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

The equation for the arclength of a clothoid, equation A.5, is used to determine $t$, the variable used for clothoid calculations.

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

The vertical angle through the loop is defined as the angle between the forward vector and the $x$-$y$ plane. It is found in the same manner as the azimuth angle is determined for a curve.

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

The projection of the loop in the $x$-$z$ plane is determined by the Fresnel Integrals. The $y$ coordinate varies linearly with the arclength of the loop.

$$x = a_{1} \ensuremath\int_{0}^{t} \cos(\frac{\pi}{2} u^{2})\,du$$ (2.53)

$$y = s_{proj} \tan\alpha$$ (2.54)

$$z = a_{1} \ensuremath\int_{0}^{t} \sin(\frac{\pi}{2} u^{2})\,du$$ (2.55)

The calculation of the forward and radial vectors is nearly identical to the calculation in the curve. An abbreviated description will be shown here.

Derivatives with respect to arclength of the position vector must be taken. The required chain rule relationships are first determined.

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

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

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

$$= \cos\phi \cos\alpha$$ (2.56)

$$f_{y} = \ensuremath \frac{d(s_{proj} \tan\alpha)}{ds_{proj}} \ensuremath \frac{ds_{proj}}{ds}$$

$$= \sin\alpha$$ (2.57)

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

$$= \sin\phi \cos\alpha$$ (2.58)

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

$$\begin{eqnarray*} r_{x} & = & \ensuremath \frac{d\left( \cos\phi \cos\alpha \ri... ...\phi}{ds} \\ & = & \frac{\pi t}{a_{1}} \cos\phi \cos^{2}\alpha \end{eqnarray*}$$

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

$$r_{x} = -\sin\phi$$ (2.59)

$$r_{y} = 0$$ (2.60)

$$r_{z} = \cos\phi$$ (2.61)

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

$$k = \frac{\pi t}{a_{1}} \cos^{2}\alpha$$ (2.62)