Tiara-Barrel Calibration

Deriving E(Q,P)
$$ E = [(aU+b)+(cD+d)](1-e(k^{2}-P^{2}) $$

$$ Q = U + D \qquad\qquad\qquad P = \frac{U-D}{U+D} = \frac{U-D}{Q} $$

$$ \therefore \qquad PQ = U-D = U-D+D-D = Q-2D $$

$$ \therefore \qquad D = \frac{Q}{2}(1-P) \qquad \& \qquad U = \frac{Q}{2}(1+P) $$

$$ \therefore \qquad D = \frac{Q}{2}(1-P) \qquad \& \qquad U = \frac{Q}{2}(1+P) $$

$$ E = \left [ \left (a\left ( \frac{Q}{2}(1+P)\right ) +b \right )+ \left (c \left ( \frac{Q}{2}(1+P)\right )+d \right ) \right](1-e(k^{2}-P^{2}) $$

$$ \therefore \qquad E = \left [ \left (\frac{aQ}{2} + \frac{aQP}{2} + b + \frac{cQ}{2} + \frac{cQP}{2} + d \right ) \right](1-e(k^{2}-P^{2}) $$

$$ \therefore \qquad E = \frac{Q}{2} (P(a-c)+a+c)(1-e(k^{2}-P^{2}) $$

$$ \therefore \qquad Q = \frac{E}{1-e(k^{2}-P^{2})} \frac{2}{(P(a-c)+a+c)} $$