α similar to the one used in the clothoid. The new curve will be called Symmetrically Projected Transition Curve (SPTC). The analysis below will be based on the relation:

$$\mathit{RX}=R_{x}x=A^2$$

(20)

where, R is the curvature radius at the end of the curve and Χ its projection on axis x. Moreover, R _x is the curvature radius at an intermediate point with respective projection x.

With reference to Fig. 2 the following relation is obtained:

Fig. 2

Transition curve graph in detail

$$\mathit{dx}=d\ell cos\tau =R_x\mathit{d\tau }cos\tau$$

(21)

which, using the relation (20), results in the following

$$\mathit{dx}=\left(A^2/x\right)cos\mathit{\tau d\tau }$$

(22)

or

$$\mathit{xdx}=A^{2}cos\mathit{\tau d\tau }$$

(23)

By integration of Eq. (23)

$${\displaystyle \int \mathit{xdx}}=A^2{\displaystyle \int cos\mathit{\tau d\tau }}+C$$

(24)

the following relation is obtained

$$x^2/2=A^{2}sin\tau +C$$

(25)

However, at the beginning of the transition curve x  =  0 and τ  =  0. Thus, the angle τ (in rad) is expressed with the relation:

$$sin\tau =\frac{x^2}{2A^2}=\frac{x}{2R_x}$$

(26)

which is similar to (5) and to (14) and expresses the angle τ versus the projection of an intermediate point of the curve on the axis x and of the constant Α or of the curvature radius R _x on that point.

The relation (26) is also valid at the end of the transition curve and becomes:

$$sin\tau =\frac{X^2}{2A^2}=\frac{X}{2R}$$

(27)

where Χ is the projection of the end of the curve and R the respective curvature radius.

For the calculation of y for every x from the Fig. 2 there is the relation:

$$\mathit{dy}=tan\mathit{\tau dx}=\frac{sin\tau }{\sqrt{1-{sin}^2\tau }}\mathit{dx}$$

(28)

which, based on (26) becomes:

$$\mathit{dy}=\frac{\frac{x^2}{2A^2}}{\sqrt{1-\frac{x^4}{4A^4}}}\mathit{dx}$$

(29)

Integration of Eq. (29) yields:

$$y={\displaystyle \underset{0}{\overset{x}{\int }}\frac{\frac{x^2}{2A^2}}{\sqrt{1-\frac{x^4}{4A^4}}}\mathit{dx}}$$

(30)

In order to calculate the integral a Tailor series analysis is performed, beginning from the following simple relation:

$$g\left(x\right)=\frac{1}{\sqrt{1-x}}={\left(1-x\right)}^{-1/2}=1+\frac{1}{2}\frac{x^1}{1!}+\frac{3}{4}\frac{x^2}{2!}+..$$

(31)

which is written as follows:

$$g\left(x\right)={\displaystyle \sum _{i=0}^n{a}_i\frac{x^i}{i!}=}{\displaystyle \sum _{i=0}^n{b}_i}$$

(32)

with coefficients a _i , b _i of the form:

$$a_i=a_{i-1}\frac{2i-1}{2},a_0=1$$

(33)

$$b_i=b_{i-1}\frac{2i-1}{2i}x,b_0=1$$

(34)

If x is replaced with x⁴ the following relation is obtained:

$$g\left(x\right)=\frac{1}{\sqrt{1-x^4}}={\displaystyle \sum _{i=0}^n{a}_i\frac{x^{4i}}{i!}=}{\displaystyle \sum _{i=0}^n{b}_i}$$

(35)

where coefficients a _i are the same as in Eq. (33) and coefficients b _i are given by:

$$b_i=b_{i-1}\frac{2i-1}{2i}{x}^4$$

(36)

Subsequently, function h(x) is defined:

$$h\left(x\right)=\frac{x^2}{\sqrt{1-x^4}}={\displaystyle \sum _{i=0}^n{b}_i}$$

(37)

with b _i as (36) and b₀ = x². Using relation (37) and by replacing x with $x/\left(A\sqrt{2}\right)$ relation (30) becomes:

$$y={\displaystyle \underset{0}{\overset{x}{\int }}\frac{\frac{x^2}{2A^2}}{\sqrt{1-\frac{x^4}{4A^4}}}\mathit{dx}}={\displaystyle \underset{0}{\overset{x}{\int }}f\left(x\right)\mathit{dx}}$$

(38)

where the equation:

$$f\left(x\right)={\displaystyle \sum _{i=0}^n{b}_i}$$

(39)

with coefficients b _i of the form:

$$b_i=b_{i-1}\frac{2i-1}{2i}\frac{x^4}{4A^4},b_0=\frac{x^2}{2A^2}$$

(40)

The integration will give an equation of the form:

$$y={\displaystyle \underset{0}{\overset{x}{\int }}\frac{\frac{x^2}{2A^2}}{\sqrt{1-\frac{x^4}{4A^4}}}\mathit{dx}={\displaystyle \sum _{i=0}^n\frac{c_i}{4i+3}}}$$

(41)

with coefficients:

$$c_i=c_{i-1}\frac{2i-1}{2i}\frac{x^4}{4A^4},c_0=\frac{x^3}{2A^2}$$

(42)

By taking into account only the first four terms, the following relation emerges:

$$y=\frac{x^3}{2A^2}\left[\frac{1}{3}+\frac{1}{14}{\left(\frac{x^2}{2A^2}\right)}^2+\frac{3}{88}{\left(\frac{x^2}{2A^2}\right)}^4+\frac{1}{48}{\left(\frac{x^2}{2A^2}\right)}^6+..\right]$$

(43)

The above relation gives the value of y at intermediate points of the transition curve versus the projection x. If only the first term is used relation (10) is obtained. Thus, cubic parabola is a first approximation of the proposed transition curve.

