Connecting Elements to Create a Track

Each element is specified in terms of local coordinates. In order to create a track, it is necessary to connect these elements together in a global setting. To do this, simple coordinate transformations are used.

Let $(x_{i},y_{i},z_{i})$ represent the local coordinates of the $i^{th}$ element, and let $(X_{i},Y_{i},Z_{i})$ represent the global coordinates of that element. Let

and $\theta$ represent the arclength and azimuth angle at the current position, and let $s_{f}$ and $\theta_{f}$ represent the arclength and azimuth angle, respectively, at the end of that element. The position transformation from local to global coordinates requires a rotation about the local

axis to line up the

axes with the azimuth angle at the end of the previous element, as well as a translation to line up the local origin with the end of the previous element. This is accomplished as shown below, where the subscript

is used to denote which element the quantities refer to.

$\displaystyle X_{i}(s)$	$\textstyle =$	$\displaystyle X_{i-1}(s_{f}) + x_{i}(s)\cos(\theta_{f_{i-1}}) - y_{i}(s)\sin(\theta_{f_{i-1}})$	(2.147)
$\displaystyle Y_{i}(s)$	$\textstyle =$	$\displaystyle Y_{i-1}(s_{f}) + x_{i}(s)\sin(\theta_{f_{i-1}}) + y_{i}(s)\cos(\theta_{f_{i-1}})$	(2.148)
$\displaystyle Z_{i}(s)$	$\textstyle =$	$\displaystyle Z_{i-1}(s_{f}) + z_{i}(s)$	(2.149)

The transformation is somewhat simpler for the forward and radial vectors than it is for the position, since they are not dependent on an origin. Only a rotation is required to line up these vectors properly. The forward vector transformation is shown here.

$\displaystyle f_{x_{i}}(s)$	$\textstyle =$	$\displaystyle f_{x_{i}}(s)\cos(\theta_{f_{i-1}}) - f_{y_{i}}(s)\sin(\theta_{f_{i-1}})$	(2.150)
$\displaystyle f_{y_{i}}(s)$	$\textstyle =$	$\displaystyle f_{x_{i}}(s)\sin(\theta_{f_{i-1}}) + f_{y_{i}}(s)\cos(\theta_{f_{i-1}})$	(2.151)
$\displaystyle f_{z_{i}}(s)$	$\textstyle =$	$\displaystyle f_{z_{i}}(s)$	(2.152)

$\displaystyle r_{x_{i}}(s)$	$\textstyle =$	$\displaystyle r_{x_{i}}(s)\cos(\theta_{f_{i-1}}) - r_{y_{i}}(s)\sin(\theta_{f_{i-1}})$	(2.153)
$\displaystyle r_{y_{i}}(s)$	$\textstyle =$	$\displaystyle r_{x_{i}}(s)\sin(\theta_{f_{i-1}}) + r_{y_{i}}(s)\cos(\theta_{f_{i-1}})$	(2.154)
$\displaystyle r_{z_{i}}(s)$	$\textstyle =$	$\displaystyle r_{z_{i}}(s)$	(2.155)

Refer to appendix B for a more general description of rotational transformations.