# Sensitivity Analysis for Gaussian-Associated Features
## 9 Numerical Example: Measuring a Sphere to Determine Datum Location  
In several applications a high-quality sphere or tooling ball is used to determine a location in specifying a frame of reference. Here we give example calculations for measurement strategies involving:  
* Six points:four equally spaced around the equator, and one at each pole. * Five points: as for six points but missing the point at the South pole
* Nine points: four equally spaced around the equator and another four equally spaced at elevation angle \(\phi>0\) and additional point at the North pole. We assume that the associated point-cloud variance matrix is given by \(V_{X}=\sigma_{R}^{2}I\). We are interested in strategies such that all coordinates of the sphere center have approximately the same uncertainty. This is achieved by the six-point strategy which is symmetrical in each axis. However, due constraints associated mounting the reference sphere, it is quite unlikely that such a six-point strategy is feasible. The other point distributions all involve points on a hemisphere and are usually possible to implement. For the nine-point strategy, the measurement strategy depends on the elevation angle \(\phi\). If \(J_{\phi}\) is the Jacobian matrix associated with the nine points, and \(H_{\phi}=J_{\phi}^{\top}J_{\phi}\), then  
\[H_{\phi}=\left[ \begin{array}{cccc}4-2\sin^{2}\phi&0&0&0\\ 0&4-2\sin^{2}\phi&0&0\\ 0&0&1+4\sin^{2}\phi&1+4\sin\phi\\ 0&0&1+4\sin\phi&9\end{array} \right].\]  
The variance matrix associated with \(a\) depends on matrix \(V_{\phi}=H_{\phi}^{-1}\) and the requirement that the uncertainties in the sphere center coordinates are equal leads to the equation  
\[38\sin^{2}\phi-8\sin\phi-28=0,\]  
a quadratic equation in \(\sin\phi\), from which we deduce  
\[\sin\phi=(8+\sqrt{4320})/76,\] \begin{table}
\begin{tabular}{c c c c c c c} \hline
**2\(\alpha\)/deg** & \(s(x_{A})\) & \(s(y_{A})\) & \(as(u_{A})\) & \(as(v_{A})\) & \(s(r_{0})\) & \(s(\phi)\) \\ \hline
360 & 4.06 & 4.06 & 0.04 & 0.04 & 1.73 & 0.03 \\ \hline
270 & 5.21 & 3.69 & 0.04 & 0.05 & 1.97 & 0.03 \\ \hline
180 & 9.33 & 4.06 & 0.04 & 0.10 & 3.98 & 0.07 \\ \hline
160 & 11.37 & 4.34 & 0.05 & 0.12 & 5.14 & 0.09 \\ \hline
140 & 14.37 & 4.73 & 0.05 & 0.15 & 6.89 & 0.12 \\ \hline
120 & 19.01 & 5.30 & 0.06 & 0.20 & 9.64 & 0.17 \\ \hline
100 & 26.72 & 6.15 & 0.06 & 0.28 & 14.25 & 0.25 \\ \hline
80 & 40.94 & 7.48 & 0.08 & 0.43 & 22.79 & 0.39 \\ \hline
60 & 71.67 & 9.77 & 0.10 & 0.75 & 41.31 & 0.72 \\ \hline
40 & 159.51 & 14.43 & 0.15 & 1.67 & 94.27 & 1.63 \\ \hline
20 & 633.90 & 28.59 & 0.30 & 6.62 & 380.32 & 6.59 \\ \hline \end{tabular}
\end{table} 
Table 7: Arc segment of a cone. Square roots \(s(a)\) of the diagonal elements of \(V_{a\phi}\) as a function of \(\alpha\). For points \(x_{1:m}\) approximately uniformly distributed over a segment of a cone with cone angle \(\phi=45\) degrees, \(-\alpha\leq\theta_{i}\leq\alpha\leq\pi\) and \(-a\leq z_{i}\leq a\), and point-cloud variance matrix \(V_{X}=\sigma_{R}^{2}I\), the uncertainties \(u(\mathbf{a})=\sigma_{R}s(\mathbf{a})/\sqrt{m}\). i.e., \(\phi=75.95^{\circ}\). For this value of \(\phi\), the four points on the upper circle are at height \(0.97r_{0}\) and radius approximately \(r_{0}/4\), so quite near the North pole. Table 8 gives estimates \(s(a)=\sigma_{R}u(a)\) for the three measurement strategies discussed above. The first column in the pairs of columns gives the uncertainties calculated using an analytic approximation based on assuming a uniform distribution of points on the sphere or hemisphere while the second column gives the uncertainties based on the actual distribution of points. For the six-point strategy, the uncertainties in the sphere center coordinates are seen to be the same and that the analytical approximation to the uncertainties is exact. For the five-point strategy, the uncertainty associated with \(z_{0}\) is larger than those associated with \(x_{0}\) and \(y_{0}\) by a factor of two.