For Υ at the end of the transition curve the following relation will be valid:

$$Y=\frac{X^2}{2R}\left[\frac{1}{3}+\frac{1}{14}{\left(\frac{X}{2R}\right)}^2+\frac{3}{88}{\left(\frac{X}{2R}\right)}^4+\frac{1}{48}{\left(\frac{X}{2R}\right)}^6+\dots \right]$$

(44)

The length ℓ on the curve will be calculated with the use of the relation:

$$d\ell =\mathit{dx}/cos\tau =\mathit{dx}\sqrt{1-{sin}^2\tau }=\sqrt{1-\frac{x^4}{4A^4}}\mathit{dx}$$

(45)

An analysis similar to the previous one also results in:

$$\ell ={\displaystyle \underset{0}{\overset{x}{\int }}\frac{\mathit{dx}}{\sqrt{1-\frac{x^4}{4A^4}}}}={\displaystyle \sum _{i=0}^n\frac{b_i}{4i+1}}$$

(46)

with the coefficients given by the relations:

$$b_i=b_{i-1}\frac{2i-1}{2i}\frac{x^4}{4A^4},b_0=x$$

(47)

By employing the first four terms, relation (48) emerges which gives the value of length ℓ on the transition curve versus the projection x on intermediate points:

$$\ell =x\left[1+\frac{1}{10}{\left(\frac{x^2}{2A^2}\right)}^2+\frac{1}{24}{\left(\frac{x^2}{2A^2}\right)}^4+\frac{5}{208}{\left(\frac{x^2}{2A^2}\right)}^6+\cdots \right]$$

(48)

At the end of the transition curve length L is:

$$L=X\left[1+\frac{1}{10}{\left(\frac{X}{2R}\right)}^2+\frac{1}{24}{\left(\frac{X}{2R}\right)}^4+\frac{5}{208}{\left(\frac{X}{2R}\right)}^6+\cdots \right]$$

(49)

which coincides with relation (15) if only its first two terms are considered important.

3 Comparison of the curves

Using the relations of the previous paragraphs all the transition curves were calculated and selected segments of the transition curves (R ≤ 1500 m) are presented in Fig. 3, to demonstrate the differences between them. These differences are presented in detail in Fig. 4 (R ≤ 1110 m).

Fig. 3

Comparison of transition curves used

Fig. 4

Comparison of transition curves in detail

Besides the cubic parabola and the symmetrically projected transition curve, two clothoids also are illustrated. The curvature radius of the circle in which all curves end is R = 1000 m. Τhe parameter for the first three curves is Α = 1000 m. For the fourth curve the following parameter value was used Α = 1013.278 m, corresponding to the curve’s length L = 1026.732 m, so that its projection at the end is Χ = 1000 m, exactly as the projection of the cubic parabola.

In Table 1 the details of all four curves are shown. It should be noted that radius values of about R = 1000 m are high for ordinary roads but are common to railway alignments.

Table 1 Basic characteristics of transition curves used

From Fig. 4 it is shown that the symmetrically projected transition curve is very close to the clothoid with the same parameter. The length is their difference and this is due to the fact that the length of the clothoid, which is L = 1000 m, is equal to the projection of the symmetrically projected transition curve. Namely with the second curve there is a variation from the radius R=∞ to the radius R = 1000 m at a length greater by 28 m.

In the following tables some comparative characteristic quantities will be presented for five different kinds of transition curves. Quantities for the clothoid, the cubic parabola and the curve SPTC are presented with accuracy, using 10 terms of the series for clothoid and SPTC. Two approximations of the curve SPTC (with three and two terms respectively) are presented on the last two columns of each table.

All curves have parameter Α = 1000 m. The clothoid is calculated according to relation (9) using as value of ℓ the one corresponding to a projection equal to the first column of the Table. This is calculated according to relation (8). The clothoid was used in this way so that the resulted quantities are directly comparative to the quantities of the other curves.

In Table 2 the values of the displacement Υ (tangent offset) for various values of Χ (tangent distance) are shown. The deviation of the cubic parabola from the other curves for large values of Χ, ratios Χ/Α > 0.5, as well as the affinity of the clothoid with the SPTC are obvious. The last two columns give an indication of the areas where their use is satisfactory for simplified calculations without the use of a computer.

