Straight Elements

A straight element is a section of track with zero curvature. It is used to make hills, drops, and flat sections. It is specified with two parameters, as shown in figure 2.1.

Figure 2.1: Straight Element Parameters.
The height and vertical angle of a straight element are shown here.

vertical angle $(\phi)$

The vertical angle remains constant throughout the straight element. For a flat section, it is zero. For an uphill section, it may vary from (just greater than) zero to (just less than) ninety degrees. For a downhill section, it may vary from (just less than) zero to (just greater than) negative ninety degrees, although in reality these extremes are rarely found. ( $-\frac{\pi}{2} < \phi < \frac{\pi}{2}$)

height $(h)$

The elevation change from one end of the straight element to the other.^2.2

The required geometric parameters, as explained in section 2.1, can be determined from the parameters given above.

The length of the straight element is calculated from the geometry depicted in the figure.

$$l = \frac{h}{\sin\phi}$$ (2.5)

Since the vertical angle remains constant, the initial and final vertical angles are the same.

$$\phi_{0} = \phi_{f} = \phi$$ (2.6)

As stated in section 1.2, all elements begin in the $x$-$z$ plane. Since there is no curvature in a straight section, the entire element lies in the $x$-$z$ plane. Therefore, both the initial and final azimuth angles are zero.

$$\theta_{f} = 0$$ (2.7)

The required geometric functions can now be determined in terms of $s$, the length along the element. At the origin, the value of $s$ is zero.

The position vector components can be determined from the geometry depicted in figure 2.1. Since there is no curvature, $y$ remains zero throughout the element.

$$x = s \cos\phi$$ (2.8)

$$y = 0$$ (2.9)

$$z = s \sin\phi$$ (2.10)

The forward vector is found by taking the derivative of the position vector with respect to distance along the element, $s$.

$$f_{x} = \cos\phi$$ (2.11)

$$f_{y} = 0$$ (2.12)

$$f_{z} = \sin\phi$$ (2.13)

The radial vector is found by taking the derivative of the forward vector with respect to distance along the element.

$$r_{x} = 0$$ (2.14)

$$r_{y} = 0$$ (2.15)

$$r_{z} = 0$$ (2.16)

The curvature is the magnitude of the radial vector.

$$k = 0$$ (2.17)

^2.2 Note that if $\phi = 0$, the length (instead of the height) will be the required second parameter.