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# Table 2 Dimensions extracted from the accelerometer data

Name Formula Description
x x raw accelerometer reading in the x axis
y y raw accelerometer reading in the y axis
z z raw accelerometer reading in the z axis
rxyz $$\left\Vert \begin{array}{c}x\\ {}y\\ {}z\end{array}\right\Vert$$ magnitude of resultant vector
rxy $$\left\Vert \begin{array}{c}x\\ {}y\end{array}\right\Vert$$ magnitude of resultant vector in x,y plane
rxz $$\left\Vert \begin{array}{c}x\\ {}z\end{array}\right\Vert$$ magnitude of resultant vector in x,z plane
ryz $$\left\Vert \begin{array}{c}y\\ {}z\end{array}\right\Vert$$ magnitude of resultant vector in y,z plane
ϑxy $$\arctan \left(\frac{y}{x}\right)$$ angle of resultant vector in x,y plane
ϑxz $$\arctan \left(\frac{z}{x}\right)$$ angle of resultant vector in x,z plane
ϑyz $$\arctan \left(\frac{z}{y}\right)$$ angle of resultant vector in y,z plane
ϑz $$\arctan \left(\frac{z}{\left\Vert \begin{array}{c}x\\ {}y\end{array}\right\Vert}\right)$$ angle of resultant vector with x,y plane collapsed to 1 line
ϑy $$\arctan \left(\frac{y}{\left\Vert \begin{array}{c}x\\ {}z\end{array}\right\Vert}\right)$$ angle of resultant vector with x,z plane collapsed to 1 line
ϑx $$\arctan \left(\frac{x}{\left\Vert \begin{array}{c}y\\ {}z\end{array}\right\Vert}\right)$$ angle of resultant vector with y,z plane collapsed to 1 line
Ωx $$\arcsin \left(\frac{yz}{\left\Vert \begin{array}{c}x\\ {}y\\ {}0\end{array}\left\Vert \begin{array}{c}\\ {}\\ {}\end{array}\right\Vert \begin{array}{c}x\\ {}0\\ {}z\end{array}\right\Vert}\right)$$ solid angle of resultant pyramid base projected along x axis
Ωy $$\arcsin \left(\frac{xz}{\left\Vert \begin{array}{c}x\\ {}y\\ {}0\end{array}\left\Vert \begin{array}{c}\\ {}\\ {}\end{array}\right\Vert \begin{array}{c}0\\ {}y\\ {}z\end{array}\right\Vert}\right)$$ solid angle of resultant pyramid base projected along y axis
Ωz $$\arcsin \left(\frac{xy}{\left\Vert \begin{array}{c}x\\ {}0\\ {}z\end{array}\left\Vert \begin{array}{c}\\ {}\\ {}\end{array}\right\Vert \begin{array}{c}0\\ {}y\\ {}z\end{array}\right\Vert}\right)$$ solid angle of resultant pyramid base projected along z axis
Vxyz xyz volume of resultant cuboid
Ax yz area of resultant pyramid base projected along x axis
Ay xz area of resultant pyramid base projected along y axis
Az xy area of resultant pyramid base projected along z axis
Axyz $$\frac{1}{2}\left\Vert \left[\begin{array}{c}x\\ {}y\\ {}0\end{array}\right]\times \left[\begin{array}{c}x\\ {}0\\ {}z\end{array}\right]\right\Vert$$ area of resultant triangle 