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Circular Region: $(a_{1}t_{1_{f}} \leq s_{proj} \leq a_{1}t_{1_{f}} + r \delta )$

The azimuth angle at any point of the circular region is determined with the following.

\begin{displaymath}
\theta = \gamma_{1} + \frac{s_{proj} - a_{1}t_{1_{f}}}{r}
\end{displaymath}

The end of the first clothoid occurs when $t=t_{1_{f}}$. This point is labeled $(x_{1_{f}},y_{1_{f}})$ and is depicted in figure 2.4.

\begin{eqnarray*}
x_{1_{f}} & = & a_{1} \ensuremath\int_{0}^{t_{1_{f}}} \cos(\f...
...1} \ensuremath\int_{0}^{t_{1_{f}}} \sin(\frac{\pi}{2} u^{2})\,du
\end{eqnarray*}



The center of the circular region can now be found from the following relationship.

\begin{eqnarray*}
x_{c} & = & x_{1_{f}} - r \sin\gamma_{1} \\
y_{c} & = & y_{1_{f}} + r \cos\gamma_{1}
\end{eqnarray*}



The positions in the circular region are then determined with the following. Note that $z$ is calculated in the same manner as before, as it varies linearly with arclength along the curve.
$\displaystyle x$ $\textstyle =$ $\displaystyle x_{c} + r \sin\theta$ (2.31)
$\displaystyle y$ $\textstyle =$ $\displaystyle y_{c} - r \cos\theta$ (2.32)
$\displaystyle z$ $\textstyle =$ $\displaystyle s_{proj} \tan\phi$ (2.33)

Derivatives with respect to arclength will need to be taken to determine the remaining functions. Since the position vector is expressed in terms of $\theta$, the chain rule is applied.

\begin{eqnarray*}
\ensuremath \frac{d\theta}{ds} & = & \ensuremath \frac{d\thet...
...\ensuremath \frac{ds_{proj}}{ds} \\
& = & \frac{1}{r} \cos\phi
\end{eqnarray*}



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

$\displaystyle f_{x}$ $\textstyle =$ $\displaystyle \ensuremath \frac{dx}{ds}$  
  $\textstyle =$ $\displaystyle \ensuremath \frac{d\left( x_{c} + r \sin\theta \right)}{d\theta} \ensuremath \frac{d\theta}{ds}$  
  $\textstyle =$ $\displaystyle \cos\theta \cos\phi$ (2.34)
$\displaystyle f_{y}$ $\textstyle =$ $\displaystyle \ensuremath \frac{dy}{ds}$  
  $\textstyle =$ $\displaystyle \ensuremath \frac{d\left( y_{c} - r \cos\theta \right)}{d\theta} \ensuremath \frac{d\theta}{ds}$  
  $\textstyle =$ $\displaystyle \sin\theta \cos\phi$ (2.35)
$\displaystyle f_{z}$ $\textstyle =$ $\displaystyle \ensuremath \frac{dz}{ds}$  
  $\textstyle =$ $\displaystyle \ensuremath \frac{d(s_{proj} \tan\phi)}{ds_{proj}} \ensuremath \frac{ds_{proj}}{ds}$  
  $\textstyle =$ $\displaystyle \sin\phi$ (2.36)

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

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



The unit vector in the direction of the radial vector must be determined.
$\displaystyle r_{x}$ $\textstyle =$ $\displaystyle -\sin\theta$ (2.37)
$\displaystyle r_{y}$ $\textstyle =$ $\displaystyle \cos\theta$ (2.38)
$\displaystyle r_{z}$ $\textstyle =$ $\displaystyle 0$ (2.39)

The curvature is the magnitude of the (non-unit) radial vector.
\begin{displaymath}
k = \frac{1}{r} \cos^{2}\phi
\end{displaymath} (2.40)

Note the similarity of the above to equation 2.30. The curvature of a clothoid, $\frac{\pi t}{a}$ has been replaced by the curvature of a circle, $\frac{1}{r}$. The $\cos^{2}\phi$ term exists in both curvature equations, due to the linearly varying elevation along the curve.


next up previous contents
Next: Clothoid 2: Up: Curve Elements Previous: Clothoid 1:   Contents
Darla Weiss 2000-02-13