Clothoid 1: $(0 \leq s_{proj} \leq a_{1}t_{1_{f}} )$

The variable $t$ is used for clothoid calculations; however, $t$ is actually a function of the projected arclength, $s_{proj}$. In this way, $t$ is used both as a function and as a variable. Since $t$ is a function of $s_{proj}$, which is itself a function of $s$, which is a function of $S$, $S$ is ultimately the independent variable of all functions, as discussed in section 1.2.

The equation of arclength of a clothoid is used to determine $t$.

$$t = \frac{s_{proj}}{a_{1}}$$

The azimuth angle along the curve is found from the value of $t$, using equation A.7.

$$\theta = \frac{\pi t^{2}}{2}$$

The projection of the curve in the $x$-$y$ plane is determined by the Fresnel Integrals, which are described in appendix A. These integrals, when multiplied by the scaling factor $a$, will produce the coordinates of the points on a clothoid. The $z$ coordinates of the curve vary linearly with the arclength along the curve.

$$x = a_{1} \ensuremath\int_{0}^{t} \cos(\frac{\pi}{2} u^{2})\,du$$ (2.21)

$$y = a_{1} \ensuremath\int_{0}^{t} \sin(\frac{\pi}{2} u^{2})\,du$$ (2.22)

$$z = s_{proj} \tan\phi$$ (2.23)

Derivatives with respect to arclength will need to be taken to determine the remaining functions. Since the position vector is expressed in terms of $t$, the chain rule is applied. The chain rule will also be applied to quantities expressed in terms of $\theta$, so the required relationships are developed here.

$$\begin{eqnarray*} \ensuremath \frac{dt}{ds} & = & \ensuremath \frac{dt}{ds_{pro... ...ensuremath \frac{dt}{ds} \\ & = & \frac{\pi t}{a_{1}} \cos\phi \end{eqnarray*}$$

The forward vector is the derivative of the position vector with respect to arclength.

$$f_{x} = \ensuremath \frac{dx}{ds}$$

$$= \ensuremath \frac{d\left[ a_{1} \ensuremath\int_{0}^{t} \cos(\frac{\pi}{2} u^{2})\,du \right]}{dt} \ensuremath \frac{dt}{ds}$$

$$= a_{1} \cos \left( \frac{\pi t^{2}}{2} \right) \left( \frac{1}{a_{1}} \right) \cos\phi$$

$$= \cos\theta \cos\phi$$ (2.24)

$$f_{y} = \ensuremath \frac{dy}{ds}$$

$$= \ensuremath \frac{d\left[ a_{1} \ensuremath\int_{0}^{t} \sin(\frac{\pi}{2} u^{2})\,du \right]}{dt} \ensuremath \frac{dt}{ds}$$

$$= a_{1} \sin \left( \frac{\pi t^{2}}{2} \right) \left( \frac{1}{a_{1}} \right) \cos\phi$$

$$= \sin\theta \cos\phi$$ (2.25)

$$f_{z} = \ensuremath \frac{dz}{ds}$$

$$= \ensuremath \frac{d(s_{proj} \tan\phi)}{ds_{proj}} \ensuremath \frac{ds_{proj}}{ds}$$

$$= \tan\phi \cos\phi$$

$$= \sin\phi$$ (2.26)

The radial vector is the derivative of the forward vector with respect to arclength.

$$\begin{eqnarray*} r_{x} & = & \ensuremath \frac{df_{x}}{ds} \\ & = & \ensurem... ...}{ds} \\ & = & \ensuremath \frac{d(\sin\phi)}{ds} \\ & = & 0 \end{eqnarray*}$$

The unit vector in the direction of the radial vector must be determined.

$$r_{x} = -\sin\theta$$ (2.27)

$$r_{y} = \cos\theta$$ (2.28)

$$r_{z} = 0$$ (2.29)

The curvature is the magnitude of the (non-unit) radial vector.

$$k = \frac{\pi t}{a_{1}} \cos^{2}\phi$$ (2.30)

For a curve with no elevation change, equation 2.30 reduces to $\frac{\pi t}{a_{1}}$, which is the equation for the curvature of a planar clothoid.^2.4 The only difference in curvature between a two-dimensional curve and a three-dimensional curve is the $\cos^{2}\phi$ term, which has a constant value. This simple result is due to the method of linearly stretching the two-dimensional curve to create the three-dimensional element, as explained in section 1.4.

^2.4 This is shown toward the beginning of this section and in appendix A.