next up previous contents
Next: Bibliography Up: Dynamic Simulation and Analysis Previous: Clothoids   Contents


Rotation Transformations

Many calculations of a roller coaster track require performing a rotation of a specified angle about a specified axis, as shown in figure B.1. It is desirable to know the location of the points in the rotated frame in terms of the unit vectors describing the non-rotated frame. This appendix will illustrate the calculation performed.

Figure B.1: Rotation Transformations.
This figure illustrates the rotation of the position vector $\mathbf{r}$ through a specified angle $\theta$.
Rotation Transformations

Let point $P$ be specified in the $x$-$y$ coordinate system by the vector $\mathbf{r}$.

\begin{displaymath}
\mathbf{r} = x_{P} \ensuremath \mathbf{\widehat{i}} + y_{P} \ensuremath \mathbf{\widehat{j}}
\end{displaymath} (B.1)

The $x$-$y$ axes are then rotated about the $z$ axis through an angle $\theta$, resulting in the $x'$-$y'$ axes. Point $P$ is rotated as well, producing point $P'$ and position vector $\mathbf{r'}$, which can be represented two ways.

\begin{eqnarray*}
\mathbf{r'} & = & x_{P} \ensuremath \mathbf{\widehat{i'}} + y...
...h \mathbf{\widehat{i}} + y_{P}' \ensuremath \mathbf{\widehat{j}}
\end{eqnarray*}



From the figure, it can be seen that:

\begin{eqnarray*}
\ensuremath \mathbf{\widehat{i'}} & = & \ensuremath \mathbf{\...
...hat{i}} \sin\theta + \ensuremath \mathbf{\widehat{j}} \cos\theta
\end{eqnarray*}



The new position vector, $\mathbf{r'}$, can now be written as:

\begin{displaymath}
\mathbf{r'} = x_{P} ( \ensuremath \mathbf{\widehat{i}} \cos...
...i}} \sin\theta + \ensuremath \mathbf{\widehat{j}} \cos\theta )
\end{displaymath}

The new coordinates can now be written in terms of the non-rotated axes.
$\displaystyle x_{P}'$ $\textstyle =$ $\displaystyle x_{P} \cos\theta - y_{P} \sin\theta$ (B.2)
$\displaystyle y_{P}'$ $\textstyle =$ $\displaystyle x_{P} \sin\theta + y_{P} \cos\theta$ (B.3)


next up previous contents
Next: Bibliography Up: Dynamic Simulation and Analysis Previous: Clothoids   Contents
Darla Weiss 2000-02-13