Bozine Spungiformulae mad boz lab

$$\Delta\phi=\phi_2-\phi_1,\qquad \Delta\lambda=\lambda_2-\lambda_1$$ $$a=\sin^2\!\left(\frac{\Delta\phi}{2}\right)+\cos\phi_1\cos\phi_2\sin^2\!\left(\frac{\Delta\lambda}{2}\right)$$ $$d=2R\arctan2\!\left(\sqrt{a},\sqrt{1-a}\right)$$

$$\begin{aligned} &U_1=\arctan\big((1-f)\tan\phi_1\big),\quad U_2=\arctan\big((1-f)\tan\phi_2\big)\\ &\lambda=\Delta\lambda,\quad \text{iterate } \lambda \leftarrow f(\lambda)\\ &\sin\sigma = \sqrt{(\cos U_2\sin\lambda)^2 + (\cos U_1\sin U_2-\sin U_1\cos U_2\cos\lambda)^2}\\ &\cos\sigma = \sin U_1\sin U_2 + \cos U_1\cos U_2\cos\lambda\\ &\sigma = \operatorname{atan2}(\sin\sigma,\cos\sigma)\\ &s = b A (\sigma - \Delta\sigma) \end{aligned}$$

$$x = R\lambda,\qquad y = R\ln\!\left(\tan\left(\frac{\pi}{4}+\frac{\phi}{2}\right)\right)$$

$$\begin{aligned} x &= k_0 N\left(A + \frac{1-T+ C}{6}A^3 + \frac{5-18T+T^2+72C-58e'^2}{120}A^5\right) \\ y &= k_0\left(M - M_0 + N\tan\phi\left(\frac{A^2}{2} + \frac{5-T+9C+4C^2}{24}A^4+\cdots\right)\right) \end{aligned}$$

$$n = \frac{\ln(\cos\phi_1/\cos\phi_2)}{\l