Pong

## June10th 2015 GPS Smoothing Example Continued

In the more general case, we use a parabolic curve with the following parametric form:

$\vec{ c }(t) = \vec{ a } + \vec{ b }t + \vec{ c }t^2$

Minimizing the weighted distance to the 3D curve yields the following solution:

$\left( \begin{array}{c} a_x \\ a_y \\ a_z \\ b_x \\ b_y \\ b_z \\ c_x \\ c_y \\ c_z \end{array} \right) = \left( \begin{array}{c c c c c c c c c} \sum_i w_i^2 & 0 & 0 & \sum_i w_i^2 t_i & 0 & 0 & \sum_i w_i^2 t_i^2 & 0 & 0 \\ 0 & \sum_i w_i^2 & 0 & 0 & \sum_i w_i^2 t_i & 0 & 0 & \sum_i w_i^2 t_i^2 & 0 \\ 0 & 0 & \sum_i w_i^2 & 0 & 0 & \sum_i w_i^2 t_i & 0 & 0 & \sum_i w_i^2 t_i^2 \\ \sum_i w_i^2 t_i & 0 & 0 & \sum_i w_i^2 t_i^2 & 0 & 0 & \sum_i w_i^2 t_i^3 & 0 & 0 \\ 0 & \sum_i w_i^2 t_i & 0 & 0 & \sum_i w_i^2 t_i^2 & 0 & 0 & \sum_i w_i^2 t_i^3 & 0 \\ 0 & 0 & \sum_i w_i^2 t_i & 0 & 0 & \sum_i w_i^2 t_i^2 & 0 & 0 & \sum_i w_i^2 t_i^3 \\ \sum_i w_i^2 t_i^2 & 0 & 0 & \sum_i w_i^2 t_i^3 & 0 & 0 & \sum_i w_i^2 t_i^4 & 0 & 0 \\ 0 & \sum_i w_i^2 t_i^2 & 0 & 0 & \sum_i w_i^2 t_i^3 & 0 & 0 & \sum_i w_i^2 t_i^4 & 0 \\ 0 & 0 & \sum_i w_i^2 t_i^2 & 0 & 0 & \sum_i w_i^2 t_i^3 & 0 & 0 & \sum_i w_i^2 t_i^4 \end{array} \right)^{-1} \left( \begin{array}{c} \sum_i w_i^2 p_{i_x} \\ \sum_i w_i^2 p_{i_y} \\ \sum_i w_i^2 p_{i_z} \\ \sum_i w_i^2 p_{i_x} t_i \\ \sum_i w_i^2 p_{i_y} t_i \\ \sum_i w_i^2 p_{i_z} t_i \\ \sum_i w_i^2 p_{i_x} t_i^2 \\ \sum_i w_i^2 p_{i_y} t_i^2 \\ \sum_i w_i^2 p_{i_z} t_i^2 \end{array} \right)$