Table 2 Displacement Y (tangent offset) for various values of X (tangent distance)

In Table 3 the values of angle τ in degrees are presented. The value of angle τ was calculated by using relation (5) for the clothoid, relation (14) for the cubic parabola and relation (26) for SPTC. The deviation of the cubic parabola from the other curves for large values of Χ, ratios Χ/Α > 0.5, as well as the affinity of the clothoid with the SPTC are obvious.

Table 3 Deflection angle τ for various values of X

In Table 4 the lengths L on the curve corresponding with the values of Χ are shown. The length L of the clothoid was calculated by using relation (8). The length L of the cubic parabola was calculated by using the commonly used approximate relation (15) instead of (19) which is more accurate. The length of SPTC was calculated by using relation (49). On the last two columns of Table 4, the approximations with four and three terms respectively are presented. The approximation with two terms is equal to the approximate length of cubic parabola. The deviation of the cubic parabola from the other curves for large values of Χ, ratios Χ/Α > 0.7, as well as the affinity of the clothoid with the SPTC are obvious. The use of relation (19), that gives more accurate results, will yield smaller values of the length of cubic parabola and thus the deviation of cubic parabola from the other curves would be greater.

Table 4 Lengths L (on curve) corresponding with the values of X

In Table 5 the values of the deviation ΔR for different values of Χ are shown. The values of the deviation on the second column of the Table result from the approximate relation:

Table 5 Deviation ΔR for different values of X

$$\mathit{\Delta R}=X^2/\left(24R\right)$$

(50)

which is the one most frequently used [1], [3] (Esveld 2001, Lamm et al. 1999). The following accurate relation was used for the other columns [3] (Lamm et al. 1999):

$$\mathit{\Delta R}=Y-R\left(1-cos\tau \right)$$

(51)

The divergence of the real deviation of the cubic parabola from the approximate relation (50) for large values of the ratio Χ/Α > 0.5 is obvious. The alignment design with the simultaneous use of the cubic parabola and of the approximate value of the deviation can lead to serious mistakes.

The use of the cubic parabola for ratios Χ/Α ≤ 0.3 does not lead to mistakes but does not differentiate it from other curves as results from Tables 2, 3, 4 and 5. This means that apart from the simplicity, which characterizes it, it has nothing to offer compared to the clothoid.

In Table 6 the values of quantity ΔΧ versus Χ are presented. Quantity ΔΧ (Fig. 5) represents the distance of the projection of the center of the curvature at the end of the curve from the middle of the linear segment, which defines the projection of the whole curve. The two projections are viewed on axis Χ.

Table 6 Displacement ΔX versus projection X

Fig. 5

Displacement ΔX of a transition curve

In the second column of the Table the value of ΔΧ is given based on the approximate relation:

$$\mathit{\Delta X}=X^3/\left(16R^2\right)$$

(52)

which is only valid for the cubic parabola [1] (Esveld 2001). The exact values in the following columns are calculated based on the relation:

$$\mathit{\Delta X}=X/2-R_{X}sin{\tau }_X$$

(53)

where with indicator X are symbolized the radius R and angle τ for the particular point of the transition curve that its projection is at a distance Χ from its starting point [1] (Esveld 2001). It is worth noting the good approximation of the ΔΧ of the cubic parabola from the relation (52) for values of the ratio Χ/Α<=0.5, as shown by the comparison of the second with the fourth column of the Table 6.

The most remarkable observation than can be made in the table is the fact that ΔΧ always zero for the SPTC curve whereas in the other curves it has a value which increases as Χ/Α increases. Thus, the SPTC curve is symmetrically projected on its basic tangent. This property can be easily explained by examining relation (27) and Fig. 6. This property contributes to the simplicity of the alignment design. That is another reason to prefer the SPTC curve. The comparison of all transition curves for different ratios of transition curve length versus arc radius, leads to remarkable conclusions. The results are presented in Tables 2, 3, 4, 5 and 6 to demonstrate the usability of each transition curve. In particular they allow the designer to know the accuracy of the calculations.

Fig. 6

Symmetrically Projected Transition Curve (SPTC)

4 Conclusions

A comparison of the cubic parabola transition curve and the clothoid revealed that for a ratio X/A ≤ 0.3 the curves have no significant differences. Thus, beyond simplicity reasons the cubic parabola has no other obvious advantage compared to the clothoid. However, referring to cubic parabola calculations, for a ratio X/A ≥ 0.5 and taking in to account the approximate calculation procedure of ΔR it can lead to alignment design errors. Consequently, the usage limits for each transition curve should be well known. A new transition curve was also proposed in this work. The new curve is called Symmetrically Projected Transition Curve (SPTC). SPTC was found, in most cases, to have better performance than cubic parabola. By employing its approximate forms it can offer the correct solutions in cases where the use of the cubic parabola was preferred, due to simplicity reasons, to the clothoid. The SPTC will be more attractive than the clothoid to the engineers who prefer the cubic parabola as a transition curve because it is of the form y=f(x). SPTC can also give the relation between tangent distance and tangent offset of the clothoid because the two curves are identical for X/A ≤ 0.5. Symmetry is an important characteristic of the SPTC and contributes to simplicity, accuracy and audit ability of the designed alignment. Finally SPTC can also be used as a transition curve between two adjacent circular arcs in the same direction.